Stability for parabolic quasiminimizers. Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson

Transcription

1 Stability for parabolic quasiminimizers Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson REPORT No. 1, 213/214, fall ISSN X ISRN IML-R /14- -SE+fall

2 STABILITY FOR PARABOLIC QUASIMINIMIZERS YOHEI FUJISHIMA, JENS HABERMANN, JUHA KINNUNEN AND MATHIAS MASSON Abstract. This paper studies parabolic quasiminimizers which are solutions to parabolic variational inequalities. We show that, under a suitable regularity condition on the boundary, parabolic Q-quasiminimizers related to the parabolic p-laplace equations with given boundary values are stable with respect to parameters Q and p. The argument is based on variational techniques, higher integrability results and regularity estimates in time. This shows that stability does not only hold for parabolic partial differential equations but it also holds for variational inequalities. 1. Introduction This paper investigates a stability question for parabolic quasiminimizers related to the parabolic p-laplace equations u t div( u p 2 u) =, 2N N + 2 < p <, on a space time cylinder = (, T ) in R N+1. A function u is called a parabolic Q-quasiminimizer, if for some Q 1 we have u t φ + 1 u p Q (u φ) p, p supp φ p supp φ for all φ Co ( ). Here, the data on the parabolic boundary par = {} (, T ) are taken in an appropriate sense. Parabolic quasiminimizers with Q = 1 are called minimizers and in that case there is a one to one correspondence between minimizers and solutions: every minimizer is a solution of the corresponding partial differential equation. However, when Q > 1 being a quasiminimizer is not only a local property (see [8]) and, consequently, there is no connection to the partial differential equation and only variational methods are available. Parabolic quasiminimizers were introduced in [22], and later they have been studied, for example, in [2, 5, 7, 15, 16, 17, 18, 25, 26]. We consider stability of parabolic quasiminimizers with respect to parameters p and Q. More precisely, assume that we have sequences Q i Q and p i p as i and let u i be a parabolic Q i -quasiminimizer of the 2 Mathematics Subject Classification. 35K92, 35B35. This research is supported by the Academy of Finland. Part of the work was done during a visit to the Institut Mittag-Leffler (Djursholm, Sweden). 1

3 2 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON variational inequality with exponent p i with the same initial and boundary conditions. It is well known that the quasiminimizers with Q i > 1 are not unique. To ensure the existence of a limit function, we assume that the sequence parabolic quasiminimizers converges pointwise. This extra condition is redundant in the case that the sequence converges to a minimizer, since minimizers are known to be unique. According to our main results (Theorems 2.1 and 2.2) the limit function u is a parabolic Q-quasiminimizer with the same boundary and initial values as all terms of the sequence. Furthermore, if Q = 1, then the limit function is a minimizer and u i converges to u in the parabolic Sobolev space. Stability of elliptic quasiminimizers has been studied in [6, 7] and [14]. For elliptic equations, see [11, 12, 13, 23, 24]. We investigate the stability question from a purely variational point of view, by using the assumption that the complement of satisfies a uniform capacity density condition. This paper extends the results of [9] for parabolic quasiminimizers and the argument is based on local and global higher integrability results for the gradients of parabolic quasiminimizers, see [19] and [2]. In particular, the paper [9] only covers the degenerate case p 2 and hence our results are new even for the parabolic p-laplace equation in the singular case 2N/(N + 2) < p < 2. New arguments are needed to compensate the lack of the partial differential equation and the results show that the class of parabolic quasiminimizers is stable under parturbations of the parameters. A careful analysis of the regularity in time plays a decisive role in the argument. 2. Notation and basic definitions 2.1. Notation. Let N 1, R N be a bounded open set, 1 < p <, and T >. The usual first order Sobolev space W 1,p () is equipped with the norm u W 1,p () := u L p () + u L p (). The Sobolev space with zero boundary values Wo 1,p () is defined as a completion of Co () with respect to the norm W 1,p (). The parabolic Sobolev spaces L p (, T ; W 1,p ()) and L p (, T ; Wo 1,p ()) consist of all measurable functions u : (, T ) R such that u(, t) W 1,p () and u(, t) Wo 1,p () for almost every t (, T ), respectively, and ( T ) 1/p u L p (,T ;W 1,p ()) := u(, t) p W 1,p () dt <. Functions in the parabolic space L p (, T ; W 1,p ()), for which there exist t 1, t 2 (, T ) with t 1 < t 2 such that u(x, t) = for almost every x when t [t 1, t 2 ], are denoted by L p c(, T ; W 1,p ()). Moreover, we define the space Lip c (, T ; Wo 1,p ()) as the space of all functions u L p c(, T ; W 1,p ()) for which u(, t) W 1,p () Lip(, T ).

4 STABILITY FOR PARABOLIC QUASIMINIMIZERS 3 For x R N and r >, we define the p-capacity of a closed set E B r (x) with respect to B r (x) by cap p (E, B r (x)) { } := inf u(y) p dy : u Co (B r (x)) with u 1 in E. B r(x) Here B r (x) denotes the open ball with the radius r > and the center at x. The set R N \ is called uniformly p-thick if there exist positive constants µ and r such that (2.1) cap p ((R N \ ) B r (x), B 2r (x)) µ cap p (B r (x), B 2r (x)) for all x R N \ and r (, r ) Parabolic quasiminimizers. Let N 1, R N be a bounded open set, 1 < p <, T >, and Q 1. A function u L p (, T ; W 1,p ()) is called a (local) parabolic Q-quasiminimizer if it satisfies (2.2) u( t φ) + 1 u p Q (u φ) p, p supp φ p supp φ for all test functions φ C o ( ), where denotes the parabolic cylinder (, T ) and supp denotes the support of the function. In addition, for g L p (, T ; W 1,p ()), we say that u L p (, T ; W 1,p ()) is a (global) parabolic Q-quasiminimizer with initial and boundary value g if u satisfies (2.2), (2.3) u(, t) g(, t) W 1,p o () for almost every t (, T ), and 1 (2.4) lim h h h u g 2 =. Now we are ready to state our main results. Theorem 2.1. Let T > and N 2. For p > 2N/(N +2), let R N be a bounded open set such that R N \ is a uniformly p-thick. Let p i > 2N/(N+2) and Q i 1, i = 1, 2,..., be real numbers such that (2.5) lim p i = p and lim Q i = Q. For i N, let u i L p i (, T ; W 1,p i ()) be a parabolic Q i -quasiminimizer with initial and boundary value g C 1 ( ) and suppose that there exists a measurable function u such that (2.6) lim u i (x, t) = u(x, t) for almost every (x, t).

5 4 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Then u L p (, T ; W 1,p ()) and u satisfies (2.7) u(, t) g(, t) Wo 1,p () for almost every t (, T ), 1 h (2.8) lim u g 2 =, h h and moreover (2.9) u( t φ) + 1 u p Q (u φ) p p p for all φ C o ( ). supp φ supp φ Observe that since quasiminimizers are not unique, we have to assume some kind of convergence as in (2.6). Otherwise the limit function does not need to exist. If u i converge to the minimizer as i, then we obtain the strong convergence of u i. Theorem 2.2. Assume the same conditions as in Theorem 2.1. If Q = 1, then (2.1) u i u in L p (, T ; W 1,p ()) as i. 3. Preliminary results In this section we give some preliminary results on properties of quasiminimizers. In particular, we study regularity for parabolic quasiminimizers in time variable, the Sobolev space with zero boundary values, a Hardy type estimate for u Wo 1,p (), and global higher integrability for the gradients of parabolic quasiminimizers Regularity for parabolic quasiminimizers. We denote by, the paring between (L p (, T ; Wo 1,p ())) and L p (, T ; Wo 1,p ()). We say that the function v (L p (, T ; Wo 1,p ())) is the weak derivative of the function u L p (, T ; W 1,p ()) if v, ϕ = u( t ϕ), for all ϕ Co ( ), and v is denoted by t u. We first prove the following lemma, which improves regularity for parabolic quasiminimizers in the time variable. Similar phenomenon has been observed already in [22]. Lemma 3.1. Let N 1, R N be a bounded open set, p > 1, T >, and Q 1. Let u L p (, T ; W 1,p ()) be a parabolic Q-quasiminimizer. Then t u (L p (, T ; Wo 1,p ())) and t u, ϕ 2p Q p u p 1 L p (,T ;W 1,p ()) ϕ L p ( ), for all ϕ L p (, T ; Wo 1,p ()).

6 STABILITY FOR PARABOLIC QUASIMINIMIZERS 5 Proof. Since the desired inequality holds if u L p (,T ;W 1,p ()) =, we can assume without loss of generality that u L p (,T ;W 1,p ()). Let ϕ Co ( ) be a test function satisfying ϕ L p ( ) = 1, and set φ = u L p (,T ;W 1,p ())ϕ. Then, by (2.2) we have T u L p (,T ;W 1,p ()) u t ϕ Q (u u p L p (,T ;W 1,p ())ϕ) p (3.1) 2p 1 Q p ( u p + u p L p (,T ;W 1,p ()) ϕ p ) 2p Q p u p L p (,T ;W 1,p ()). On the other hand, replacing φ by φ in (2.2), we obtain T (3.2) u L p (,T ;W 1,p ()) u t ϕ 2p Q p u p L p (,T ;W 1,p ()). Since ϕ L p ( ) = 1, by (3.1) and (3.2) we have T (3.3) u t ϕ 2p Q L p (,T ;W 1,p ()) ϕ L p ( ), p u p 1 for all ϕ Co ( ). Since Co ( ) is dense in L p (, T ; Wo 1,p ()), by (3.3) we obtain t u (L p (, T ; Wo 1,p ())) and the desired inequality. This completes the proof. Next we consider a regularization of quasiminimizers with respect to the time variable. Let φ Co ( ) with supp φ = [t 1, t 2 ]. For < ε < min{t 1, T t 2 }, we take a mollifier ξ ε Co (R) satisfying supp ξ ε [ ε, ε], ξ ε, and ξ ε L p (R) = 1 and denote (3.4) [φ] ε (x, t) := ε ε φ(x, t s)ξ ε (s)ds in. Since ξ ε (s) is an even function, by (2.2), after a change of variables and an integration by parts in the time derivative term, we obtain ( t [u] ε )φ + 1 u p p supp[φ] ε (3.5) Q (u [φ] ε ) p, p supp[φ] ε for all φ Co ( ). In the above inequality [u] ε denotes the smoothing of u according to (3.4). By Lemma 2, Corollary 1 and Remark 2 in [16] we know that, for a function ψ L p c(, T ; Wo 1,p ()) and for any ε >, there exists a

7 6 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON function ϕ Lip c ( ) and hence also a function ϕ C o ( ) such that ψ ϕ L p (,T ;W 1,p ()) < ε, ψ ϕ L 2 ( ) < ε, and supp ϕ \ supp ψ < ε. Using this density result, it is straightforward to show that inequality (3.5) also holds for any φ L p c(, T ; Wo 1,p ()). On the other hand, since ξ ε L p (R N ) = 1, by the Hölder inequality and the Fubini theorem we have T [φ]ε (x, t) T ε p (2ε) p 1 φ(x, t s) p ξ p ε (s) ds (3.6) and (3.7) T ε ε T +ε [ ξε(s) p T ε ε = φ(x, t) p, φ(x, t s) ] p φ(x, τ) p dx dτ, [ φ] ε (x, t) p φ(x, t) p φ p L p (,T ;W 1,p ()), for all sufficiently small ε >. Using these properties, we prove the following lemma. Lemma 3.2. Let N 1, R N be a bounded open set, p > 2N N+2, T > and Q 1. Let η W 1, ( ) be a function such that η(x, t) = for a.e. (x, t) near t = and t = T. Assume that u L p (, T ; W 1,p ()) is a parabolic Q-quasiminimizer. Then t (ηu) (L p (, T ; Wo 1,p ())) and t v, ηu = t (ηu), v, for every function v L p (, T ; W 1,p o ()) with t v (L p (, T ; W 1,p o ())). Proof. For simplicity, we set X = L p (, T ; W 1,p ( )). Let {η n } n=1 C ( ) be a sequence such that η n η W 1, ( ) as n. Since η n ϕ C o ( ) for all ϕ C o ( ), by Lemma 3.1 we have (3.8) t u, η n ϕ 2p Q p u p 1 X (η nϕ) L p ( ) 2p Q p u p 1 X η n W 1, ( ) ϕ X, ds

8 STABILITY FOR PARABOLIC QUASIMINIMIZERS 7 for all ϕ Co ( ). Moreover, since t u, η n ϕ = u(ϕ t η n + η n t ϕ) and uϕ( t η n ) tη n L ( ) u L 2 ( ) ϕ L 2 ( ) C 1 t η n L ( ) u X ϕ X, which follows from the continuous embedding W 1,p () L 2 (), as p > 2N/(N + 2), by (3.8) we obtain η n u( t ϕ) C 1 t η n L ( ) u X ϕ X + 2p Q p u p 1 X η n W 1, ( ) ϕ X. Here, C 1 is the constant which depends on N, p and on the domain. Then, letting n, by the Poincaré inequality we see that there exists a constant C 2, depending only on p, N, and Q and η W 1, ( ), such that (3.9) ( ηu( t ϕ) C 2 u X + u p 1 X ) ϕ X, for all ϕ C o ( ). This implies that t (ηu) (L p (, T ; W 1,p o ( ))). Let {v n } n=1 C o ( ) be a sequence such that v n v L p (,T ;W 1,p ( )) as n. Letting ε > be a sufficiently small constant, we have [ηu] ε C o ( ) since η = a.e. near t = and t = T. Then we get by integration by parts that t v n, [ηu] ε = v n, t [ηu] ε, and passing to the limit ε, by Lemma 3.1 we obtain (3.1) t v n, ηu = ( t v n )ηudxdt = t (ηu), v n. Furthermore, by (3.9) we have ( ) t v n t v, ηu C 2 u X + u p 1 v n v X. This together with Lemma 4.1 and (3.1) yields X t v, ηu = t (ηu), v, and we conclude the proof of Lemma 3.2.

9 8 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON 3.2. Sobolev space with zero boundary values. In this subsection we recall some properties of the Sobolev space with zero boundary values and a Hardy type estimate. We begin with a stability result for Sobolev spaces with zero boundary values under suitable assumptions on regularity of the boundary. For the proof of Proposition 3.3, we refer to [3]. Proposition 3.3. Let N 1, R N be a bounded open set, and p > 1. Assume that R N \ is uniformly p-thick. Then there exists a positive constant ε such that W 1,p ε o () W 1,p () = Wo 1,p (). We also have the following Hardy type estimate, see [1] and [1]. Proposition 3.4. Let N 1, R N be a bounded domain, and p > 1. Assume that R N \ is uniformly p-thick. Then there exists a positive constant C, depending only on N, p and µ, such that for all u Wo 1,p (). ( u(x) dist(x, R N \ ) ) p dx C u p W 1,p (), Furthermore, the following proposition gives a sufficient condition for a function u W 1,p () such that u Wo 1,p (). For the proof of Proposition 3.5, see [4]. Proposition 3.5. Let N 1, p > 1, and be an open subset of R N. If u W 1,p () satisfies ( ) u(x) p dist(x, R N dx <, \ ) then u Wo 1,p () Global higher integrability. One of the fundamental ingredients for our stability proofs is global higher integrability of the gradient u on the domain, which follows from the parabolic Q quasiminimizing property. For the proof of Proposition 3.6, see [2]. Proposition 3.6. Let N 2, p > 2N/(N + 2), and R N be a bounded open set such that R N \ is uniformly p-thick with parameters µ and r. For Q 1 and g C 1 ( ), let u L p (, T ; W 1,p ()) be a parabolic Q- quasiminimizer satisfying (2.3) and (2.4). Then there exists a positive constant δ, depending only on N, p, Q, µ, and r, such that Furthermore u L p+δ (, T ; W 1,p+δ ()). u p+δ is bounded from above by some positive constant depending only on N, p, Q, µ, r, δ, g, and u L p ( ).

10 STABILITY FOR PARABOLIC QUASIMINIMIZERS 9 4. Uniform estimate for quasiminizers In this section we study a uniform estimate of u i L p i(,t ;W 1,p i()) with respect to i N. We first study the Caccioppoli type estimate, and prove the following lemma, which is an extension of Lemma 3.1 of [9]. Lemma 4.1. Assume the same conditions as in Theorem 2.1. Then, for any δ >, there exists a positive constant C, depending only on N, δ and the upper bound p of {p i }, and in particular independent of i, such that sup t (,T ) C u i (, t) g(, t) 2 dx + t g p i/(p i 1) + Q g p i + δ u i g p i, for all i N. Here Q denotes the upper bound of {Q i }. Proof. For < s < T and < h < min{s/4, (T s)/2}, we set χ h s (t) :=, t h, (t h)/h, h < t 2h, 1, 2h < t s 2h, (s t h)/h, s 2h < t s h,, s h < t T. Then, since the support of the function χ h s is compact in (, T ) and by the lateral boundary condition (2.3), we can take the test function φ h ε (x, t) := χ h s ([u i ] ε [g] ε ) Lip c (, T ; W 1,p i o ()) in (3.5), and obtain [u i ] ε t φ h ε + 1 p i supp[φ h ε ] ε (4.1) Q i ( u i [φ h ) ε ] ε p i, p i supp[φ h ε ]ε where ε > is a sufficiently small constant and [ ] ε is defined by (3.4). Set [u i ] ε t φ h ε (4.2) = ([u i ] ε [g] ε ) t φ h ε + [g] ε t φ h ε =: I 1 + I 2.

11 1 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON By an integration by parts we have [ I 1 = [ui ] ε [g] ε 2 t χ h s + χ h s ([u i ] ε [g] ε ) t ([u i ] ε [g] ε ) ] T = [u i ] ε [g] ε 2 t χ h s + 1 χ h s t [u i ] ε [g] ε 2 2 = 1 [u i ] ε [g] ε 2 t χ h s. 2 Thus, letting ε and thereafter h, we obtain, using also the initial condition (2.4): ε I 1 1 s h u i (x, t) g(x, t) 2 2h s 2h + 1 2h (4.3) u i (x, t) g(x, t) 2 2h h h u i (, s) g(, s) 2 dx, for almost all s (, T ). Furthermore, by integration by parts and the Young inequality, for any δ >, we can find a positive constant C δ, independent of i, such that T h I 2 = χ h s ([u i ] ε [g] ε ) t [g] ε [u i ] ε [g] ε t [g] ε h T δ [u i ] ε [g] ε p i + C δ t [g] ε p i/(p i 1). T Since g C 1 ( ), we obtain (4.4) lim I 2 δ 2 p 1 u i g p i + C δ t g p i/(p i 1). ε,h On the other hand, since lim ( u i [φ h ) ε ] ε p i ε,h supp[φ h ε ]ε supp(u i g) g p i, by (4.1) (4.4) there exists a positive constant C, independent of i and δ, such that u i (, s) g(, s) 2 dx + T p C δ t g p i/(p i 1) + C g p i + Cδ u i g p i, which holds for almost all s (, T ). Here the constant C δ depends on p and δ. Therefore, for any δ >, taking a sufficiently small δ > satisfying Cδ < δ and passing over to the supremum over s (, T ) on the left hand side, we obtain the desired inequality.

12 STABILITY FOR PARABOLIC QUASIMINIMIZERS 11 As a corollary of Lemma 4.1, we obtain a uniform estimate of the norm u i L p i(,t ;W 1,p i()). Corollary 4.2. Assume the same conditions as in Theorem 2.1. Then (4.5) ( ) sup u i (, t) 2 dx + u i p i + <. i N sup t (,T ) Furthermore there exists a positive constant δ such that ( (4.6) sup u i p+δ + u i p+δ) <. i N Proof. Corollary 4.2 can be proved by the same argument as Corollary 3.2 in [9] with the aid of Lemma 4.1, and hence we omit the details of the proof. Additionally we note that the uniform bound of the sup-term in the case p < 2 can also be obtained via the Caccioppoli type inequality in Lemma 4.1 and the fact that g C 1 ( ). By Corollary 4.2 we can obtain the strong convergence of u i and weak convergence of the derivatives in L p+δ ( ), and moreover also in the case p < 2 the strong convergence in L 2. Lemma 4.3. Assume the same conditions as in Theorem 2.1. Then there exist a subsequence {u i } i=1 and a positive constant δ such that we have u L p+δ (, T ; W 1,p+δ ()) and (4.7) (4.8) u i u in L p+δ ( ) L 2 ( ), u i u weakly in L p+δ ( ), as i. Furthermore t u (L p+δ (, T ; Wo 1,p+δ ())) and (4.9) t u i t u in the weak- topology on (L p+δ (, T ; Wo 1,p+δ ())) as i. Proof. We can prove (4.7) and (4.8) by the similar argument as in Lemma 3.3 in [9]. Since lim p i = p and lim Q i = Q, by Lemma 3.1 there exist positive constants C and δ, independent of i, such that t u i, φ 2p i Q i u i p i 1 p L p i(,t ;W 1,p i()) φ L p i( ) i 2p i Q i u i p i 1 p L p+δ (,T ;W 1,p+δ ()) φ L p+δ ( ) ( T ) 1/p i 1/(p+δ) i C u i p i 1 L p+δ (,T ;W 1,p+δ ()) φ L p+δ ( ), for all φ C o ( ) and sufficiently large i N. This together with (4.6) implies that (4.1) sup t u i (L i N p+δ (,T ;Wo 1,p+δ <. ( ))

13 12 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Then, in view of the Rellich-Kondrachov theorem together with (4.6) and (4.1), by taking a subsequence if necessary we can find ũ L p+δ ( ) with ũ L p+δ ( ) such that u i ũ in L p+δ ( ), u i ũ weakly in L p+δ ( ), as i (see [2] and [21]). Since u i u almost everywhere in as i, we have ũ = u, and obtain (4.7) and (4.8). Moreover, applying [21, Corollary 8], with the choices X := W 1,p (), B := L 2 () and Y := W 1,p () = ( Wo 1,p () ) and having the inclusions W 1,p () L 2 () W 1,p (), where the first inclusion is compact, since p > 2N/(N + 2), the uniform bounds (4.6) and (4.1) allow us to conclude also the strong convergence u i u in L 2 ( ) (in fact u i u in L q (, T ; L 2 ()) for any q < ). Next we prove (4.9). Taking a subsequence if necessary, we see that there exists a function v (L p+δ (, T ; Wo 1,p+δ ( )) such that (4.11) t u i v in the weak- topology on (L p+δ (, T ; W 1,p+δ o ( )), as i. On the other hand, since it follows from (4.7) that lim tu i, φ = lim u i t φ = u t φ, for all φ Co ( ), by (4.11) we have v, φ = lim t u i, φ = u t φ = t u, φ and t u, φ v (L p+δ (,T ;W 1,p+δ o ( )) φ L p+δ ( ), for all φ Co ( ). This implies that t u (L p+δ (, T ; Wo 1,p+δ ( )) and v = t u. Thus we obtain (4.9). 5. Proof of Theorem 2.1 In this section we complete the proof of our first main result. Proof of Theorem 2.1. We first prove (2.7) and (2.8). Let ε > be a sufficiently small constant to be chosen later. Taking a sufficiently large i, we have p ε < p i < p + ε and u i (t) g(t) Wo 1,p ε () for almost all

14 STABILITY FOR PARABOLIC QUASIMINIMIZERS 13 t (, T ), and by Proposition 3.4 we obtain ( ) ui (x, t) g(x, t) p ε dist(x, R N \ ) (5.1) C 1 T C 2 T u i (t) g(t) p ε W 1,p ε () dt ( ) u i (t) p ε W 1,p ε () + g p ε dt. W 1,p ε () At this point we use the Caccioppoli type estimate in terms of Lemma 4.1 to control the W 1,p ε norm of u i as follows ( ) (p ε)/pi u i p ε 1 (p ε)/p i [ T 1 (p ε)/p i C t g p i/(p i 1) + C g p i ] + δ (p ε)/pi u i g p i [ T C 1 (p ε)/p i u i p ε L p i( ) + tg (p ε)/(p i 1) L p i /(p i 1) ( ) ] + g p ε, L p i(,t ;W 1,p i()) for a constant C which depends on N, p and δ and for arbitrary δ >. Combining this with (5.1) and using once again Hölder s inequality, we arrive at ( ) ui (x, t) g(x, t) p ε dist(x, R N \ ) [ ] C 1 + u i p ε L p+ε ( ) + g p ε L p+ε (,T ;W 1,p+ε ()) + tg (p ε)/(p i 1) L (p ε)/(p ε 1) ( ). Here we used in the last step also that (p ε)/(p ε 1) > p i /(p i 1). The above estimate holds for all i N, and C is a positive constant which depends only on N, p, δ and and blows up as. Moreover, the right hand side of the above inequality is uniformly bounded, since p i > 2N/(N + 2) > 1 and the uniform bound of u i L p+ε ( ). Since u i u for almost every (x, t), by (4.6), (5.1), and the Fatou lemma we obtain T ( ) u(x, t) g(x, t) p ε dist(x, R N <. \ ) This implies that ( ) u(x, t) g(x, t) p ε dist(x, R N dx <, \ )

15 14 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON for almost every t (, T ), and by Proposition 3.5 we obtain u(t) g(t) Wo 1,p ε () for almost every t (, T ). On the other hand, since R N \ is uniformly p-thick, we can apply Proposition 3.3 to obtain W 1,p ε o () W 1,p () = Wo 1,p (), for some ε >. Now, choosing ε small enough, for example ε := ε /2, we conclude that u(t) g(t) Wo 1,p () for almost every t (, T ), and obtain (2.7). In order to prove (2.8), for τ (, T ), we take parameters h, k > such that 2h < τ k < τ and set, t h, χ h,k,τ (t) = (t h)/h, h < t 2h, 1, 2h < t τ k, (τ t)/k, τ k < t τ,, τ < t T. Consider the function χ h,k,τ ([u i] ε [g] ε ) in (3.5), and let ε. Then, by the similar argument as in Lemma 4.1 we obtain 1 k τ τ k C sup i N u i g 2 1 h τ o 2h h u i g 2 ( + u i g p i + t g p i/(p i 1) + g p i ), where C is a positive constant. We first pass to the limit h in the inequality above, and then pass to the limit i. Since u i u in L p+δ ( ) and in L 2 ( ) as i by (4.7) and u i t= = g, we obtain (5.2) 1 k τ τ k C sup i N u g 2 τ Furthermore, since 1 lim k k τ τ k ( ) + u i g p i + t g p i/(p i 1) + g p i. u g 2 = u(x, τ) g(x, τ) 2 dx, by (4.6) and (5.2) we obtain u(x, τ) g(x, τ) 2 dx =. This implies that lim τ 1 lim h h h and we conclude that (2.8) holds. u(x, t) g(x, t) 2 =,

16 STABILITY FOR PARABOLIC QUASIMINIMIZERS 15 We next prove (2.9). Fix α >. Let φ Co ( ) and set K = supp φ. Since K is a compact subset of, we can take open sets O 1 and O 2 such that K O 1 O 2 and (5.3) u p+δ < α, O 2 \K where δ > is the constant given in Lemma 4.3. Let ε > be a sufficiently small constant, and take a test function ϕ i,ε := φ + η([u i ] ε [u] ε ), where η Co ( ) is a cut-off function such that η 1, η 1 in a neighborhood of K, and η in \ O 1. Then, since ϕ i,ε is a valid test function in (3.5), we obtain [u i ] ε t (ϕ i,ε ) + 1 p i supp[ϕ i,ε ] ε (5.4) Q i p i Let β >. By (4.8) we have u p β lim inf u i p β K K [ ( lim inf K ( T ) β/p (lim inf supp[ϕ i,ε ] ε (u i [ϕ i,ε ] ε ) p i. ) (p β)/pi ( T ) 1 (p β)/p i K ) (p β)/p. Since β > is arbitrary and K supp[ϕ i,ε ] ε for every i, we obtain (5.5) u p lim inf. K supp[ϕ i,ε ] ε Let I i,ε := [u i ] ε ( t (ϕ i,ε φ)) = [u i ] ε t (η[u i u] ε ). Then we have I i,ε = [u i u] ε t (η[u i u] ε ) + [u] ε t (η[u i u] ε ) =: J i,ε + K i,ε. ]

17 16 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Since η Co ( ), we have J i,ε = ( t η)[u i u] 2 ε + 1 η 2 t [u i u] 2 ε = ( t η)[u i u] 2 ε 1 ( t η)[u i u] 2 ε 2 = 1 ( t η)[u i u] 2 ε, 2 and by (4.7) we obtain (5.6) lim lim ε J i,ε =. Furthermore, putting K i,ε = ( t η)[u] ε [u i u] ε + T ( t [u i u] ε )η[u] ε = ( t η)[u] ε [u i u] ε ( t [u i u] ε )η(u [u] ε ) + ( t [u i u] ε )ηu =: K 1 i,ε + K 2 i,ε + K 3 i,ε, by (3.6), (3.7), and Lemma 3.1 we can find a positive constant C, independent of ε and i, such that and By (4.7) we obtain K 1 i,ε sup t η u L 2 ( ) u i u L 2 ( ) K 2 i,ε C u i u p 1 L p (,T ;W 1,p ()) ([u] ε u) L p ( ). (5.7) lim lim ε ( K 1 i,ε + K 2 i,ε ) =. On the other hand, Lemma 3.2 implies that K 3 i,ε = t (ηu), [u i u] ε t (ηu), u i u = t (u i u), ηu, as ε and, consequently, by (4.9) we have lim lim ε K3 i,ε =. This together with (5.6) and (5.7) implies that (5.8) lim lim ε [u i ] ε t ϕ i,ε = u t φ.

18 STABILITY FOR PARABOLIC QUASIMINIMIZERS 17 Therefore, letting ε in (5.4), by (5.5) and (5.8) we obtain u t φ + o(1) + 1 u p p K (5.9) Q i (u i φ η(u i u)) p i, p i O 1 for all sufficiently large i N. By the definition of η and K we have η 1 in a neighborhood of K and φ in \ K, and obtain (u i φ η(u i u)) p i O (5.1) 1 = (u φ) p i + (u i η(u i u)) p i. Since K O 1 \K (u i η(u i u)) (1 η) u i + η u i u + η u in O 1 \ K, we have (u i η(u i u)) p i (5.11) O 1 \K C 1 O 1 \K ((1 η) p i + η p i u i u p i + η p i u p i ), where C 1 is a positive constant independent of i. By the Hölder inequality we have ( ) pi /(p+δ) η p i u p i u p+δ O 1 \ K 1 pi/(p+δ). O 1 \K O 1 \K This together with (5.3) implies that there exists a positive constant C 2 such that (5.12) lim sup η p i u p i < C 2 α p/(p+δ). O 1 \K Furthermore, by the Hölder inequality and (4.7) there exists a positive constant C such that (5.13) η p i u i u p i C u i u p/(p+δ) L p+δ ( ), O 1 \K as i. Note that the constant C in the above estimate depends also on η and therefore also on φ, but it is independent of i. In order to finish the proof of (2.9), it remains to estimate the quantity lim O 1 \K (1 η) p i dz. For this aim we modify the arguments of [7] and [14], and prove the following auxiliary result.

19 18 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Lemma 5.1. Assume the same conditions as in Theorem 2.1. Let D be a compact subset of. For any s (, s ), define := {z : dist p (z, D) < s}, where dist p denotes the parabolic distance in R N+1 and s := dist p (D, R N (, ) \ ). Then there exists a positive constant C such that lim sup C u p, for almost every s (, s ). Proof. Let < σ < s < s and η C o ( ) be a cut-off function such that (5.14) η 1, η 1 in D(σ), η in \. Let ε > be a sufficiently small constant, and take the test function φ i,ε = η([u i ] ε [u] ε ). Then, by (3.5) we have [u i ] ε ( t φ i,ε ) + 1 p i supp[φ i,ε ] ε (5.15) Q i (u i [φ i,ε ] ε ) p i. p i supp[φ i,ε ] ε By the same argument as above we see that (5.16) lim lim [u i ] ε t φ i,ε =. ε To proceed with the argument, we denote the integral in the line above with I ε (i) and moreover I (i) := lim ε I ε (i). With this short hand notation (5.16) writes as lim I (i) = lim lim ε I ε (i) =. Therefore, with this notation, letting ε in (5.15), we have + I (i) 2Q (u i φ i ) p i, for all sufficiently large i N, where φ i := η(u i u). This together with (5.14) implies that + I (i) 2Q (u i φ i ) p i D(σ) C C ((1 η) p i + η p i u i u p i + η p i u p i ) \D(σ) + C for some constant C which is independent of i. ( η p i u i u p i + η p i u p i ),

20 Then, adding STABILITY FOR PARABOLIC QUASIMINIMIZERS 19 C D(σ) to the both hand sides of the above inequality, we obtain (1 + C) + I (i) (5.17) D(σ) C For s (, s ), we set ( + η p i u i u p i + η p i u p i ). Ψ(s) := lim sup. Since D(s ) if s < s, the function Ψ is nondecreasing and finite because of the higher integrability of u i, and we conclude that the set of points of discontinuity of Ψ is at most countable. Let s (, s ) be a point of continuity of Ψ. Then, by (5.17) we have (1 + C)Ψ(σ) CΨ(s) + lim sup η p i u i u p i +C u p. Note that to obtain the preceding estimate, we have to see that (5.18) u p i u p, as i. To see this, we use the elementary estimate ξ a ξ b [ 1 ε ξ max{a,b}+ε + 1 ( 1 ε a + 1 ) ] a b, b for all ξ R k, a, b > and ε >, which we take from [9, Proof of Th. 3.4]. We let i be large enough to have p i p + δ/2 and apply the preceding estimate with the choices ξ = Du, a = p i, b = p and ε = δ/2, where δ > denotes the higher integrability exponent Proposition 3.6, to obtain u p i u p [ 2 δ u p+δ + C ] p i p, where the constant C depends on p and. Letting i, the right hand side tends to zero and therefore we conclude the desired convergence (5.18). On the other hand, as in (5.13), we can obtain lim sup η p i u i u p i =.

21 2 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Therefore we have (1 + C)Ψ(σ) CΨ(s) + C u p for all σ (, s). Since s is a point of continuity of Ψ, we obtain (1 + C)Ψ(s) CΨ(s) + C u p, which gives Ψ(s) C This completes the proof of Lemma 5.1. u p. We continue the proof of Theorem 2.1. Since η 1 in a neighborhood of K, we can take a compact set D O 1 \ K satisfying (5.19) (1 η) p i. O 1 \K Then we see that O 2 \ K for all sufficiently small s >, and by Lemma 5.1 we can take constants C 3 and s > such that lim sup C 3 u p. D This together with (5.3) and (5.19) implies that lim sup (1 η) p i lim sup O 1 \K C 3 u p ( C 3 u p+δ O 2 \K C 3 ( T ) 1 p/(p+δ) α p/(p+δ), ) p/(p+δ) ( T ) 1 p/(p+δ) and by (5.11) we have (u i η(u i u)) p i C 1 C 2 α p/(p+δ) +C 1 C 3 ( T ) 1 p/(p+δ) α p/(p+δ), O 1 \K where all the appearing constants are independent of α. Therefore, by the arbitrariness of α > and (5.9) we obtain u( t φ) + 1 u p Q (u φ) p p p K for all φ C o ( ) with K = supp φ. Thus we obtain (2.9), and complete the proof of Theorem 2.1. Finally we prove our second main result. K

22 STABILITY FOR PARABOLIC QUASIMINIMIZERS 21 Proof of Theorem 2.2. Let h > and ε > be sufficiently small constants, and take the test function in (3.5) with χ h (t) := φ h ε (x, t) := χ h [u i u] ε, t h, (t h)/h, h < t 2h, 1, 2h < t T 2h, (T t h)/h, T 2h < t T h,, T h < t T. Then, by (3.5) we obtain [u i ] ε t φ h ε + 1 p i supp[φ h ε (5.2) ]ε Q i ( ui [φ h ) ε ] ε p i. p i supp[φ h ε ] ε Denote [u i ] ε t φ h ε = [u i u] ε t φ h ε + [u] ε t φ h ε =: I 1 + I 2. As in the proof of Lemma 4.1, since u i t= = u t= = g, we obtain (5.21) lim lim I 1. h ε Furthermore, we have I 2 = ( t [u] ε )χ h [u i u] ε = t [u] ε, χ h [u i u] ε = t [u] ε t u, χ h [u i u] ε t u, χ h [u i u] ε χ h (u i u) t u, χ h (u i u), and it follows from Lemma 3.1, (3.6), and (3.7) that there exists a constant C > such that and t ([u] ε u), χ h [u i u] ε C [u] ε u p 1 L p (,T ;W 1,p ()) (u i u) L p ( ) t u,χ h [u i u] ε χ h (u i u) C u p 1 L p (,T ;W 1,p ()) {(u i u) [u i u] ε } L p ( ), as ε. These together with Lemma 3.2 imply that (5.22) lim ε I 2 = t u, χ h (u i u) = t {χ h (u i u)}, u.

23 22 FUJISHIMA, HABERMANN, KINNUNEN AND MASSON Since t {χ h (u i u)}, u t (u i u), u C (1 χ h )(u i u) p 1 L p (,T ;W 1,p ()) u L p ( ), as h, by (4.9) and (5.22) we obtain (5.23) lim h lim ε I 2 = t (u i u), u as i. Therefore, applying the similar argument as in the proof of Theorem 2.1, by (5.2) (5.23) we conclude that lim sup lim sup Q i u p i (5.24) T = u p. This is the only point, where Q i 1 comes into play. Let γ > be a sufficiently small constant and i N be sufficiently large such that p i p < γ. Then, by the Hölder inequality we have (5.25) u i p = u i p γ u i γ ( ) (p γ)/pi ( ) (pi p+γ)/p i u i γp i/(p i p+γ). Let q i be the constant such that γp i p i p + γ q i = p + δ, that is, q i = (p + δ) p i p + γ γp i, where δ > is given in Lemma 4.3. Then, since q i > 1 for sufficiently large i, by the Hölder inequality we have ( ) (pi p+γ)/p i u i γp i/(p i p+γ) ( ) (pi p+γ)/(p i q i ) u i γp iq i /(p i p+γ) ( T ) (p i p+γ)(1 1/q i )/p i ( ) δ/(p+δ) = u i p+δ ( T ) (p i p+γ)/p i δ/(p+δ). This together with (4.6) implies that there exists a positive constant C, which is independent of γ, such that ( ) (pi p+γ)/p i lim sup u i γp i/(p i p+γ) (5.26) C γ ( T ) γ(1/p 1/(p+δ)).

24 STABILITY FOR PARABOLIC QUASIMINIMIZERS 23 Then, by (5.24), (5.25), and (5.26) we obtain ( ) (p γ)/p lim sup u i p u p C γ ( T ) γ(1/p 1/(p+γ)). Since γ > is arbitrary, we conclude that (5.27) lim sup u i p u p. On the other hand, since it follows from corollary 4.2 that (5.28) u i u weakly in L p ( ) as i, we have and by (5.27) we obtain u p lim inf T u i p, lim u i L p ( ) = u L p ( ). This together with (5.28) implies that u i u in L p ( ) as i. Thus we complete the proof of Theorem 2.2. References [1] A. Ancona, On strong barriers and an inequality of Hardy for domains in R n, J. London Math. Soc. 34 (1986), [2] J. Habermann, Global gradient estimates for non quadratic vector valued parabolic quasi minimizers, submitted for publication. [3] L. I. Hedberg and T. Kilpeläinen, On the stability of Sobolev spaces with zero boundary values, Math. Scand. 85 (1999), [4] T. Kipeläinen, J. Kinnunen, and O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 (2), [5] J. Kinnunen, N. Marola, M. Miranda jr., and F. Paronetto, Harnack s inequality for parabolic De Giorgi classes in metric spaces, Adv. Differential Equations, 17 (212), [6] J. Kinnunen, N. Marola and O. Martio, Harnack s principle for quasiminimizers, Ric. Mat. 56 (27), [7] J. Kinnunen and O. Martio, Potential theory of quasiminimizers, Ann. Acad. Sci. Fenn. Math. 28 (23), [8] J. Kinnunen and M. Masson, Parabolic comparison principle and quasiminimizers in metric measure spaces, to appear in Proc. Amer. Math. Soc. [9] J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations, Adv. Calculus of Variations 3 (21), [1] J. L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 38 (1988), [11] G. Li and O. Martio, Stability of solutions of varying degenerate elliptic equations, Indiana Univ. Math. J., 47 (1998) (3), [12] P. Lindqvist, Stability for the solutions of div( u p 2 u) = f with varying p, J. Math. Anal. Appl., 127 (1987)(1), [13] P. Lindqvist, On nonlinear Rayleigh quotients, Potential Anal. 2(1993)(3), [14] O. E. Maasalo and A. Zatorska-Goldstein, Stability of quasiminimizers of the p- Dirichlet integral with varying p on metric spaces, J. London Math. Soc. (2) 77 (28),

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