REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents
|
|
- Ambrose Underwood
- 5 years ago
- Views:
Transcription
1 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 of solutions to the equation p u = f(x), under suitable assumptions on the source term. As an application we deduce some strong comparison principles for the p-laplacian, in the case of vanishing source terms. Contents. Introduction 2. Local regularity 7 3. Local summability of the weight 2 4. The strong comparison principle 7 5. Examples 9 References 20. Introduction In this paper we study the regularity of weak solutions to (.) p u = f(x) in a domain R n, n 2. The issue of regularity, as we will see, is also related to the validity of the strong comparison principle. A solution u to (.) can be defined e.g. assuming that u W,p () in the weak distributional meaning. This is also the space where it is natural to prove the existence of the solutions under suitable assumptions. Nevertheless, under our assumptions on the source term f, it follows Date: November 23, Mathematics Subject Classification. 35B5,35J92,35B65. Key words and phrases. p-laplace equations, regularity theory, Harnack inequality, comparison principles.
2 2 BERARDINO SCIUNZI by [8, 29] that u C,α () for some 0 < α <, see also [4] for the regularity up to the boundary. On the contrary, solutions to p-laplace equations generally are not of class C 2 (). We recall in Section 5 two leading examples (Example 5. and Example 5.2) for the reader s convenience, that show the existence of solutions that are not in C 2 (). Nevertheless it is possible to show that, in the case p > 2 and strictly positive (or negative) source terms, any solution is not of class C 2 at its critical points. We add the details of this fact in Proposition 5.4. Therefore two crucial issues arises from this fact: - the study of integrability properties of the second derivatives of the solutions, - the study of the maximal exponent for the Hölder continuity of the gradients. One of the first regularity results regarding the second derivatives of the solutions states that, under suitable assumptions on f, we have: u p 2 u W,2 loc (). This result can be found in [5] for the general case f L s () with s > max{2, n }. Let us mention that this is also implicit in [8, 3, 29]. We p also refer the readers to [24] for W 2,2 estimates in the case of singular ( < p < 2) quasilinear elliptic equations involving singular potentials. In [5, 25] it has been shown that: if u C () is a solution to (.), then i) u W 2,2 loc ii) u W 2,q loc away from zero. () if < p < 3. p () for any q < p 2 if p 3 and if f is strictly bounded Actually in [5] positive solutions to p u = f(u) were considered. Anyway the arguments of [5] apply also in our context providing i) and ii) that are in any case a consequence of the results in this paper. Note that the degenerate nonlinear nature of the p-laplacian causes that the Calderón-Zygmund theory can not be extended simply to the case of p-laplace equations. We refer the readers to [7, 8] for a very interesting extension of the Calderón-Zygmund theory to the quasilinear case. It is worth emphasizing that, as it follows observing the -D solution in Example 5.2, the exponent q in ii) here above is optimal. On the contrary the radial solutions in Example 5. are more regular. This phenomenon (see Remark 5.3) highlights the important role played by the critical set Z u of the solution u: (.2) Z u := { u = 0}.
3 REGULARITY AND COMPARISON PRINCIPLES 3 The critical set Z u is in fact the set of points where the p-laplace operator is singular ( < p < 2) or degenerate (p > 2). Away from the critical set standard regularity theory applies. It is an open problem to understand if the positivity assumption on the source term in ii) can be removed. We will show here that in general it is not necessary. One of the purposes of this paper is to point out the strong relation between the study of the integrability properties of the second derivatives of the solutions and the study of the maximal exponent for the Hölder continuity of the gradient, starting from a recent important result of Eduardo V. Teixeira [27] which is based on previous estimates obtained in []. Namely, exploiting the results in [] and also some techniques from [], it has been shown in [27] that, if u W,p () is a solution to (.) and (.3) f L s () with s > n, then (.4) u C,min{α M, s n (p )s }, that is u C,α s n for any α (0, α M ) (0, ] where α (p )s M is the maximal exponent for the C,α regularity of p-harmonic functions. We also refer the interested readers to [9,, 9, 20, 28]. Let us point out that (.3) is in particular implied by stronger assumptions that we will consider in our results. It is in any case important to state explicitly (.3) since the parameter s will appear in our statements and in some cases it could be different by the one obtained by embedding theorems. Let us also mention that in [27] more general problems are considered including in particular operators with variable coefficients. Improving the technique in [5] and exploiting (.4), we get some weighted estimates that in particular show that the positivity of f is not necessary to get ii). Let us set (.5) 0 < µ := min{α M, s n }(p ), (p )s with s > n as in (.3). Also set (.6) 0 < µ := min{α M (p ), }, that is the limiting case for s. We have the following: Theorem.. Let u W,p () be a solution to (.). Assume that (.3) holds and f W,n (). Consider a critical point x 0 Z u with. Then: if < p < 3
4 4 BERARDINO SCIUNZI we have that (.7) B ρ(x 0 ) D 2 u 2 x x 0 η C with C = C(p, n, f, u, x 0 ) and for any η such that η < η := n ( 2 min{α M, s n (p )s } 2), with s > n given by (.3). In particular u W 2,2 loc (). If else p 3 and f satisfies (I µ ) (see (.2) and (.3)), then, for any q < p, we p 2 have: D 2 u q (.8) x x 0 τ C, for any B ρ(x 0 ) (.9) τ < τ := n 2 q 2 µ, with µ defined in (.5). Theorem. is a consequence of Corollary 2.2 and Corollary 3.2 that improves the results in [5]. In fact it is not required the strict positivity of f as in [5] and the exponents to the singular weights in (.7) and (.8) are optimal, as it can be deduced from the examples in Section 5. To guess this, the reader should evaluate the integral in (.8) in the case of the solution of Example 5.2. As an application it is interesting to state Theorem. for the case of Hénon-type equations: (.0) p u = x σ g(u), σ 0. We have the following: Theorem.2. Let u C () be a solution to (.0) with 0 and 0 σ < µ. Let x 0 Z u with. Then, if < p < 3 we have that (.7) holds for u for any η such that η < η := n ( 2 min{α M, (p ) } 2). In particular u W 2,2 p loc (). If else p 3 then, for any q <, we p 2 have that (.8) holds true for u for any τ such that: (.) τ < τ := n 2 q 2 µ, with µ defined in (.5).
5 REGULARITY AND COMPARISON PRINCIPLES 5 Theorem.2 follows by applying Theorem. with f(x) := x σ g(u(x)) L (). The regularity theory developed in Section 2 is very much related to study of the summability properties of u. This is a consequence of regularity estimates at first and later it allows to improve the regularity results, as it is the case of Corollary 3.2. The summability of u near the critical set Z u is in some sense a measure of the degeneracy of our equation at its critical points. This is an information that allows to get weighted Sobolev inequalities and that consequently turns out to be crucial in many applications. This is the case in particular when dealing with the strong comparison principle for p-laplacian. Consider a differential operator L : D R (to be understood in the weak sense if necessary), defined on its domain D = D() (a function space over some domain ). We say that the strong comparison principle holds for L in if, for any (connected) subdomain and any u, v D such that Lu Lv, then the assumption u v in implies the alternative u < v or u v in. The semilinear case is well understood and we refer the readers to [0]. Note that, for example, in the case Lu := u g(x, u) with g locally Lipschitz continuous w.r.t. the second variable, the strong comparison principle reduces to the maximum principle. It is crucial here the fact that the Laplace operator is linear. Nevertheless, also in the quasilinear case, the situation is well understood far away from the critical set Z u. Namely, in the case Lu := p u g(x, u) and p > with g locally Lipschitz continuous w.r.t. the second variable, the strong comparison principle holds in any connected component of \ Z u (see [3, 23]). Near the critical set, the nonlinear nature of the p-laplacian is in addition to the degeneracy of the operator and very few is known, even in the p-harmonic case Lu := p u and assuming furthermore that both u and v are p-harmonic. Let us mention [2] for some results in the p-harmonic case in dimension two. In the general case Lu := p u g(x, u) it has been proved in [6] that the strong comparison principle holds (also over the critical set) provided that u or v are a solution of the equation, 2n2 < p < and the n2 right hand side g(x, u) is strictly positive.
6 6 BERARDINO SCIUNZI It is a common feeling that, at least in the case p > 2, the strong comparison principle could fail if the source term change sign. However, as we will see here, the strict positivity of the source term is not necessary. Namely we will provide some conditions on the (possibly vanishing) source term under which the strong comparison principle holds. We will in particular assume that f satisfies the condition (I µ ) here below. Namely, for µ defined as in (.5) we state: Condition (I µ ): for some 0 < µ < µ, we have: (.2) f W,m n () for some m > 2( µ) and, given any x 0, there exist C(x 0, µ) > 0 and ρ(x 0, µ) > 0 such that, B ρ(x0,µ)(x 0 ) and (.3) f(x) C(x 0, µ) x x 0 µ in B ρ(x0,µ)(x 0 ). We have the following: Theorem.3. Let u, v C () such that (.4) p u f(x) p v f(x) in. Assume that u or v is a solution to (.) with 2n2 n2 < p < and assume that f satisfies (.3) and (I µ ). Then, if u v in a (connected) subdomain, it follows that u < v in, unless u v in. Let us point out that, even if f may be deleted in (.4), we need in any case that one of the functions we are comparing is a solution to the equation. As an example, we state here below a strong comparison principle for Hénon type problems. We have the following: Theorem.4. Let u, v C () such that (.5) p u x σ g(u) p v x σ g(v) in, with 0, 0 σ < µ (see (.6)) and g( ) nonnegative locally Lipschitz continuous with g(s) > 0 for s > 0. Assume that u or v is a (nontrivial) non-negative solution to (.0) with 2n2 < p <. n2 Then, if u v in a (connected) subdomain, it follows that u < v in, unless u v in. Theorem.3 and Theorem.4 follow by a Harnack-type comparison inequality, see Theorem 4.2, that can be proved exploiting the iterative technique in [30] which goes back to [7, 2, 22] and was first used to prove Hölder continuity properties of solutions of some strictly elliptic linear operators. To apply this technique in our case the key tool is
7 REGULARITY AND COMPARISON PRINCIPLES 7 the weighted Sobolev inequality in Theorem 4. that can be obtained once the regularity results of Theorem. are available. The paper is organized as follows. In Section 2 we prove some weighted regularity estimates for the solutions to (.), that we exploit in Section 3 to prove summability properties of u avoiding the assumption that f is strictly bounded away from zero needed in [5]. In Section 4 we prove Theorem.3 and Theorem.4. Finally in Section 5 we collect some examples showing which is the best regularity expected for the solutions to (.). 2. Local regularity Let u W,p () be a solutions to (.), that is (2.6) u p 2 ( u, φ) = f(x)φ for every φ W,p 0 (). Note that actually u C,α () under our assumptions. We will get our regularity results exploiting the linearized operator at u. To define it let us start recalling some known facts about weighted Sobolev spaces. It is natural to distinguish the case < p < 2 from the case p 2. The case p 2. The weighted Sobolev space (with weight ρ) W,2 (, ρ) can be defined as the set of those functions having distributional derivative for which the norm: (2.7) v ρ = v L 2 () v L 2 (,ρ) is bounded. We are interested in particular to the case ρ = u p 2 which is the weight that is naturally associated to our problem. Furthermore we can also define the weighted Sobolev space H,2 (, ρ) as the closure of C () in W,2 (, ρ) w.r.t. the norm in (2.7). Note that, since p 2, then ρ L () so that C () W,2 (, ρ) and furthermore it follows that H,2 (, ρ) W,2 (, ρ). A posteriori, according to Theorem 3. (see also Theorem 4.), we have that ρ L (). This allows to exploit the result of Meyers and Serrin [6], as remarked in [30] and to deduce that in any domain actually H,2 (, ρ) = W,2 (, ρ). Note now that H,2 0 (, ρ) may be defined as the closure of Cc () in W,2 (, ρ) w.r.t. the norm in (2.7). H,2 0 (, ρ) also coincides with the completion of Cc () w.r.t. the norm in (2.7). This will be a consequence of the regularity results (namely Theorem 3.) that we are going to prove. To do this, at the beginning, we have to choose a space where to define the linearized operator. We define (in the case p 2) the linearized operator in the space H,2 (, ρ) and,
8 8 BERARDINO SCIUNZI for any v H,2 (, ρ), we consider the linearized equation L u (v, ) = 0 associated to (.) at a given solution u defined by (2.8) L u (v, ϕ) = u p 2 ( v, ϕ) (p 2) u p 4 ( u, v)( u, ϕ) for every ϕ H,2 0 (, ρ), where we have posed: f i (x) = f x i (x). It follows by [5] that, setting u i (x) = u x i (x), then u i H,2 (, ρ) and (2.9) L u (u i, ϕ) = 0 ϕ H,2 0 (, ρ). The case < p < 2. In this case, a posteriori, according to Theorem 3. (see also Theorem 4.), the above construction can be carried out as well, since we will prove that ρ L (). At the beginning, to achieve Theorem 3., we start defining the linearized operator in the space A u := { v H 0() : v L 2 (,ρ 2 ) < }. It follows by [5] that u i A u and (2.20) L u (u i, ϕ) = 0 ϕ H,2 0 (). We have the following: Theorem 2.. Let u W,p () be a solution to (.), < p <. Assume that (.3) holds and let x 0 Z u with. Fix 0 β <, 0 µ < µ and γ < n 2 for n 3 while γ = 0 if n = 2. Then, if f W,m () for some m > (2.2) B ρ(x 0 ) n 2 µ, it follows that: u p 2 β u i,j 2 C i, j {,..., n}, x x 0 µ x y γ with C = C(x 0, ρ, f, n, p, β, γ, µ) that does not depend on y. Proof. We start recalling that, since we assumed that (.3) is satisfied, then the results in [27] apply and (.4) holds. Let us fix some notations. We consider G ε (t) = (2t 2ε)χ [ε, 2ε] (t) tχ [2ε, ) (t) for t 0, while G ε (t) = G ε ( t) for t 0 (χ [a,b] ( ) denoting the characteristic function of a set). Consider a cut-off function φ ρ, in such a way that φ ρ Cc (), φ ρ = in B ρ (x 0 ) and φ ρ 2. For shortness we will write φ instead ρ of φ ρ. Also, for 0 β <, γ < (n 2) if n 3, γ = 0 if n = 2 and µ < µ fixed, we set (2.22) T ε (t) = G ε(t) t β and H δ (t) = G δ(t) t γ, f i ϕ
9 REGULARITY AND COMPARISON PRINCIPLES 9 and consider the test function ϕ = T ε (u i ) Hδ( x y ) x x 0 µ φ 2. Note that ϕ can be plugged into (2.9) (see also (2.20)) since it is regularized near the critical set Z u by the definition of T ε (note also that x 0 Z u ). Moreover ϕ is regularized near any y because of the definition of H δ. Finally ϕ is smooth elsewhere by standard regularity theory. Therefore we have (2.23) u p 2 u i 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ (p 2) u p 4 ( u, u i ) 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ u p 2 φ 2 ( u i, x H δ ( x y )) T ε (u i ) x x 0 µ (p 2) u p 4 φ 2 ( u, u i )( u, x H δ ( x y )) T ε (u i ) x x 0 µ µ u p 2 ( u i, x x 0 ) T ε (u i ) Hδ( x y ) φ2 x x 0 µ µ(p 2) u p 4 ( u, u i )( u, x x 0 ) T ε (u i ) Hδ( x y ) φ2 x x 0 µ 2 u p 2 ( u i, φ) T ε (u i ) Hδ( x y ) φ x x 0 µ 2(p 2) u p 4 ( u, u i )( u, φ) T ε (u i ) Hδ( x y ) φ x x 0 µ = f i (x) T ε (u i ) Hδ( x y ) φ 2. x x 0 µ The main idea in the following computations is to estimate all the terms in the previous inequality and then close the estimates via Hölder s inequality. We start observing that min{(p ), } u p 2 u i 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2 u p 2 u i 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2 (p 2) u p 4 ( u, u i ) 2 T ε(u i ) Hδ( x y ) x x 0 µ φ 2
10 0 BERARDINO SCIUNZI and get by (2.23) that C u p 2 u i 2 T ε(u i ) Hδ( x y ) φ 2 x x 0 µ u p 2 φ 2 u i x H δ ( x y ) T ε (u i ) x x 0 µ u p 2 u i T ε (u i ) Hδ( x y ) (2.24) φ2 x x 0 µ u p 2 u i φ T ε (u i ) Hδ( x y ) φ x x 0 µ f i (x) T ε (u i ) Hδ( x y ) φ 2. x x 0 µ Here and in the following we denote with C = C(x 0, ρ, f, n, p, β, γ, µ) a generic constant that we allow to vary each line. For ε > 0 fixed, we exploit the dominated convergence theorem to let δ 0 and get C u p 2 u i 2 T ε(u i ) φ 2 x x 0 µ x y γ (2.25) u p 2 u i T ε (u i ) x x 0 µ φ2 x y γ u p 2 u i T ε (u i ) φ 2 x x 0 µ x y γ u p 2 u i T ε (u i ) φ φ x x 0 µ x y γ f i (x) T ε (u i ) x x 0 µ x y γ φ2. We use Young s inequality ab ϑa 2 b2 and the fact that T 4ϑ ε(t) t β and get: u p 2 u i T ε (u i ) x x 0 µ φ2 x y γ u p 2 2 u i χ { ui ε} φ x y γ 2 x x 0 µ 2 u i β 2 u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C ( u u(x 0 ) 4ϑ x y γ2 x x 0 µ p β u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C 4ϑ, u p 2 2 u i 2 β 2 x y γ2 2 x x 0 µ 2 ) p β φ 2 φ
11 REGULARITY AND COMPARISON PRINCIPLES since we assumed that γ < n 2 and µ < min{α M, s n }(p ). (p )s Exploiting again Young s inequality we get: u p 2 u i T ε (u i ) x x 0 µ x y γ φ2 u p 2 2 u i χ { ui ε} φ x y γ 2 x x 0 µ 2 u i β 2 u p 2 u i 2 χ { ui ε} ϑ x y γ x x 0 µ u i β φ2 C 4ϑ ϑ C ϑ ϑ u u(x 0 ) p β φ 2 x y γ x x 0 µ2 u p 2 u i 2 χ { ui ε} x y γ x x 0 µ u i β φ2 x y γ x x 0 2 φ2 u p 2 u i 2 χ { ui ε} x y γ x x 0 µ u i β φ2 C 4ϑ. u p 2 2 u i 2 β 2 x y γ 2 x x 0 µ2 2 since we assumed that γ < n 2 and µ < min{α M, s n }(p ). We (p )s now use the fact that φ and get: ρ u p 2 u i T ε (u i ) φ φ x x 0 µ x y γ = 2 u p 2 2 u i χ { ui ε} u p 2 2 u i 2 β 2 φ ρ x x 0 µ 2 x y γ 2 u i β 2 x x 0 µ 2 x y γ 2 u p 2 u i 2 χ { ui ε} ϑ x x 0 µ x y γ u i β φ2 C ( u u(x0 ) ) p β ϑ x y γ x x 0 µ p β u p 2 u i 2 χ { ui ε} ϑ φ 2 C x y γ u i β ϑ, since we assumed that γ < n 2 and µ < min{α M, s n }(p ). (p )s Finally we have: f i (x) T ε (u i ) x x 0 µ x y γ φ2 ( ) ( C f m m ( ) m ) m C x x 0 µ x y γ φ
12 2 BERARDINO SCIUNZI by assump- since µ <, γ < n 2 and f W,m () for some m > tion. n 2 µ Taking into account (2.25), exploiting the above estimates and evaluating T ε, we get u p 2 u i 2 ( G u i β x x 0 µ x y γ ε (u i ) β G ε(u i ) Cϑχ { ui ε}) φ 2 C. u i Since for for ϑ small (G ε(u i ) β Gε(u i) u i Cϑχ { ui ε}) is strictly positive we conclude that u p 2 β u i 2 x x 0 µ x y γ φ2 and the thesis follows by the definition of φ. u p 2 u i 2 u i β x x 0 µ x y γ φ2 C, Corollary 2.2. Let u W,p () be a solution to (.). Assume that (.3) holds and let < p < 3. Assume that f W,m () for some m n 2 µ and consider a critical point x 0 Z u with. Then we have D 2 u 2 (2.26) x x 0 C, η B ρ (x 0 ) with C = C(p, n, f, u, x 0 ) and for any η such that η < η := n ( 2 min{α M, s n (p )s } 2). Since µ, the result holds in particular if f W,n (). As a consequence we also get that u W 2,2 loc (). Proof. The proof follows by directly by Theorem 2. and (.4) since we have that p 3 < 0 by assumption. 3. Local summability of the weight We prove in this section a summability property of u, extending the results of [5] to the case of vanishing source terms. We will assume that f satisfies the condition (I µ ) and prove the following: Theorem 3.. Let < p < and let u W,p () be a solution of (.). Assume that f satisfies (.3) and assume that (I µ ) holds. Then, for any x 0 Z u and for some ρ = ρ(x 0 ) > 0, we have (3.27) C. u (p ) x y (n 2) B ρ (x 0 )
13 REGULARITY AND COMPARISON PRINCIPLES 3 Namely B ρ (x 0 ) u t x y C, γ with 0 t < p, γ < n 2 if n 3 and γ = 0 if n = 2 and C = C(t, γ, p, f, ρ, x 0 ) not depending on y. Proof. Consider ϕ = u t ε x x 0 µ δ Gδ(f) H δ ( x y ) φ 2 f with ε, δ > 0, H δ and G δ defined according to (2.22), t < p and 0 < µ < µ with µ defined in (.5) such that (.2) and (.3) hold, by (I µ ). Here, as above, we assume that the ball is contained in, and we consider a cut-off function φ = φ ρ, in such a way that φ ρ Cc (), φ ρ = in B ρ (x 0 ) and φ ρ 2. In particular ρ we may and do assume that (I µ ) holds in. We will write φ instead of φ ρ. We use ϕ as test function in (2.6) and get (3.28) = 2 f(x)ϕ = u p 2 ( u, ϕ) = t u t u p 2 ( u, u ) Gδ(f) H ( u t ε) 2 x x 0 µ δ φ 2 δ f µ x x 0 µ u p 2 ( u, x x 0 ) Gδ(f) H ( x x 0 µ δ) 2 u t δ φ 2 ε f u p 2 ( u, x H δ ( x y )) u t ε u p 2 ( u, φ) u t ε ( Gδ (t) t x x 0 µ δ Gδ(f) f x x 0 µ δ Gδ(f) H δ φ f ) χ {δ f 2δ} t=f u t ε u p 2 ( u, f) H x x 0 µ δ φ 2, δ φ 2
14 4 BERARDINO SCIUNZI that gives C f ϕ N u p u t u i ( u t ε) H δ ( x y ) 2 x x 0 µ δ G δ(f) φ 2 f i= u p x x 0 µ u t ε G δ(f) H δ ( x y ) φ 2 f u p H δ ( x y ) u t ε u p φ u t ε u p f u t ε x x 0 µ δ G δ(f) φ 2 f x x 0 µ δ G δ(f) H δ ( x y )φ f x x 0 µ δ Hδ( x y ) φ 2, f where we have estimated the last term exploiting the fact that ( G δ(t) ) 4 t=f for δ f 2δ. t f Here and in the following we denote with C = C(x 0, ρ, f, n, p, t, γ, µ) a generic constant that we allow to vary each line. We now let δ 0 and, by the dominated convergence theorem and (.2), (.3) (see (I µ )), we get: (3.29) C u t ε x y γ φ2 f u t ε x x 0 µ N u p u t u i ( u t ε) 2 i= u p x x 0 µ u t ε x y γ φ2 x x 0 µ x y γ φ2 u p u t ε x x 0 µ φ2 x y γ u p φ u t ε u p f u t ε x x 0 µ x y φ γ x x 0 2µ x y γ φ2. x y γ φ2
15 REGULARITY AND COMPARISON PRINCIPLES 5 The reader can easy check that, since µ < and γ < n 2 and ε > 0, then the dominated convergence theorem applies. Let us now estimate the terms on the righthand side of (3.29). We have: C ϑ N u p u t u i ( u t ε) 2 i= x x 0 µ x y γ φ2 φ 2 ( u t ε) 2 x y γ 2 x x 0 u pt 2 D 2 u µ ( u t ε) 3 2 x y γ 2 ( u t ε) x y γ C ϑ ϑ ( u t ε) x y γ φ 2 x x 0 µ u (p 2) (2t p) D2 u 2 φ 2 x y γ x x 0 µ φ 2 x x 0 µ C ϑ, exploiting Theorem 2. (note that (.2) is enough) with β = 2 t p. Note that the assumption t < p implies β <. The assumption t < p also yields C C u p x x 0 µ u t ε x y γ φ2 x x 0 µ x y γ φ2 since µ < (see (.5)) and γ < n 2. Moreover u p u t ε x x 0 µ φ2 x y γ C C, x x 0 µ φ2 x y γ since µ < and γ < n 2. Also we have: u p φ u t ε C ρ C x x 0 µ φ φ x x 0 µ x y γ x x 0 µ φ x y γ C, x y γ φ
16 6 BERARDINO SCIUNZI since t < p, µ <, γ < n 2. Note that here the constant that we get depends on ρ as stated in the theorem. We also have u p f u t ε ( C f ) f m x x 0 2µ x y γ φ2 x x 0 2µ x y γ φ2 m ( ( ) m ) m C, x x 0 2µ x y γ exploiting that f W,m () for some m > n (see (.2)) and the 2( µ) fact that t < p and µ < and γ < n 2 (γ = 0 is n = 2). Collecting the above estimates, by (3.29), we get ( Cϑ) ( u t ε) x y (3.30) γ x x 0 µ φ2 C. We now take ϑ small, say ϑ small such that in (3.30) we have ( Cϑ) 2, and get ( u t ε) x y γ x x 0 µ φ2 C, that gives the thesis by letting ε 0 and exploiting Fatou s Lemma. Corollary 3.2. Let p 3 and let u W,p () be a solution of (.). Assume that (.3) and (I µ ) hold and assume that f W,m () for some m > n 2 µ. Then, for x 0 Z u (and for some ρ = ρ(x 0 ) > 0) and for any q < p p 2, we have: D 2 u q (3.3) x x 0 τ C, for any B ρ (x 0 ) (3.32) τ < τ := n 2 q 2 µ, with µ defined in (.5). The result holds in particular if f W,n (). Proof. We can rewrite any τ < τ as: τ := γ q 2 (µ ε)
17 REGULARITY AND COMPARISON PRINCIPLES 7 for some γ < n 2 if n 3, γ = 0 if n = 2 and ε > 0 (small). Then, for any 0 β < and < q < 2, we have D 2 u q x x 0 τ dx B ρ(x 0 ) ( C, B ρ(x 0 ) B ρ(x 0 ) u (p 2 β) q 2 D 2 u q x x 0 q 2 (γµ ε) u (p 2 β) D 2 u 2 x x 0 γµ ε dx u (p 2 β) q 2 x x 0 γ 2 q 2 ) q 2 ( Bρ(x0) dx ) 2 q u (p 2 β) q 2 q x x0 γ dx 2 where we used Holder inequality with exponents 2 and 2, Theorem q 2 q 2. with y = x 0 (since we assumed that f W,m () for some m > n 2 µ ) and Theorem 3.. Note that, to apply Theorem 3., we can choose 0 < β < such that (p 2 β) q 2 q assumption q < p p 2. < p because of the 4. The strong comparison principle The summability properties of u obtained in Theorem 3. allows to prove a weighted Sobolev inequality that we recall here below: Theorem 4. ([5]). Let p > 2 and u C () be a solution of (.). Assume that f satisfies (.3) and (I µ ) (see (.2) and (.3)). Setting ρ = u p 2, for any, we have (4.33) with t = p p 2 and y. r, p 2 p ρ t x y C, γ < r <, γ < n 2 if n 3 while γ = 0 if n = 2, Assuming in the case n 3 with no loose of generality that γ > n 2t, then it follows that the space H,2 0 (, ρ) is continuously embedded in L q, for q < 2 (p), where 2 (p) = 2 n p 2 p n. More precisely there exists a constant C S such that (4.34) w L q ( ) C S w L 2 (,ρ) = C S ( w 2 ρ for any w H,2 0 (, ρ) and q < 2 (p). Moreover the embedding is compact. ) 2,
18 8 BERARDINO SCIUNZI Proof. Once Theorem 3. is proved then (4.33) follows by a simple covering argument. We can therefore exploit (4.33) to repeat verbatim the proof of Theorem 3. in [5]. Finally the embedding is compact by [2, Lemma 5.]. As a consequence we have Theorem 4.2. Let u, v C () such that p u f(x) p v f(x) in. Assume that u or v is a solution to (.) with 2n2 n2 < p <, and assume that f satisfies (.3) and (I µ ) (see (.2) and (.3)). Then, for x 0 Z u fixed, if we assume that u v in B 5ρ (x 0 ), it follows that (4.35) sup (v u) C H inf (v u) B ρ (x 0 ) for some constant C H = C H (u, v, f, n, p, ρ, x 0 ). Proof. The proof follows by the Moser-type iteration scheme, as developed in [30]. The technique in [30] actually is mainly based on Sobolev embedding and works in our case thanks to Theorem 4.. The details can be found in [6, Theorem 3.3]. We are now ready to prove Theorem.3 and Theorem.4. Proof of Theorem.3: The proof is a consequence of the Harnack inequality in Theorem 4.2. Let us set C u,v := { x : u(x) = v(x) }. Since u and v are continuous, it follows that C u,v is closed. Let us consider now x 0 C u,v and ρ = ρ(x 0 ) > 0 such that B 5ρ (x 0 ). Note that actually u(x 0 ) = v(x 0 ) since u touches v from below at x 0. If u(x 0 ) 0 then, eventually reducing ρ, we have that u = v in B ρ (x 0 ) by the classical strong comparison principle [3] (see also [0]). If else u(x 0 ) = 0 = v(x 0 ), eventually reducing ρ, we exploit Theorem 4.2 to prove that u = v in B ρ (x 0 ). Therefore C u,v is also open and the thesis follows recalling that is connected. Proof of Theorem.4: Let us consider the case when u is a nonnegative solution to the equation (.0) and set f(x) := x σ g(u(x)). Since p u 0, by the strong maximum principle [32] it follows that u is strictly positive, since we assumed that it is not trivial. Therefore,
19 REGULARITY AND COMPARISON PRINCIPLES 9 g(u(0)) > 0 since g(s) > 0 for s > 0 and, by the assumption 0 σ < µ, it follows that f satisfies the condition (I µ ) (it is easy to check the validity of (.2)). Therefore Theorem 3. holds and Theorem 4. follows as well. This allows to repeat verbatim the proof of [6, Theorem 3.3] and deduce also in this case the validity of the Harnack inequality, namely Theorem 4.2. The proof now can be finished arguing exactly as in the proof of Theorem Examples Example 5.. Let < p < and consider u C,α (B) weak solution to p u = u q in B (5.36) u > 0 in B u = 0 on B, where B is the unit ball in R n centered at zero. Then, by [4, 5], it follows that u is radial and radially decreasing with Z u := { u = 0} {0}. Therefore, arguing as in [26], by l Hopital s rule we have u x p and D 2 u x 2 p p. Example 5.2. Let p >. Then the function solves and u(x,..., x N ) = x p p u =, Z u = {x = 0}. Remark 5.3. It is easy to see that, for p 3, the solution given in Example 5.2 is no more regular than what we obtained in Corollary (3.2). On the contrary, the solutions in Example 5. are more regular (just use polar coordinates to see this). We might guess that the geometry of the critical set Z u plays a role. Proposition 5.4. Let u C,α () be a weak solution of (.) in R n. Suppose that x 0 Z u, namely u(x 0 ) = 0, and suppose that: for some ρ, ϑ > 0 we have that f ϑ in B ρ (x 0 ). Then (in a neighborhood of x 0 ) we have α p. In particular, for p > 2, we have that u / C 2 (). p
20 20 BERARDINO SCIUNZI Proof. Let us consider the balls B ρ (x 0 ) and centered in x 0 with 2ρ ρ. Consider a cut-off function φ ρ such that, φ ρ 0 in, φ ρ Cc () and { φρ in B ρ (5.37) n ρ (x 0 ) φ ρ c in B ρ n 2ρ (x 0 ) \ B ρ (x 0 ). Using φ ρ as test-function in (.) we get (5.38) u p 2 ( u, φ ρ ) dx = We have that: u p 2 ( u, φ ρ ) dx so that (5.39) On the other hand f φ ρ dx. u u(x 0 ) p φ ρ dx c ρα(p ) ρ n ρ n c ρ α(p ) u p 2 ( u, φ ρ ) dx ρ 0 0 if (α(p ) ) > 0. (5.40) f φ ρ dx ϑ φ ρ dx ϑ B > 0 for ρ sufficiently small. Therefore, for (α(p ) ) > 0, we get a contradiction by (5.38), (5.39) and (5.40). As a consequence we get that near x 0 necessarily (α(p ) ) 0 and the thesis. Note now that, if p > 2, then < so that u / p C2 (). References [] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations. Ann. of Math., 30 (), 989, pp [2] D. Castorina, P. Esposito and B. Sciunzi, Degenerate elliptic equations with singular nonlinearities. Calc. Var. and PDE, 34 (3), 2009, pp [3] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5(4), 998, pp [4] L. Damascelli, F. Pacella, Monotonicity and symmetry of solutions of p- Laplace equations, < p < 2, via the moving plane method. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(4), 998, pp [5] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-laplace equations. J. Differential Equations, 206(2), 2004, pp [6] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations. Calc. Var. Partial Differential Equations, 25 (2), 2006, pp
21 REGULARITY AND COMPARISON PRINCIPLES 2 [7] E. De Giorgi, Sulla differenziabilità e l analicità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino, serie 3a, 3, 957, pp [8] E. Di Benedetto, C α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 983, pp [9] F. Duzar, G. Mingione Gradient estimates via nonlinear potentials. Am. J. Math., 33, 20, pp [0] D. Gilbarg, N. and Trudinger, S, Elliptic partial differential equations of second order. Reprint of the 998 Edition, Springer. [] T. Kuusi, G. Mingione, Universal potential estimates. J. Funct. Anal., 262, 202, pp [2] T. Ivaniec and J. Manfredi, Regularity of p-harmonic functions on the plane. Revista Matematica Iberoamericana, 5(-2), 989, pp. 9. [3] O. A. Ladyzhenskaya, N. N. Ural tseva, Linear and quasilinear elliptic equations. Academic Press, 968. [4] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 2(), 988, pp [5] H. Lou, On singular sets of local solutions to p-laplace equations, Chinese Annals of Mathematics, Series B29. 5, 2008, pp [6] N.G. Meyers, G. and J. Serrin, H = W. Proc. Nat. Acad. Sci. U.S.A., 5, 964, pp Papers in memory of Ennio De Giorgi. [7] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math., 5 (4), 2006, pp [8] G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci., (5)., 6(2), 2007, pp [9] G. Mingione, Gradient estimates below the duality exponent. Math. Ann., 346, 200, pp [20] G. Mingione, Gradient potential estimates. J. Eur. Math. Soc., 3, 20, pp [2] J.K. Moser, On Harnack s theorem for elliptic differential equations. Comm. on Pure and Applied Math., 4, 96, pp [22] J. Nash, Continuity of solutions of elliptic and parabolic equations. Amer. J. Math., (80) 958, pp [23] P. Pucci, J. Serrin, The maximum principle. Birkhauser, Boston (2007). [24] P. Pucci and R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations. Indiana Univ. Math. J., 57(7), 2008, pp [25] B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations. NoDEA Nonlinear Differential Equations Appl., 4(3-4) (2007), [26] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math. J., 49(3), 2000, pp [27] E. V. Teixeira, Regularity for quasilinear equations on degenerate singular sets. Math. Ann. DOI: 0.007/s [28] E. V. Teixeira, Sharp regularity for general Poisson equations with borderline sources. J. Math. Pures Appl., (9). 99(2), 203, pp [29] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations. 5(), 984, pp [30] N.S. Trudinger, Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3), 27, 973, pp [3] N.S. Trudinger, On the regularity of generalized solutions of linear nonuniformly elliptic equations, Arch. Rational Mech. Anal., 42, 97, pp
22 22 BERARDINO SCIUNZI [32] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 2(3), 984, pp Dipartimento di Matematica, UNICAL,, Ponte Pietro Bucci 3B,, Arcavacata di Rende, Cosenza, Italy address: sciunzi@mat.unical.it
SYMMETRY 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 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 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 informationNon-homogeneous semilinear elliptic equations involving critical Sobolev exponent
Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Yūki Naito a and Tokushi Sato b a Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan b Mathematical
More informationContinuity of Solutions of Linear, Degenerate Elliptic Equations
Continuity of Solutions of Linear, Degenerate Elliptic Equations Jani Onninen Xiao Zhong Abstract We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the
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 informationSome aspects of vanishing properties of solutions to nonlinear elliptic equations
RIMS Kôkyûroku, 2014, pp. 1 9 Some aspects of vanishing properties of solutions to nonlinear elliptic equations By Seppo Granlund and Niko Marola Abstract We discuss some aspects of vanishing properties
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 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 informationVANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N
VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on
More informationA LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j
Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp. 1 7. ISSN 107-6691. URL: http://ejde.math.swt.edu (147.6.103.110) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu
More informationSymmetry of entire solutions for a class of semilinear elliptic equations
Symmetry of entire solutions for a class of semilinear elliptic equations Ovidiu Savin Abstract. We discuss a conjecture of De Giorgi concerning the one dimensional symmetry of bounded, monotone in one
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 informationu( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)
M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation
More informationEXISTENCE 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 informationAN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS. To the memory of our friend and colleague Fuensanta Andreu
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value
More informationHARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of
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 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 informationSymmetry of nonnegative solutions of elliptic equations via a result of Serrin
Symmetry of nonnegative solutions of elliptic equations via a result of Serrin P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract. We consider the Dirichlet problem
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 informationNONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationOPTIMAL REGULARITY FOR A TWO-PHASE FREE BOUNDARY PROBLEM RULED BY THE INFINITY LAPLACIAN DAMIÃO J. ARAÚJO, EDUARDO V. TEIXEIRA AND JOSÉ MIGUEL URBANO
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 18 55 OPTIMAL REGULARITY FOR A TWO-PHASE FREE BOUNDARY PROBLEM RULED BY THE INFINITY LAPLACIAN DAMIÃO J. ARAÚJO, EDUARDO
More informationRegularity estimates for fully non linear elliptic equations which are asymptotically convex
Regularity estimates for fully non linear elliptic equations which are asymptotically convex Luis Silvestre and Eduardo V. Teixeira Abstract In this paper we deliver improved C 1,α regularity estimates
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
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 informationA RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS. Zhongwei Shen
A RELATIONSHIP BETWEEN THE DIRICHLET AND REGULARITY PROBLEMS FOR ELLIPTIC EQUATIONS Zhongwei Shen Abstract. Let L = diva be a real, symmetric second order elliptic operator with bounded measurable coefficients.
More informationTHE HARNACK INEQUALITY FOR -HARMONIC FUNCTIONS. Peter Lindqvist and Juan J. Manfredi
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 04, pp. 1-5. Published April 3, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
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 informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationSYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS
Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT
More informationRegularity of the p-poisson equation in the plane
Regularity of the p-poisson equation in the plane Erik Lindgren Peter Lindqvist Department of Mathematical Sciences Norwegian University of Science and Technology NO-7491 Trondheim, Norway Abstract We
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 informationAN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value
More informationON A CONJECTURE OF P. PUCCI AND J. SERRIN
ON A CONJECTURE OF P. PUCCI AND J. SERRIN Hans-Christoph Grunau Received: AMS-Classification 1991): 35J65, 35J40 We are interested in the critical behaviour of certain dimensions in the semilinear polyharmonic
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 informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationarxiv: v1 [math.ap] 7 Dec 2015
Cavity problems in discontinuous media arxiv:52.02002v [math.ap] 7 Dec 205 by Disson dos Prazeres and Eduardo V. Teixeira Abstract We study cavitation type equations, div(a i j (X) u) δ 0 (u), for bounded,
More informationUniqueness of ground states for quasilinear elliptic equations in the exponential case
Uniqueness of ground states for quasilinear elliptic equations in the exponential case Patrizia Pucci & James Serrin We consider ground states of the quasilinear equation (.) div(a( Du )Du) + f(u) = 0
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 informationEverywhere differentiability of infinity harmonic functions
Everywhere differentiability of infinity harmonic functions Lawrence C. Evans and Charles K. Smart Department of Mathematics University of California, Berkeley Abstract We show that an infinity harmonic
More informationRecent developments in elliptic partial differential equations of Monge Ampère type
Recent developments in elliptic partial differential equations of Monge Ampère type Neil S. Trudinger Abstract. In conjunction with applications to optimal transportation and conformal geometry, there
More informationSymmetry and monotonicity of least energy solutions
Symmetry and monotonicity of least energy solutions Jaeyoung BYEO, Louis JEAJEA and Mihai MARIŞ Abstract We give a simple proof of the fact that for a large class of quasilinear elliptic equations and
More informationON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT
PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationis a weak solution with the a ij,b i,c2 C 1 ( )
Thus @u @x i PDE 69 is a weak solution with the RHS @f @x i L. Thus u W 3, loc (). Iterating further, and using a generalized Sobolev imbedding gives that u is smooth. Theorem 3.33 (Local smoothness).
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 informationThe De Giorgi-Nash-Moser Estimates
The De Giorgi-Nash-Moser Estimates We are going to discuss the the equation Lu D i (a ij (x)d j u) = 0 in B 4 R n. (1) The a ij, with i, j {1,..., n}, are functions on the ball B 4. Here and in the following
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationSymmetry and non-uniformly elliptic operators
Symmetry and non-uniformly elliptic operators Jean Dolbeault Ceremade, UMR no. 7534 CNRS, Université Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France E-mail: dolbeaul@ceremade.dauphine.fr
More informationA Product Property of Sobolev Spaces with Application to Elliptic Estimates
Rend. Sem. Mat. Univ. Padova Manoscritto in corso di stampa pervenuto il 23 luglio 2012 accettato l 1 ottobre 2012 A Product Property of Sobolev Spaces with Application to Elliptic Estimates by Henry C.
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationANNALES DE L I. H. P., SECTION C
ANNALES DE L I. H. P., SECTION C E. DI BENEDETTO NEIL S. TRUDINGER Harnack inequalities for quasi-minima of variational integrals Annales de l I. H. P., section C, tome 1, n o 4 (1984), p. 295-308
More informationBesov regularity of solutions of the p-laplace equation
Besov regularity of solutions of the p-laplace equation Benjamin Scharf Technische Universität München, Department of Mathematics, Applied Numerical Analysis benjamin.scharf@ma.tum.de joint work with Lars
More informationCOMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO
COMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO KEVIN R. PAYNE 1. Introduction Constant coefficient differential inequalities and inclusions, constraint
More informationHARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.
Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES
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 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 informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationHARNACK INEQUALITY FOR QUASILINEAR ELLIPTIC EQUATIONS WITH (p, q) GROWTH CONDITIONS AND ABSORPTION LOWER ORDER TERM
Electronic Journal of Differential Equations, Vol. 218 218), No. 91, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu HARNACK INEQUALITY FOR QUASILINEAR ELLIPTIC EQUATIONS
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 informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationI Results in Mathematics
Result.Math. 45 (2004) 293-298 1422-6383/04/040293-6 DOl 10.1007/s00025-004-0115-3 Birkhauser Verlag, Basel, 2004 I Results in Mathematics Further remarks on the non-degeneracy condition Philip Korman
More informationREGULARITY RESULTS FOR THE EQUATION u 11 u 22 = Introduction
REGULARITY RESULTS FOR THE EQUATION u 11 u 22 = 1 CONNOR MOONEY AND OVIDIU SAVIN Abstract. We study the equation u 11 u 22 = 1 in R 2. Our results include an interior C 2 estimate, classical solvability
More informationA NOTE ON ZERO SETS OF FRACTIONAL SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY. 1. Introduction
A NOTE ON ZERO SETS OF FRACTIONAL SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY ARMIN SCHIKORRA Abstract. We extend a Poincaré-type inequality for functions with large zero-sets by Jiang and Lin
More informationA Dirichlet problem in the strip
Electronic Journal of Differential Equations, Vol. 1996(1996), No. 10, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationThe Harnack inequality for second-order elliptic equations with divergence-free drifts
The Harnack inequality for second-order elliptic equations with divergence-free drifts Mihaela Ignatova Igor Kukavica Lenya Ryzhik Monday 9 th July, 2012 Abstract We consider an elliptic equation with
More informationHölder estimates for solutions of integro differential equations like the fractional laplace
Hölder estimates for solutions of integro differential equations like the fractional laplace Luis Silvestre September 4, 2005 bstract We provide a purely analytical proof of Hölder continuity for harmonic
More informationTOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity
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 informationarxiv: v3 [math.ap] 28 Feb 2017
ON VISCOSITY AND WEA SOLUTIONS FOR NON-HOMOGENEOUS P-LAPLACE EQUATIONS arxiv:1610.09216v3 [math.ap] 28 Feb 2017 Abstract. In this manuscript, we study the relation between viscosity and weak solutions
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 SOLUTIONS FOR BIHARMONIC ELLIPTIC PROBLEMS WITH THE SECOND HESSIAN
Electronic Journal of Differential Equations, Vol 2016 (2016), No 289, pp 1 16 ISSN: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu MULTIPLE SOLUTIONS FOR BIHARMONIC ELLIPTIC PROBLEMS
More informationExistence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities
Journal of Physical Science Application 5 (2015) 71-81 doi: 10.17265/2159-5348/2015.01.011 D DAVID PUBLISHING Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities
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 informationS chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.
Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable
More informationIntegro-differential equations: Regularity theory and Pohozaev identities
Integro-differential equations: Regularity theory and Pohozaev identities Xavier Ros Oton Departament Matemàtica Aplicada I, Universitat Politècnica de Catalunya PhD Thesis Advisor: Xavier Cabré Xavier
More informationTHE HOT SPOTS CONJECTURE FOR NEARLY CIRCULAR PLANAR CONVEX DOMAINS
THE HOT SPOTS CONJECTURE FOR NEARLY CIRCULAR PLANAR CONVEX DOMAINS YASUHITO MIYAMOTO Abstract. We prove the hot spots conjecture of J. Rauch in the case that the domain Ω is a planar convex domain satisfying
More informationA comparison theorem for nonsmooth nonlinear operators
A comparison theorem for nonsmooth nonlinear operators Vladimir Kozlov and Alexander Nazarov arxiv:1901.08631v1 [math.ap] 24 Jan 2019 Abstract We prove a comparison theorem for super- and sub-solutions
More informationCOINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 95, Number 3, November 1985 COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE p-harmonic OPERATOR SHIGERU SAKAGUCHI Abstract. We consider the obstacle
More informationBoundary value problems for the infinity Laplacian. regularity and geometric results
: regularity and geometric results based on joint works with Graziano Crasta, Roma La Sapienza Calculus of Variations and Its Applications - Lisboa, December 2015 on the occasion of Luísa Mascarenhas 65th
More informationTHE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)
Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev
More informationLiouville Theorems for Integral Systems Related to Fractional Lane-Emden Systems in R N +
Liouville Theorems for Integral Systems Related to Fractional Lane-Emden Systems in R Senping Luo & Wenming Zou Department of Mathematical Sciences, Tsinghua University, Beijing 00084, China Abstract In
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationON THE NATURAL GENERALIZATION OF THE NATURAL CONDITIONS OF LADYZHENSKAYA AND URAL TSEVA
PATIAL DIFFEENTIAL EQUATIONS BANACH CENTE PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WASZAWA 1992 ON THE NATUAL GENEALIZATION OF THE NATUAL CONDITIONS OF LADYZHENSKAYA
More informationWELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL
More informationRegularity of Weak Solution to Parabolic Fractional p-laplacian
Regularity of Weak Solution to Parabolic Fractional p-laplacian Lan Tang at BCAM Seminar July 18th, 2012 Table of contents 1 1. Introduction 1.1. Background 1.2. Some Classical Results for Local Case 2
More informationNonlinear Analysis 72 (2010) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage:
Nonlinear Analysis 72 (2010) 4298 4303 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Local C 1 ()-minimizers versus local W 1,p ()-minimizers
More informationMINIMAL SURFACES AND MINIMIZERS OF THE GINZBURG-LANDAU ENERGY
MINIMAL SURFACES AND MINIMIZERS OF THE GINZBURG-LANDAU ENERGY O. SAVIN 1. Introduction In this expository article we describe various properties in parallel for minimal surfaces and minimizers of the Ginzburg-Landau
More informationUniform estimates for Stokes equations in domains with small holes and applications in homogenization problems
Uniform estimates for Stokes equations in domains with small holes and applications in homogenization problems Yong Lu Abstract We consider the Dirichlet problem for the Stokes equations in a domain with
More informationHomogenization and error estimates of free boundary velocities in periodic media
Homogenization and error estimates of free boundary velocities in periodic media Inwon C. Kim October 7, 2011 Abstract In this note I describe a recent result ([14]-[15]) on homogenization and error estimates
More informationA Remark on -harmonic Functions on Riemannian Manifolds
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 07, pp. 1-10. Published June 15, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationEXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS
EXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS Adriana Buică Department of Applied Mathematics Babeş-Bolyai University of Cluj-Napoca, 1 Kogalniceanu str., RO-3400 Romania abuica@math.ubbcluj.ro
More informationEquilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains
Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains J. Földes Department of Mathematics, Univerité Libre de Bruxelles 1050 Brussels, Belgium P. Poláčik School
More informationUniform estimates for Stokes equations in domains with small holes and applications in homogenization problems
Uniform estimates for Stokes equations in domains with small holes and applications in homogenization problems Yong Lu Abstract We consider the Dirichlet problem for the Stokes equations in a domain with
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More information