ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA
|
|
- Scot Alexander
- 5 years ago
- Views:
Transcription
1 ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA FRANCESCO PETITTA Abstract. Let R N a bounded open set, N 2, and let p > 1; we study the asymptotic behavior with respect to the time variable t of the entropy solution of nonlinear parabolic problems whose model is { u t (x, t) p u(x, t) = µ in (, T ), u(x, ) = u (x) in, where T > is any positive constant, u L 1 () a nonnegative function, and µ M () is a nonnegative measure with bounded variation over = (, T ) which does not charge the sets of zero p-capacity; moreover we consider µ that does not depend on time. In particular, we prove that solutions of such problems converge to stationary solutions. Introduction A large number of papers was devoted to the study of asymptotic behavior for solution of parabolic problems under various assumptions and in different contexts: for a review on classical results see [12],[1], [22], and references therein. More recently in [13] the same problem was studied for bounded data and a class of operators rather different to the one we will discuss. Moreover, in [16] and [19] it was used an approach similar to our one, to face, respectively, the quasilinear case with natural growth terms of the type g(u) u 2 and the linear case with general measure data. Let R N be a bounded open set, N 2, and let p > 1; we are interested in the asymptotic behavior with respect to the time variable t of the entropy solution of parabolic problems whose model is u t (x, t) p u(x, t) = µ in (, T ), (.1) u(x, ) = u (x) in, u(x, t) = on (, T ), where T > is any positive constant, u L 1 () a nonnegative function, and µ M () is a nonnegative measure with bounded variation over = (, T ) which does not charge the sets of zero p-capacity in accordance with its definition given in [1]; 2 Mathematics Subject Classification. 35B4, 35K55. Key words and phrases. Asymptotic behavior, nonlinear parabolic equations, measure data. 1
2 2 FRANCESCO PETITTA moreover we suppose that µ does not depend on the time variable t (i.e. there exists a bounded Radon measure ν on such that, for any Borel set B, and < t, t 1 < T, we have µ(b (t, t 1 )) = (t 1 t )ν(b)). Actually we shall investigate the limit as T tends to infinity of the solution u(x, T ) of the problem (.1). Observe that by virtue of uniqueness results concerning entropy solutions of (.1) we have u T (x, t) = u T (x, t) a.e. in, for all T > T and t (, T ), where the index T and T indicate that we deal with the solution of problem (.1) respectively on T and T ; so the function u(x, t) = u T (x, t), T > t is well defined for all t > ; we are interested in the asymptotic behavior of u(x, t) as t tends to infinity. Actually, for a larger class of problem than (.1), we shall prove that, as t tends to infinity, u(x, t) converges in L 1 () to v(x), the entropy solution of the corresponding elliptic problem. p v(x) = µ in, (.2) v(x) = on. 1. General assumptions and main result Let a : R N R N be a Carathéodory function (i.e. a(, ξ) is measurable on, ξ R N, and a(x, ) is continuous on R N for a.e. x ) such that the following holds: (1.1) a(x, ξ) ξ α ξ p, (1.2) a(x, ξ) β[b(x) + ξ ], (1.3) (a(x, ξ) a(x, η)) (ξ η) >, for almost every x, for all ξ, η R N with ξ η, where p > 1 and α, β are positive constants and b is a nonnegative function in L p (). For every u W 1,p (), let us define the differential operator A(u) = div(a(x, u)), that, thanks to the assumption on a, turns out to be a coercive monotone operator acting from the space W 1,p () into its dual W 1,p (). We shall deal with the solutions of the initial boundary value problem u t + A(u) = µ in (, T ), (1.4) u(x, ) = u (x) in, u(x, t) = on (, T ), where µ is a nonnegative measure with bounded variation over that does not depend on time, and u L 1 ().
3 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 3 Remark 1.1. If µ L p (, T ; W 1,p ()), it is well known that problem (1.4) has a unique variational solution u L p (, T ; W 1,p ()) C([, T ]; L 2 ()) such that u t L p (, T ; W 1,p ()) and u(x, ) =, that is T u t, ϕ W 1,p (),W 1,p () dt + a(x, u) ϕ dxdt = T f, ϕ W 1,p (),W 1,p () dt, for all ϕ L p (, T ; W 1,p ()) (see [17] for the case p 2 and [15] for 1 < p < 2). Let us define the concept of elliptic p-capacity associated to our problem (for further details about elliptic p-capacity see [14]). Definition 1.2. Let K be a compact set, and W (K, ) = {ϕ C () : ϕ χ K }, where χ K represents the characteristic function of K. We define the elliptic p-capacity of the compact set K with respect to as the quantity { } cap e p(k) = inf ϕ p dx : ϕ W (K, ), where we set inf = + as usual. p-capacity of U as follows: If U is an open set, we define the elliptic cap e p(u) = sup { cap e p(k) : K U, Kcompact }. Finally, if B is a Borel set we define cap e p(b) = inf { cap e p(u) : B U }. We also recall the notion of parabolic p-capacity associated to our problem (for further details see for instance [1]). Definition 1.3. Let = T = (, T ) for any fixed T >, and let us call V = W 1,p () L 2 () endowed with its natural norm W 1,p () + L 2 () and { } W = u L p (, T ; V ), u t L p (, T ; V ), endowed with its natural norm u W = u L p (,T ;V ) + u t L p (,T ;V ). So, if U is an open set, we define the parabolic p-capacity of U as cap p (U) = inf{ u W : u W, u χ U a.e. in }, where again we set inf = + ; then for any Borel set B we define cap p (B) = inf{cap p (U), U open set of, B U}.
4 4 FRANCESCO PETITTA Let us denote with M () the set of all measures with bounded variation over that do not charge the sets of zero elliptic p-capacity, that is if µ M (), then µ(e) =, for all E such that cap e p(e) = ; analogously we define M () the set of all measures with bounded variation over that does not charge the sets of zero parabolic p-capacity, that is if µ M () then µ(e) =, for all E such that cap p (E) =. In [3] (for more details see also [6]) the concept of entropy solution of the elliptic boundary value problem associated to (1.4) was introduced: let µ M () be a measure with bounded variation over which does not charge the sets of zero elliptic p-capacity; we call v an entropy solution for the boundary value problem (1.5) { A(v) = µ in, v = on, if v is finite a.e. and its truncated function T k (v) W 1,p () (recall that T k (s) = max( k, min(k, s)), see for instance [9] for more details), for all k >, and it holds (1.6) a(x, v) T k (v ϕ) dx T k (v ϕ) dµ, for all ϕ W 1,p () L (), for all k > ; notice that the integral in the right hand side of (1.6) is well defined by virtue of the decomposition result of [6] (see below) and it turns out to be independent on the different decompositions of µ. Moreover observe that the gradient of such a solution v is not in general well defined in the sense of distributions, anyway it is possible to give a sense to (1.6), using the notion of approximated gradient of v, defined as the a.e. unique measurable function that coincides a.e. with T k (v), over the set where v k, for every k > (see for instance [3]). Such a solution exists and is unique for all measures in M () (see [6]), and turns out to be a distributional solution of problem (1.5); moreover such a solution satisfies (1.6) with the equality sign (see [9]). An analogous definition will be given in the parabolic case following [21]. Definition 1.4. Let k > and define Θ k (z) = z T k (s) ds, as the primitive function of the truncation function; let µ M () and u L 1 (), then we say that u(x, t) C(, T ; L 1 ()) is an entropy solution of the problem (1.7) u t + A(u) = µ in (, T ), u(x, ) = u (x) in, u(x, t) =, on (, T ),
5 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 5 if, for all k >, we have that T k (u) L p (, T ; W 1,p ()), and it holds Θ k (u ϕ)(t ) dx Θ k (u ϕ()) dx (1.8) + T + ϕ t, T k (u ϕ) W 1,p (),W 1,p () dt a(x, u) T k (u ϕ) dxdt T k (u ϕ) dµ, for any ϕ L p (, T ; W 1,p ()) L () C([, T ]; L 1 ()) with ϕ t L p (, T ; W 1,p ()). Remark 1.5. The entropy solution u of the problem (1.7) exists and is unique as shown in [21] for L 1 () data; this result was improved in many papers for more general measure data, in [2] it was proved via the notion of renormalized solution that turns out to be equivalent to the one of entropy solution with this kind of data (see [11]). Moreover, the solution u is such that a(x, u) L q 1 () for all q < 1 +, even if its (N + 1)(p 1) approximated gradient may not belong to any Lebesgue space. In [1] the authors prove the following useful result: Theorem 1.6. Let B be a Borel set in, and t < t 1 T. Then we have Proof. See [1], Theorem cap p (B (t, t 1 )) = if and only if cap e p(b) =. Thanks to this result we derive that measures of M () which do not depend on time can actually be identified with a measure in M (); recall that, in [6] is proved that, if µ M (), then it may be decomposed as µ = f div(g), where f L 1 () and g (L p ()) N. So, if µ is a measure of M () which does not depend on time and B is a Borel set in of zero elliptic p-capacity, then thanks to Theorem 1.6 we deduce that cap p (B (, T )) = and so µ(b (, T )) = ; since µ is supposed to be independent on time, we have = µ(b (, T )) = T ν(b), with ν M(), and so ν(b) =, thus ν M (). Hence, from now on, we shall always identify µ and ν. Moreover, notice that, if µ is a measure in M (), then f can be chosen to be nonnegative in the decomposition of [6]. Before passing to the statement and the proof of our main result let us state some interesting results about the entropy solution v of the elliptic problem (1.5). According to [3] (see also [6]) we have that v is in the Marcinkiewicz space M N() N p () that implies v L q () fon any q < N() 2N ; hence, if p >, we have v C(, T ; N p N+1 L1 ()).
6 6 FRANCESCO PETITTA So let us suppose p > 2N and let us observe that such a solution actually turns out to N+1 be an entropy solution of the initial boundary value problem (1.7) with initial datum u (x) = v(x), for all T >, since we have = = Θ k (v ϕ)(t ) dx d dt Θ k(v ϕ) dxdt = T Θ k (v ϕ)() dx T ϕ t, T k (v ϕ) W 1,p (),W 1,p () dt (v ϕ) t, T k (v ϕ) W 1,p (),W 1,p () dt that can be cancelled out with the analogous term in (1.8) getting the right formulation (1.6) for v. For technical reasons we shall use the stronger assumption (1.9) p > 2N + 1 N + 1 throughout this paper; notice that, according to [5] (see also [1]), in this case a solution u of problem (1.4) belongs to L r (, T ; W 1,r ) for any r < p N N. Observe that p > N+1 N+1 1 if and only if (1.9); hence, in this case, the gradient of the entropy solution u (that coincides with the distributional one) actually belong L 1 (). Moreover, this is the same assumption used in [21] since it allows, for instance, to get continuity of the solution with values in L 1 () directly by using the trace result of [2]. Moreover, observe that, if µ M (), and µ then the entropy solution v of the elliptic problem is nonnegative; indeed, choosing in (1.6) as test function ϕ = T h (v + ) (where v = v + v is the usual decomposition of a function through its positive and negative parts), we get (1.1) a(x, v) T k (v T h (v + )) dx T k (v T h (v + )) dµ; now, as we can write µ = f div(g) L 1 ()+W 1,p (), and T k (v T h (v + )) converges, as h tends to infinity, to T k ( v ) almost everywhere, -weakly in L (), and weakly in W 1,p () (see for instance [6]), we have lim T k (v T h (v + )) dµ = T k ( v ) dµ ; h +
7 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 7 On the other hand, observing that, for v h we have T k (v T h (v + )) = T k ( v ) =, we can split the left hand side of (1.1) into three terms: a(x, v) T k (v T h (v + )) dx = a(x, v) T k (v) dx + {h<v h+k} {v } a(x, v) (v h) dx + {v>h+k} and the last term is obviously zero, while, using hypothesis (1.3), a(x, v) (v h) dx, {h<v h+k} a(x, v) h dx, is positive and we can drop it; therefore, passing to the limit on h in the right hand side of (1.1), and using hypothesis (1.1), we obtain α T k (v ) p dx = α T k (v ) p dx {v } a(x, v) T k (v T h (v + )) dx ω(h), where where ω(h) is a positive quantity that vanishes as h tend to infinity; thus, passing to the limit on h, we get v a.e. in. Reasoning analogously, using a standard Landes time-regularization argument (see, for instance, [5] and [2] for the outline of this technic), one can prove that the entropy solution u(x, t) of problem (1.4) turns out to be nonnegative if µ. Now, we can state our main result for the homogeneous case, that is for problem (1.4); Section 2 will be devoted to the proof of this result, while in Section 3 we shall prove the same result for the nonhomogeneus problem with nonnegative initial data in L 1 (). Theorem 1.7. Let µ M () be independent on the variable t, p > 2N+1, and let N+1 µ ; let u(x, t) be the entropy solution of problem (1.4) with u =, and v(x) the entropy solution of the corresponding elliptic problem (1.5). Then in L 1 (). lim u(x, t) = v(x), t + 2. Proof of Theorem 1.7 First of all, let us state and prove a comparison result that plays a key role in the proof of our main result. In the proof of this result and in what follows we will use several facts proved in [21] for L 1 data and whose generalization to data in L 1 ()+L p (, T ; W 1,p ()) is quite simple; in particular, we will use the fact that the approximating sequences of variational solutions are strongly compacts in C(, T ; L 1 ()).
8 8 FRANCESCO PETITTA Lemma 2.1. Let u, v L 1 (), such that u v, and let µ M (); if u and v are, respectively, the entropy solutions of the problems u t + A(u) = µ in (, T ), (2.1) u(x, ) = u in, u(x, t) = on (, T ), and (2.2) then u v a.e. in, for all t (, T ). v t + A(v) = µ in (, T ), v(x, ) = v in, v(x, t) = on (, T ), Proof. First of all, let us suppose u, v L 2 (); we will use an approximation argument. Consider the entropy solutions of the same problems with datum F W 1,p () instead of µ; let us call also these solutions u and v. These solutions coincide with the variational ones with this kind of data. Therefore we can use the variational formulation of problems (2.1) and (2.2), integrating between and t, for any t T (we will use the notation t = (, t)). Using ϕ = (u v) + as test function, and then subtracting, we get = T (u v) t, (u v) + W 1,p (),W 1,p () dt + (a(x, u) a(x, v)) (u v) + dxdt t = 1 t d 2 dt [(u v)+ ] 2 dxdt + (a(x, u) a(x, v)) (u v) + dxdt t = 1 v) 2 [(u + ] 2 (t) dx 1 [(u v) + ] 2 () dx 2 + (a(x, u) a(x, v)) (u v) + dxdt. t Since the last term is positive we can drop it, while the second one is zero as u v. Therefore we have, for all t (, T ) so u v a.e. in, for all t (, T ). (u v) + = a.e. in,
9 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 9 Now, let us consider u and v as the entropy solutions of problems (2.1) and (2.2) with nonnegative data in M (), and u, v in L 1 () as initial data; as we can write µ = f div(g) L 1 () + W 1,p (), we can approximate the L 1 term f with a sequence f n of nonnegative regular functions that converges to f in L 1 () and such that f n L 1 () f L 1 (); moreover, let us consider two sequences of smooth functions u,n and v,n such that u,n converges to u, and v,n converges to v in L 1 (), with u,n v,n. So, we can apply the result proved above finding two sequence of solutions u n and v n of problems (2.1) and (2.2) with data f n div(g) and u,n, v,n as initial data, for which the comparison result holds true; so u n v n for all t (, T ), a.e. in. Now, it is well known (see for instance [2]) that the solutions of (2.1) and (2.2) obtained as limits of u n and v n are unique and coincide with the unique entropy solution of the limit problem with datum µ, and initial datum, respectively, u and v ; then we have that the sequences u n and v n converge, respectively, to u and v a.e. in, for all fixed t (, T ). So u v a.e. in, for all fixed t (, T ). Now, we are able to prove Theorem 1.7. For the sake of simplicity, in what follows, the convergences are all understood to be taken up to a suitable subsequence extraction. Proof of Theorem 1.7. To simplify the notation, during this proof, we will indicate by the parabolic cylinder of height one (, 1), instead of T as usual; let n N {}, and define u n (x, t) as the entropy solution of the initial boundary value problem u n t + A(u n ) = µ in (, 1), (2.3) u n (x, ) = u(x, n) in, u n (x, t) = on (, 1), Recall that, since u C(, T ; L 1 ()), then u(x, n) L 1 () is well defined; moreover, observe that, by virtue of uniqueness of entropy solution and the definition of u n, recalling that u(x, ) =, we have that u(x, n) = u n 1 (x, 1) a.e. in, for n 1. Now, applying Lemma 2.1, and recalling that v, the entropy solution of problem (1.5), is also an entropy solution of problem (1.7) with v itself as initial datum, we get immediately that (2.4) u(x, t) v(x), for all t [, T ], a.e. in, being u(x, t) solution of the same problem with u(x, ) = as initial datum. Moreover, applying again Lemma 2.1, we get that, for every n (2.5) u n (x, t) v(x), for all t [, 1], a.e. in, Finally, if we consider a parameter s > we have that both u(x, t) and u s (x, t) u(x, t + s) are solutions of problem (1.7) with, respectively, and u(x, s) as initial datum; so, again from Lemma 2.1 we deduce that u(x, t + s) u(x, t) for t, s, and so u is a monotone nondecreasing function in t. In particular u(x, n) u(x, m) for all
10 1 FRANCESCO PETITTA n, m N with n < m, and so we have u n (x, t) u n+1 (x, t), for all n and t. Therefore, from the monotonicity of u n, it follows that exists a function ũ such that, u n (x, t) converges to ũ(x, t) a.e. on as n tends to infinity. Now, let us look for some a priori estimates concerning the sequence u n. The symbol C will indicate any positive constant whose value may change from line to line. Let us fix n and take ϕ = in the entropy formulation for u n ; we get Θ k (u n )(1) + α T k (u n ) p dxdt (2.6) k( µ M () + u(x, n) L 1 ()) k( µ M () + v L 1 ()) = Ck. Therefore, for every fixed k >, from the first term on the left hand side of (2.6), recalling that u n (x, t) is monotone nondecreasing in t, we get, reasoning as in [5], that u n is uniformly bounded in L (, 1; L 1 ()) (and the bound does not depend on k), while from the second one we have that T k (u n ) is uniformly bounded in L p (, 1; W 1,p ()) for fixed k >. We can improve this kind of estimate by using a suitable Gagliardo-Nirenberg type inequality (see [2], Proposition 3.1) which asserts that, if w L q (, T ; W 1,q ()) L (, T ; L ρ ()), with q 1, ρ 1. Then w L σ () with σ = q N+ρ and Indeed, in this way we get (2.7) and so, we can write k p+ p N meas{ u n k} then, w σ dxdt C w ρq N L (,T ;L ρ ()) { u n k} T k (u n ) p+ p N dxdt Ck T k (u n ) p+ p N dxdt (2.8) meas{ u n k} C k + p N N w q dxdt.. T k (u n ) p+ p N dxdt Ck; Therefore, the sequence u n is uniformly bounded in the Marcinkiewicz space M + p N (); that implies, since in particular p > 2N, that N+1 un is uniformly bounded in L m () for all 1 m < p 1 + p (for further properties of Marcinkiewicz spaces see for instance N [23]).
11 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 11 We are interested about a similar estimate on the gradients of functions u n ; let us emphasize that these estimate hold true for all functions satisfying (2.6), so we will not write the index n. First of all, observe that (2.9) meas{ u λ} meas{ u λ; u k} + meas{ u λ; u > k}. With regard to the first term to the right hand side of (2.9) we have meas{ u λ; u k} 1 u p dx λ p (2.1) = 1 λ p { u k} u p dx = 1 λ p { u λ; u k} T k (u) p dx Ck λ p ; while for the last term in (2.9), thanks to (2.8), we can write meas{ u λ; u > k} meas{ u k} C k σ, with σ = p 1 + p. So, finally, we get N meas{ u λ} C k + Ck σ λ, p and we can have a better estimate by taking the minimum over k of the right hand side; the minimum is achieved for the value ( ) 1 σc σ+1 p k = λ σ+1, C and so we get the desired estimate (2.11) meas{ u λ} Cλ γ with γ = p( σ ) = Np+p N = p N ; this estimate is the same obtained in [4]. σ+1 N+1 N+1 Then, coming back to our case, we have found that, for every n, u n is equibounded in M γ (), with γ = p N 2N+1, and so, since p >, N+1 N+1 un is uniformly bounded in L s () with 1 s < p N. N+1 Now, we shall use the above estimates to prove some compactness results that will be useful to pass to the limit in the entropy formulation for u n. Indeed, thanks to these estimates, we can say that there exists a function u L q (, 1; W 1,q ()), for all q < p N, such that N+1 un converges to u weakly in L q (, 1; W 1,q ()). Observe that, obviously, we have u = ũ a.e. in. On the other hand from the equation we deduce that u n t L 1 () + L s (, 1; W 1,s ()) uniformly with respect to n, where s = q, for all q < p N, and so, thanks to the Aubin-Simon type result proved in [8] we have N+1 that u n actually converges to u in L 1 (). Moreover, using the estimate (2.6) on the
12 12 FRANCESCO PETITTA truncations of u n, we deduce, from the boundedness and continuity of T k (s), that, for every k > T k (u n ) T k (u), weakly in L p (, 1; W 1,p ()), T k (u n ) T k (u), strongly in L p (). Finally, the sequence u n satisfies the hypotheses of Theorem 3.3 in [5], and so we get u n u a.e. in. All these results allow us to pass to the limit in the entropy formulation of u n ; indeed, for all k >, u n satisfies (2.12) Θ k (u n ϕ)(1) dx (2.13) Θ k (u n (x, ) ϕ()) dx (2.14) + (2.15) + 1 (2.16) ϕ t, T k (u n ϕ) W 1,p (),W 1,p () dt a(x, u n ) T k (u n ϕ) dxdt T k (u n ϕ) dµ, for all ϕ L p (, 1; W 1,p ()) L () C([, 1]; L 1 ()) with ϕ t L p (, 1; W 1,p ()). Let us analyze this inequality term by term: recalling that µ can be decomposed as µ = f div(g), where f L 1 () and g (L p ()) N, then, since T k (u n ϕ) converges to T k (u ϕ) -weakly in L (), and T k (u n ϕ) converges to T k (u ϕ) also weakly in L p (, 1; W 1,p ()), we have moreover, we can write (2.15) = (2.17) T k (u n ϕ) dµ n T k (u ϕ) dµ; (a(x, u n ) a(x, ϕ)) T k (u n ϕ) dxdt + a(x, ϕ) T k (u n ϕ) dxdt,
13 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 13 and the second term to the right hand side of (2.17) converges, as n tends to infinity, to a(x, ϕ) T k (u ϕ) dxdt, while to deal with the nonnegative first term of the right hand side of (2.17), we must use the a.e. convergence of the gradients; then, applying Fatou s lemma, we get (a(x, u) a(x, ϕ)) T k (u ϕ) dxdt lim inf n (a(x, u n ) a(x, ϕ)) T k (u n ϕ) dxdt. Our goal is to prove that u = v almost everywhere in ; to do that, it is enough to prove that u does not depend on time, and that (2.12)+(2.13)+(2.14) converges to zero as n tends to infinity; indeed, if that holds true, we obtain that u satisfies the entropy formulation for the elliptic problem (1.5), and so, since the entropy solution is unique, we get that u = v a.e. in. Let us prove first that u does not depend on time; let us denote by w(x) the almost everywhere limit of the monotone nondecreasing sequence u n (x, ) = u(x, n), hence, using the comparison Lemma 2.1, we have that, for fixed t (, 1) u n (x, ) u n (x, t) = u(x, n + t) u(x, n + 1) = u n+1 (x, ), and, since both u n and u n+1 converge to w(x) that does not depend on time, such happens also for the a.e. limit of u n (x, t) that is u = w. Now, using the monotone convergence theorem, we get lim[(2.12) + (2.13)] = Θ k (w(x) ϕ(1)) dx Θ k (w(x) ϕ()) dx n = 1 d Θ(w(x) ϕ) dtdx = dt 1 (w(x) ϕ) t, T k (w(x) ϕ) W 1,p (),W 1,p () dt, while, since T k (u n ϕ) converges to T k (w ϕ) weakly in L p (, 1; W 1,p ()), we have (2.14) n 1 ϕ t, T k (w ϕ) W 1,p (),W 1,p () dt. Finally we can sum all these terms and, since w does not depend on time, we find lim n [(2.12) + (2.13) + (2.14)] = 1 w t, T k (w ϕ) W 1,p (),W 1,p () and, as we mentioned above, this is enough to prove that w(x) = v(x). dt = ;
14 14 FRANCESCO PETITTA 3. Nonhomogeneous case Now we deal with the general case of problem (1.4) with a nonhomogeneous initial datum u, that is (3.1) u t + A(u) = µ in (, T ), u(x, ) = u (x) in, u(x, t) = on (, T ), with µ M () satisfying the usual hypotheses used throughout this paper, and u a nonnegative function in L 1 (). We shall prove the following result: Theorem 3.1. Let µ M () be independent of the variable t, p > 2N+1 N+1, u L 1 () be a nonnegative function, and let µ ; let u(x, t) be the entropy solution of problem (3.1), and v the entropy solution of the corresponding elliptic problem (1.5). Then in L 1 (). lim u(x, t) = v(x), t + Most part of our work will be concerned with comparison between suitable entropy subsolutions and supersolutions of problem (1.4). The notion of entropy subsolution and supersolution for the parabolic problem will be given as a natural extension of the one for the elliptic case (see for instance [18]). Definition 3.2. A function u(x, t) C([, T ]; L 1 ()) is an entropy subsolution of problem (3.1) if, for all k >, we have that T k (u) L p (, T ; W 1,p ()), and holds Θ k (u ϕ) + (T ) dx Θ k (u ϕ()) + dx (3.2) + T + ϕ t, T k (u ϕ) + W 1,p (),W 1,p () dt a(x, u) T k (u ϕ) + dxdt T k (u ϕ) + dµ, for all ϕ L p (, T ; W 1,p ()) L () C([, T ]; L 1 ()) with ϕ t L p (, T ; W 1,p ()) and u(x, ) u (x) u (x) almost everywhere on with u L 1 ().
15 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 15 On the other hand, u(x, t) C([, T ]; L 1 ()) is an entropy supersolution of problem (3.1) if, for all k >, we have that T k (u) L p (, T ; W 1,p ()), and holds Θ k (u ϕ) (T ) dx Θ k (u ϕ()) dx (3.3) + T + ϕ t, T k (u ϕ) W 1,p (),W 1,p () dt a(x, u) T k (u ϕ) dxdt T k (u ϕ) dµ, for all ϕ L p (, T ; W 1,p ()) L () C([, T ]; L 1 ()) with ϕ t L p (, T ; W 1,p ()) and u(x, ) u (x) u (x) almost everywhere on with u L 1 (). Thanks to Definition 3.2 it is possible to improve straightforwardly the comparison Lemma 2.1, by comparing both subsolution and supersolution with the unique entropy solution of problem (3.1); actually, we shall state this result in a simpler case that will be enough to prove Theorem 3.1. Moreover, notice that the same approximation argument we will use in the proof of next result allow us to deduce that a solution of problem (3.1) turns out to be both an entropy subsolution and an entropy supersolution of the same problem. Lemma 3.3. Let µ M (), and let u and u be, respectively, an entropy subsolution and an entropy supersolution of problem (3.1), and let u be the unique entropy solution of the same problem. Then u u u. Proof. Let us prove only that u u being the other case analogous. We know that the entropy solution u is found as limit of regular functions u n solutions of approximating problems with smooth data. So, if µ = f div(g) with f L 1 () and g (L p ()) N we can choose a sequence of smooth functions f n that converges to f in L 1 (); let us call µ n = f n div(g). Moreover, let u,n be a sequence of smooth functions that converges to u in L 1 (); so, we call u n the variational solution of problem (3.4) (u n ) t + A(u n ) = µ n in (, T ), u n (x, ) = ũ,n in, u n (x, t) = on (, T ), where ũ,n = min(u,n, u ); notice that, thanks to dominated convergence theorem, ũ,n converges to u in L 1 ().
16 16 FRANCESCO PETITTA Now, we can choose T k (u u n ) + as test function obtaining T (u n ) t, T k (u u n ) + dt + a(x, u n ) T k (u u n ) + dxdt = µ n T k (u u n ) + dxdt; On the other hand being u a subsolution we have Θ k (u u n ) + (T ) dx Θ k (u u n ) + () dx + T + (u n ) t, T k (u u n ) + dt a(x, u) T k (u u n ) + dxdt T k (u u n ) + dµdt. So, we can subtract this relation from the previous one, recalling that (u u n ) + () =, to obtain Θ k (u u n ) + (T ) dx + (a(x, u) a(x, u n )) T k (u u n ) + dxdt T k (u u n ) + dµdt µ n T k (u u n ) + dxdt + Θ k (u ũ n, ) + dx. Hence, by stability results (see [21], [11], and references therein), using that ũ,n converges to u in L 1 (), and the decomposition of µ and µ n, we deduce that the right hand side of the above inequality tends to zero as n goes to infinity. Thus, the monotonicity assumption on a, and Fatou s lemma, yield that u ũ, where ũ is an entropy solution of problem (3.1) with ũ() = u. Hence, thanks to Lemma 2.1 we have ũ u, and so, finally, we get u u. Observe that, we actually proved that, for every fixed T >, a.e. on, we have u(x, T ) u(x, T ). Proof of Theorem 3.1. We will prove the result in few steps. Step 1. u v. Let us first suppose that u solves problem (3.1) with initial datum satisfying u v a. e. in, where v is the entropy solution of problem (1.5); we proved that v is also an entropy solution of the initial boundary value problem (1.4) with v itself as initial datum, and so it turns out to be a supersolution of the same problem with initial datum u. Therefore, by Lemma 3.3, we get u(x, t) v(x) a.e. in, for any t >. On the other hand, according to Lemma 2.1, the solution ǔ of problem (1.4) starting from as initial datum satisfies ǔ(x, t) u(x, t) a.e. in, for any t >.
17 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 17 Theorem 1.7 ensures that ǔ(x, t) converges toward v in L 1 () as t goes to infinity. So, by comparison, we have that the solution u(x, t) of (1.4) converges to v in L 1 () as t diverges. Step 2. u v τ. Let us take û(x, t) the solution of problem (3.1) with u = v τ as initial datum for some τ > 1, where v τ is the entropy solution of the elliptic problem (1.5) with µ τ as datum instead of µ = f div(g) (f in L 1 (), g (L p ()) N ), where { τµ if f =, µ τ = τf div(g) if f. In [6] it was proved that, in the decomposition of µ, f can be chosen to be nonnegative and so, since τ > 1, we have that µ τ µ; in fact, if f, we can write µ τ = (τ 1)f + µ µ. Thus, we can use elliptic comparison results between entropy sub and supersolutions (see [18]) to obtain v v τ. Hence, since v τ does not depend on time, we have that it is a supersolution of the parabolic problem (3.1) with u = v τ, and, recalling that v is a subsolution of the same problem (being v v τ as initial data), we can apply the Lemma 3.3 again finding that v(x) û(x, t) v τ (x) a.e. in, for all positive t. Now, thanks to the fact that the datum µ does not depend on time, we can apply the comparison result also between û(x, t + s) solution with u = û(x, s), with s a positive parameter, and û(x, t), the solution with u = v τ as initial datum; so we obtain û(x, t + s) û(x, t) for all t, s, a.e. in. So, by virtue of this monotonicity result, and recalling that û(x, t) v(x), we have that there exists a function v v such that û(x, t) converges to v a.e. in as t tends to infinity. Clearly v does not depend on t and we can develop the same argument used for the homogeneous case, by defining û n as the solution of û n t + A(û n ) = µ in (, 1), û n (x, ) = û(x, n) in, û n (x, t) = on (, 1), and starting from the analogous estimate Θ k (û n )(t) + α T k (û n ) p dxdt k( µ M ( 1 ) + u L 1 ()) = Ck, 1 to prove that we can pass to the limit in the entropy formulation, and so, by uniqueness result, we can obtain that v = v. So, we have proved that the result holds for the solution starting from u = v τ as initial datum, with τ > 1. Finally, if u v τ for τ > 1, thanks to Lemma 3.3 we have (3.5) ǔ(x, t) u(x, t) û(x, t), where ǔ, u and û solves problem (3.1) with, respectively,, u and v τ as initial data.
18 18 FRANCESCO PETITTA So, as we proved above, û(x, t) tends to v as t goes to infinity, while, according to Theorem 1.7, the same happens for ǔ(x, t) ; therefore, using (3.5), we can conclude that the result holds true for u. Step 3. u L 1 () and µ. Let us consider the general case of a solution u(x, t) with nonnegative initial datum u L 1 () and let suppose first that µ ; let us define the monotone nondecreasing (in τ) family of functions u,τ = min(u, v τ ). As we have shown above, for every fixed τ > 1, u τ (x, t), the entropy solution of problem (3.1) with u,τ as initial datum, converges to v a.e. in, as t tends to infinity. Moreover, we have also that T k (u τ (x, t)) converges to T k (v) weakly in W 1,p () as t diverges, for every fixed k >. Now, let us state the following lemma, that will be useful in the sequel: we shall prove it below, when the proof of Theorem 3.1 will be completed. Lemma 3.4. Let µ be a nonnegative measure in M (), and τ >. Moreover, let v τ be the entropy solution of problem { A(v τ ) = µ τ in, (3.6) v τ = on, then, we have almost everywhere on. lim v τ(x) = + τ So, thanks to Lebesgue theorem, and Lemma 3.4, we can easily check that u,τ converges to u in L 1 () as τ tends to infinity. Therefore, using a stability result of entropy solution (see for instance [2]) we obtain that T k (u τ (x, t)) converges to T k (u(x, t)) strongly in L p (, T ; W 1,p ()) as τ tends to infinity. Now, making the same calculations used in [21] to prove the uniqueness of entropy solutions applied to u and u τ, where u τ is considered as the solution obtained as limit of approximating solutions with smooth data, we can easily find, for any fixed τ > 1, the following estimate Θ k (u u τ )(t) dx Θ k (u u,τ ) dx, for every k, t >. Then, let us divide the above inequality by k, and let us pass to the limit as k tends to ; we obtain (3.7) u(x, t) u τ (x, t) L 1 () u (x) u,τ (x) L 1 (), for every t >. Hence, we have u(x, t) v(x) L 1 () u(x, t) u τ (x, t) L 1 () + u τ (x, t) v(x) L 1 ();
19 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 19 then, thanks to the fact that the estimate in (3.7) is uniform in t, for every fixed ε, we can choose τ large enough such that u(x, t) u τ (x, t) L 1 () ε 2, for every t > ; on the other hand, according to Step 2, there exists t such that u τ (x, t) v(x) L 1 () ε 2, for every t > t, and this proves our result if µ. Step 4. u L 1 () and µ =. If µ = we can consider, for every ε >, the solution u ε of problem u ε t + A(u ε ) = ε in (, T ), (3.8) u ε (x, ) = u in, u ε (x, t) = on (, T ); thanks to Lemma 3.3 we get that (3.9) u(x, t) u ε (x, t), for every fixed t (, T ), and almost every x, by the previous result we obtain lim T uε (x, T ) = v ε (x), where v ε is the entropy solution of the elliptic problem associated to (3.8), and, moreover, by virtue of the stability result for this problem, we know that v ε (x) converges to as ε tends to ; so, almost everywhere on, using (3.9), we have lim sup u(x, T ) v ε (x), T and, since ε is arbitrary, we can conclude by using Vitali s Theorem since v ε is strongly compact in L 1 (). Proof of Lemma 3.4. Let us first suppose that µ W 1,p (), so that we have µ = div(g) with g (L p ()) N ; v τ then solves (3.1) a(x, v τ ) ϕ dx = τ g ϕ dx, for every ϕ W 1,p (); let us take ϕ = v τ as test function in (3.1); so, using assumption (1.1), we get α v τ p dx τ g (L p ()) v τ N W 1,p () ; now, since v τ, we can divide the above expression by τ v τ W 1,p (), getting ( ) ( ( ) 1 v τ p p dx = τ vτ p ) p dx g (L p ()). N
20 2 FRANCESCO PETITTA Therefore, we have that v τ v W 1,p () and a subsequence, such that is bounded in W 1,p (), and so there exists a function v τ weakly converges in W 1,p () (and then a.e.) to v as τ tends to infinity. So, it is enough to prove that v > almost everywhere on to conclude our proof. To this aim, for every τ >, let us define a τ (x, ξ) = 1 τ a(x, ξ); we can easily check that such an operator satisfies assumptions (1.1), (1.2) and (1.3), with the same constants α and β. Notice that in the model case of the p-laplacian, v τ thanks to its homogeneity property, we have a τ a. Now, satisfies the elliptic problem div(a τ (x, v τ )) = µ in, (3.11) v τ = on, in a variational sense; indeed, for every ϕ W 1,p (), we have ( ) vτ (3.12) a τ (x, ) ϕ dx = 1 a(x, v τ ) ϕ dx = τ g ϕ dx. Moreover, thanks to Theorem 4.1 in [7], we have that the family of operators {a τ } has a G-limit in the class of Leray-Lions type operators; that is, there exists a Carathéodory function a satisfying assumptions (1.1), (1.2) and (1.3), and a sequence of indices τ(k) (called τ again), such that a τ G a; so, because of that, being v the weak limit of v τ ( ) vτ a(x, ) τ a(x, v), in W 1,p (), we get that weakly in (L p ()) N. Therefore, using this result in (3.12) we have a(x, v) ϕ dx = g ϕ dx, for every ϕ W 1,p (); and so, v is a variational solution of problem { div(a(x, v)) = µ in, v = on.
21 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 21 Then, recalling that µ and using a suitable Harnack type inequality (see for instance [24]), we deduce that v > almost everywhere on. Now, if µ M (), we have µ τ = τf div(g), with f a nonnegative function in L 1 (); we can suppose, without loss of generality, that µ τ = τχ E div(g) for a suitable set E of positive measure; indeed, f, being nonidentically zero, it turns out to be strictly bounded away from zero on a suitable E, and so there exists a constant c such that f cχ E, and then, once we proved our result for such a µ τ, we can easily prove the statement by applying again a comparison argument. Now, reasoning v τ analogously as above we deduce that solves the elliptic problem a τ (x, ( v τ )) = χ B 1 div(g) in, τ (3.13) Moreover, v τ = on. χ B 1 τ div(g) χ B, strongly in W 1,p () as τ tends to infinity. Therefore, since G-convergence is stable v τ under such type a of convergence of data, we have, that the weak limit v of in W 1,p (), solves { div(a(x, v)) = χ B in, v = on, and so we may conclude, as above, that v > almost everywhere on, that implies that v τ goes to infinity as τ tends to infinity. Remark 3.5. As we said before, in many cases, the convergences in norm to the stationary solution can be improved depending on the regularity of the limit solution (or equivalently to the regularity of the datum); indeed, consider (2.4) in the proof of Theorem 1.7, that is u(x, t) v(x), for all t (, T ), a.e. in ; so, if, for instance, µ L q () with q > N, then Stampacchia s type estimates ensure p that the solution v of the stationary problem { A(v) = µ in, v = on, is in L () and so the convergence of u(x, t) to v of Theorem 1.7 is at least -weak in L () and almost everywhere. Reasoning similarly one can refine, depending on the data, the asymptotic results of both Theorem 1.7 and Theorem 3.1.
22 22 FRANCESCO PETITTA Acknowledgments I would like to thank Luigi Orsina for fruitful discussions on the subject of this paper, and the referee for useful comments. References [1] A. Arosio, Asymptotic behavior as t + of solutions of linear parabolic equations with discontinuous coefficients in a bounded domain, Comm. Partial Differential Equations 4 (1979), no. 7, [2] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, [3] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez, An L 1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), [4] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), [5] L. Boccardo, A. Dall Aglio, T. Gallouët, L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997) no.1, [6] L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), [7] V. Chiadó-Piat, G. Dal Maso, A. Defranceschi, G-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (199), [8] A. Dall Aglio, L. Orsina, Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5 (1992), [9] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), [1] J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (23), no. 2, [11] J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA (to appear). [12] A. Friedman, Partial differential equations of parabolic type, Englewood Cliffs, N.J., Prentice-Hall [13] N. Grenon, Asymptotic behaviour for some quasilinear parabolic equations, Nonlinear Anal. 2 (1993), no. 7, [14] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, Oxford, [15] R. Landes, V. Mustonen, A strongly nonlinear parabolic initial-boundary value problem, Arkiv for Matematik, 25 (1987), no. 1, [16] T. Leonori, F. Petitta, Asymptotic behavior of solutions for parabolic equations with natural growth term and irregular data, Asymptotic Analysis 48(3) (26), [17] J. L. Lions, uelques méthodes de résolution des problèmes aux limites non linéaire, Dunod et Gautier-Villars, (1969). [18] M. C. Palmeri, Entropy subsolution and supersolution for nonlinear elliptic equations in L 1, Ricerche Mat. 53 (24), no. 2, [19] F. Petitta, Asymptotic behavior of solutions for linear parabolic equations with general measure data, C. R. Acad. Sci. Paris, Ser. I, 344 (27) [2] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura ed Appl. (IV), 177 (1999),
23 ASYMPTOTIC BEHAVIOR IN PARABOLIC PROBLEMS WITH MEASURES 23 [21] A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L 1 data, Nonlin. Anal. TMA 28 (1997), [22] S. Spagnolo, Convergence de solutions d équations d évolution, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), , Pitagora, Bologna, [23] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du seconde ordre à coefficientes discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), [24] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl Math, 2 (1967), Dipartimento di Matematica, Università La Sapienza, Piazzale A. Moro 2, 185, Roma, Italy address: petitta@mat.uniroma1.it
EXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 3, Issue 3, Article 46, 2002 WEAK PERIODIC SOLUTIONS OF SOME QUASILINEAR PARABOLIC EQUATIONS WITH DATA MEASURES N.
More informationELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC
More informationLARGE TIME BEHAVIOR FOR p(x)-laplacian EQUATIONS WITH IRREGULAR DATA
Electronic Journal of Differential Equations, Vol. 215 (215), No. 61, pp. 1 16. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LARGE TIME BEHAVIOR
More informationarxiv: v1 [math.ap] 10 May 2013
0 SOLUTIOS I SOM BORDRLI CASS OF LLIPTIC QUATIOS WITH DGRAT CORCIVITY arxiv:1305.36v1 [math.ap] 10 May 013 LUCIO BOCCARDO, GISLLA CROC Abstract. We study a degenerate elliptic equation, proving existence
More informationASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS
ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied athematics http://jipam.vu.edu.au/ Volume 4, Issue 5, Article 98, 2003 ASYPTOTIC BEHAVIOUR OF SOE EQUATIONS IN ORLICZ SPACES D. ESKINE AND A. ELAHI DÉPARTEENT
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationNonlinear equations with natural growth terms and measure data
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 183 202. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationRenormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy
Electronic Journal of Qualitative Theory of Differential Equations 26, No. 5, -2; http://www.math.u-szeged.hu/ejqtde/ Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy Zejia
More informationA CONNECTION BETWEEN A GENERAL CLASS OF SUPERPARABOLIC FUNCTIONS AND SUPERSOLUTIONS
A CONNECTION BETWEEN A GENERAL CLASS OF SUPERPARABOLIC FUNCTIONS AND SUPERSOLUTIONS RIIKKA KORTE, TUOMO KUUSI, AND MIKKO PARVIAINEN Abstract. We show to a general class of parabolic equations that every
More informationA Not So Long Introduction to the Weak Theory of Parabolic Problems with Singular Data
A Not So Long Introduction to the Weak Theory of Parabolic Problems with Singular Data Lecture Notes of a Course Held in Granada October-December 27 by Francesco Petitta Contents Chapter. Motivations
More informationNONLINEAR ELLIPTIC EQUATIONS WITH MEASURES REVISITED
NONLINEAR ELLIPTIC EQUATIONS WITH MEASURES REVISITED Haïm Brezis (1),(2), Moshe Marcus (3) and Augusto C. Ponce (4) Abstract. We study the existence of solutions of the nonlinear problem (P) ( u + g(u)
More informationA MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS
A MULTI-DIMENSIONAL OPTIMAL HARVESTING PROBLEM WITH MEASURE VALUED SOLUTIONS A BRESSAN, G M COCLITE, AND W SHEN Abstract The paper is concerned with the optimal harvesting of a marine resource, described
More informationExistence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data
Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data Olivier Guibé - Anna Mercaldo 2 Abstract In this paper we prove the existence of
More informationSuitable Solution Concept for a Nonlinear Elliptic PDE
Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine Igbida Institut de recherche XLIM, UMR-CNRS 6172 Université de Limoges 87060 Limoges, France Colloque EDP-Normandie 2011 Rouen, 25-26 Octobre
More informationWeak Solutions to Nonlinear Parabolic Problems with Variable Exponent
International Journal of Mathematical Analysis Vol. 1, 216, no. 12, 553-564 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.216.6223 Weak Solutions to Nonlinear Parabolic Problems with Variable
More informationON THE EXISTENCE OF CONTINUOUS SOLUTIONS FOR NONLINEAR FOURTH-ORDER ELLIPTIC EQUATIONS WITH STRONGLY GROWING LOWER-ORDER TERMS
ROCKY MOUNTAIN JOURNAL OF MATHMATICS Volume 47, Number 2, 2017 ON TH XISTNC OF CONTINUOUS SOLUTIONS FOR NONLINAR FOURTH-ORDR LLIPTIC QUATIONS WITH STRONGLY GROWING LOWR-ORDR TRMS MYKHAILO V. VOITOVYCH
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationGOOD RADON MEASURE FOR ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT
Electronic Journal of Differential Equations, Vol. 2016 2016, No. 221, pp. 1 19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GOOD RADON MEASURE FOR ANISOTROPIC PROBLEMS
More informationRENORMALIZED SOLUTIONS ON QUASI OPEN SETS WITH NONHOMOGENEOUS BOUNDARY VALUES TONI HUKKANEN
RENORMALIZED SOLTIONS ON QASI OPEN SETS WITH NONHOMOGENEOS BONDARY VALES TONI HKKANEN Acknowledgements I wish to express my sincere gratitude to my advisor, Professor Tero Kilpeläinen, for the excellent
More informationRégularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen
Régularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen Daniela Tonon en collaboration avec P. Cardaliaguet et A. Porretta CEREMADE, Université Paris-Dauphine,
More informationJournal of Differential Equations
J. Differential Equations 248 21 1376 14 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Renormalized and entropy solutions for nonlinear parabolic
More informationStrongly nonlinear parabolic initial-boundary value problems in Orlicz spaces
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 203 220. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationMULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS AND DISCONTINUOUS NONLINEARITIES
MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS,... 1 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo L (21), pp.??? MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS
More informationEXISTENCE OF BOUNDED SOLUTIONS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol 2007(2007, o 54, pp 1 13 ISS: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu ftp ejdemathtxstateedu (login: ftp EXISTECE OF BOUDED SOLUTIOS
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationA note on W 1,p estimates for quasilinear parabolic equations
200-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationANISOTROPIC EQUATIONS: UNIQUENESS AND EXISTENCE RESULTS
ANISOTROPIC EQUATIONS: UNIQUENESS AND EXISTENCE RESULTS STANISLAV ANTONTSEV, MICHEL CHIPOT Abstract. We study uniqueness of weak solutions for elliptic equations of the following type ) xi (a i (x, u)
More informationRENORMALIZED SOLUTIONS OF STEFAN DEGENERATE ELLIPTIC NONLINEAR PROBLEMS WITH VARIABLE EXPONENT
Journal of Nonlinear Evolution Equations and Applications ISSN 2161-3680 Volume 2015, Number 7, pp. 105 119 (July 2016) http://www.jneea.com RENORMALIZED SOLUTIONS OF STEFAN DEGENERATE ELLIPTIC NONLINEAR
More informationNonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient
Electronic Journal of Differential Equations, Vol. 2002(2002), No. 54, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonexistence
More informationA VARIATIONAL INEQUALITY RELATED TO AN ELLIPTIC OPERATOR
Proyecciones Vol. 19, N o 2, pp. 105-112, August 2000 Universidad Católica del Norte Antofagasta - Chile A VARIATIONAL INEQUALITY RELATED TO AN ELLIPTIC OPERATOR A. WANDERLEY Universidade do Estado do
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationPseudo-monotonicity and degenerate elliptic operators of second order
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 9 24. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationNonlinear aspects of Calderón-Zygmund theory
Ancona, June 7 2011 Overture: The standard CZ theory Consider the model case u = f in R n Overture: The standard CZ theory Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with
More informationEntropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L 1 Data
Entropy Solution for Anisotropic Reaction-Diffusion-Advection Systems with L 1 Data Mostafa BENDAHMANE and Mazen SAAD Mathématiques Appliquées de Bordeaux Université Bordeaux I 351, cours de la Libération,
More informationMEYERS TYPE ESTIMATES FOR APPROXIMATE SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS AND THEIR APPLICATIONS. Yalchin Efendiev.
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX MEYERS TYPE ESTIMATES FOR APPROXIMATE SOLUTIONS OF NONLINEAR ELLIPTIC EUATIONS AND THEIR APPLICATIONS
More informationSome Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces
Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationIN this paper, we study the corrector of the homogenization
, June 29 - July, 206, London, U.K. A Corrector Result for the Homogenization of a Class of Nonlinear PDE in Perforated Domains Bituin Cabarrubias Abstract This paper is devoted to the corrector of the
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationQuasilinear degenerated equations with L 1 datum and without coercivity in perturbation terms
Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 19, 1-18; htt://www.math.u-szeged.hu/ejqtde/ Quasilinear degenerated equations with L 1 datum and without coercivity in erturbation
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationVariational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian
Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus
More informationON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS
Chin. Ann. Math.??B(?), 200?, 1 20 DOI: 10.1007/s11401-007-0001-x ON SOME ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS Michel CHIPOT Abstract We present a method allowing to obtain existence of a solution for
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationRELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 265, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RELATIONSHIP
More informationarxiv: v1 [math.ap] 27 May 2010
A NOTE ON THE PROOF OF HÖLDER CONTINUITY TO WEAK SOLUTIONS OF ELLIPTIC EQUATIONS JUHANA SILJANDER arxiv:1005.5080v1 [math.ap] 27 May 2010 Abstract. By borrowing ideas from the parabolic theory, we use
More informationPositive eigenfunctions for the p-laplace operator revisited
Positive eigenfunctions for the p-laplace operator revisited B. Kawohl & P. Lindqvist Sept. 2006 Abstract: We give a short proof that positive eigenfunctions for the p-laplacian are necessarily associated
More informationThe role of Wolff potentials in the analysis of degenerate parabolic equations
The role of Wolff potentials in the analysis of degenerate parabolic equations September 19, 2011 Universidad Autonoma de Madrid Some elliptic background Part 1: Ellipticity The classical potential estimates
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationWeighted Sobolev Spaces and Degenerate Elliptic Equations. Key Words: Degenerate elliptic equations, weighted Sobolev spaces.
Bol. Soc. Paran. Mat. (3s.) v. 26 1-2 (2008): 117 132. c SPM ISNN-00378712 Weighted Sobolev Spaces and Degenerate Elliptic Equations Albo Carlos Cavalheiro abstract: In this paper, we survey a number of
More informationHeat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control
Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control
More informationOn the uniform Poincaré inequality
On the uniform Poincaré inequality Abdesslam oulkhemair, Abdelkrim Chakib To cite this version: Abdesslam oulkhemair, Abdelkrim Chakib. On the uniform Poincaré inequality. Communications in Partial Differential
More informationA Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth
A Necessary and Sufficient Condition for the Continuity of Local Minima of Parabolic Variational Integrals with Linear Growth E. DiBenedetto 1 U. Gianazza 2 C. Klaus 1 1 Vanderbilt University, USA 2 Università
More informationON THE ASYMPTOTIC BEHAVIOR OF ELLIPTIC PROBLEMS IN PERIODICALLY PERFORATED DOMAINS WITH MIXED-TYPE BOUNDARY CONDITIONS
Bulletin of the Transilvania University of Braşov Series III: Mathematics, Informatics, Physics, Vol 5(54) 01, Special Issue: Proceedings of the Seventh Congress of Romanian Mathematicians, 73-8, published
More informationA TWO PARAMETERS AMBROSETTI PRODI PROBLEM*
PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper
More informationPROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES
PROPERTIES OF CAPACITIES IN VARIABLE EXPONENT SOBOLEV SPACES PETTERI HARJULEHTO, PETER HÄSTÖ, AND MIKA KOSKENOJA Abstract. In this paper we introduce two new capacities in the variable exponent setting:
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
More informationAnisotropic Elliptic Equations in L m
Journal of Convex Analysis Volume 8 (2001), No. 2, 417 422 Anisotroic Ellitic Equations in L m Li Feng-Quan Deartment of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China lifq079@ji-ublic.sd.cninfo.net
More informationOn a weighted total variation minimization problem
On a weighted total variation minimization problem Guillaume Carlier CEREMADE Université Paris Dauphine carlier@ceremade.dauphine.fr Myriam Comte Laboratoire Jacques-Louis Lions, Université Pierre et Marie
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationLORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 27 27), No. 2, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LORENTZ ESTIMATES FOR ASYMPTOTICALLY REGULAR FULLY NONLINEAR
More informationGIOVANNI COMI AND MONICA TORRES
ONE-SIDED APPROXIMATION OF SETS OF FINITE PERIMETER GIOVANNI COMI AND MONICA TORRES Abstract. In this note we present a new proof of a one-sided approximation of sets of finite perimeter introduced in
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationBrunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian M. Novaga, B. Ruffini January 13, 2014 Abstract We prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationSOME PROBLEMS OF SHAPE OPTIMIZATION ARISING IN STATIONARY FLUID MOTION
Advances in Mathematical Sciences and Applications Vol., No. (200x), pp. Gakkōtosho Tokyo, Japan SOME PROBLEMS OF SHAPE OPTIMIZATION ARISING IN STATIONARY FLUID MOTION Luigi. Berselli Dipartimento di Matematica
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationTime Periodic Solutions To A Nonhomogeneous Dirichlet Periodic Problem
Applied Mathematics E-Notes, 8(2008), 1-8 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Time Periodic Solutions To A Nonhomogeneous Dirichlet Periodic Problem Abderrahmane
More informationESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen
W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a
More informationarxiv: v1 [math.ap] 6 Jan 2019
A NONLINEAR PARABOLIC PROBLEM WITH SINGULAR TERMS AND NONREGULAR DATA FRANCESCANTONIO OLIVA AND FRANCESCO PETITTA arxiv:191.1545v1 [math.ap] 6 Jan 219 Abstract. We study existence of nonnegative solutions
More informationMixed exterior Laplace s problem
Mixed exterior Laplace s problem Chérif Amrouche, Florian Bonzom Laboratoire de mathématiques appliquées, CNRS UMR 5142, Université de Pau et des Pays de l Adour, IPRA, Avenue de l Université, 64000 Pau
More informationGLOBAL EXISTENCE AND ENERGY DECAY OF SOLUTIONS TO A PETROVSKY EQUATION WITH GENERAL NONLINEAR DISSIPATION AND SOURCE TERM
Georgian Mathematical Journal Volume 3 (26), Number 3, 397 4 GLOBAL EXITENCE AND ENERGY DECAY OF OLUTION TO A PETROVKY EQUATION WITH GENERAL NONLINEAR DIIPATION AND OURCE TERM NOUR-EDDINE AMROUN AND ABBE
More informationREGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents
REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM BERARDINO SCIUNZI Abstract. We prove some sharp estimates on the summability properties of the second derivatives
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationOn the second differentiability of convex surfaces
On the second differentiability of convex surfaces Gabriele Bianchi, Andrea Colesanti, Carlo Pucci Abstract Properties of pointwise second differentiability of real valued convex functions in IR n are
More informationSingularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations
Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations Irena Rachůnková, Svatoslav Staněk, Department of Mathematics, Palacký University, 779 OLOMOUC, Tomkova
More informationANALYSIS OF THE TV REGULARIZATION AND H 1 FIDELITY MODEL FOR DECOMPOSING AN IMAGE INTO CARTOON PLUS TEXTURE. C.M. Elliott and S.A.
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 6, Number 4, December 27 pp. 917 936 ANALYSIS OF THE TV REGULARIZATION AND H 1 FIDELITY MODEL FOR DECOMPOSING AN IMAGE
More informationON PARABOLIC HARNACK INEQUALITY
ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy
More informationApplications of Mathematics
Applications of Mathematics Jeanette Silfver On general two-scale convergence and its application to the characterization of G-limits Applications of Mathematics, Vol. 52 (2007, No. 4, 285--302 Persistent
More informationSUBELLIPTIC CORDES ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX0000-0 SUBELLIPTIC CORDES ESTIMATES Abstract. We prove Cordes type estimates for subelliptic linear partial
More informationReduced measures for semilinear elliptic equations with Dirichlet operator
Reduced measures for semilinear elliptic equations with Dirichlet operator Tomasz Klimsiak Institute of Mathematics of Polish Academy of Sciences, Nicolaus Copernicus University in Toruń 3rd Conference
More informationΓ-convergence of functionals on divergence-free fields
Γ-convergence of functionals on divergence-free fields N. Ansini Section de Mathématiques EPFL 05 Lausanne Switzerland A. Garroni Dip. di Matematica Univ. di Roma La Sapienza P.le A. Moro 2 0085 Rome,
More informationSemilinear obstacle problem with measure data and generalized reflected BSDE
Semilinear obtacle problem with meaure data and generalized reflected BSDE Andrzej Rozkoz (joint work with T. Klimiak) Nicolau Copernicu Univerity (Toruń, Poland) 6th International Conference on Stochatic
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationWell-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L 1 -Data
CUBO A Mathematical Journal Vol.12, N ō 01, (133 148). March 2010 Well-Posedness Results for Anisotropic Nonlinear Elliptic Equations with Variable Exponent and L 1 -Data Stanislas OUARO Laboratoire d
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationKato s inequality when u is a measure. L inégalité de Kato lorsque u est une mesure
Kato s inequality when u is a measure L inégalité de Kato lorsque u est une mesure Haïm Brezis a,b, Augusto C. Ponce a,b, a Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4
More informationHESSIAN MEASURES III. Centre for Mathematics and Its Applications Australian National University Canberra, ACT 0200 Australia
HESSIAN MEASURES III Neil S. Trudinger Xu-Jia Wang Centre for Mathematics and Its Applications Australian National University Canberra, ACT 0200 Australia 1 HESSIAN MEASURES III Neil S. Trudinger Xu-Jia
More informationOn the L -regularity of solutions of nonlinear elliptic equations in Orlicz spaces
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 8. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More information