A functionu is called a parabolicQ-quasiminimizer, if for someQ≥1 we have − Z ΩT u∂tφ dx dt+ 1 p Z suppφ |∇u|pdx dt≤ Q p Z suppφ |∇(u−φ)|pdx dt, for all φ ∈ Co∞(ΩT)

YOHEI FUJISHIMA, JENS HABERMANN, JUHA KINNUNEN AND MATHIAS MASSON

Abstract. This paper studies parabolic quasiminimizers which are so- lutions to parabolic variational inequalities. We show that, under a suit- able regularity condition on the boundary, parabolicQ-quasiminimizers related to the parabolicp-Laplace equations with given boundary val- ues are stable with respect to parameters Qand p. The argument is based on variational techniques, higher integrability results and regular- ity 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 parabolicp-Laplace equations

∂u

∂t −div(|∇u|^p−2∇u) = 0, 2N

N + 2< p <∞,

on a space time cylinder Ω_T = Ω×(0, T) in R^N+1. A functionu is called a parabolicQ-quasiminimizer, if for someQ≥1 we have

− Z

ΩT

u∂_tφ dx dt+ 1 p

Z

suppφ

|∇u|^pdx dt≤ Q p

Z

suppφ

|∇(u−φ)|^pdx dt,

for all φ ∈ C_o^∞(Ω_T). Here, the data on the parabolic boundary ∂_parΩ_T = Ω× {0} ∪∂Ω×(0, T) are taken in an appropriate sense. Parabolic quasi- minimizers withQ= 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 equa- tion 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 pa- rameters p and Q. More precisely, assume that we have sequences Qi →Q and p_i → p as i → ∞ and let u_i be a parabolic Q_i-quasiminimizer of the

2000Mathematics 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

variational inequality with exponentpi 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 se- quence 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 functionu is a parabolicQ-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 anduiconverges touin 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 Ω 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, Ω⊂ R^N be a bounded open set, 1 < p < ∞, and T > 0. The usual first order Sobolev space W^1,p(Ω) is equipped with the norm

kuk_W1,p(Ω):=kuk_Lp(Ω)+k∇uk_Lp(Ω).

The Sobolev space with zero boundary valuesWo^1,p(Ω) is defined as a com- pletion of C_o^∞(Ω) with respect to the norm k · k_W1,p(Ω). The parabolic Sobolev spaces L^p(0, T;W^1,p(Ω)) and L^p(0, T;Wo^1,p(Ω)) consist of all mea- surable functions u : Ω ×(0, T) → R such that u(·, t) ∈ W^1,p(Ω) and u(·, t)∈Wo^1,p(Ω) for almost everyt∈(0, T), respectively, and

kuk_Lp(0,T;W^1,p(Ω)):=

Z T 0

ku(·, t)k^p_W1,p(Ω)dt 1/p

<∞.

Functions in the parabolic space L^p(0, T;W^1,p(Ω)), for which there exist t1, t2 ∈ (0, T) with t1 < t2 such that u(x, t) = 0 for almost every x ∈ Ω whent6∈[t₁, t₂], are denoted by L^pc(0, T;W^1,p(Ω)). Moreover, we define the space Lip_c(0, T;Wo^1,p(Ω)) as the space of all functionsu∈L^pc(0, T;W^1,p(Ω)) for which

ku(·, t)k_W1,p(Ω)∈Lip(0, T).

For x ∈R^N and r >0, we define the p-capacity of a closed set E ⊂Br(x) with respect toB_r(x) by

cap_p(E, B_r(x)) := inf

(Z

Br(x)

|∇u(y)|^pdy: u∈C_o^∞(Br(x)) with u≥1 in E )

. HereBr(x) denotes the open ball with the radius r >0 and the center atx.

The setR^N\Ω is calleduniformlyp-thickif there exist positive constantsµ and r0 such that

(2.1) cap_p((R^N \Ω)∩B_r(x), B_2r(x))≥µcap_p(B_r(x), B_2r(x)) for all x∈R^N \Ω and r∈(0, r0).

2.2. Parabolic quasiminimizers. LetN ≥1, Ω⊂R^N be a bounded open set, 1 < p < ∞, T > 0, and Q ≥ 1. A function u ∈ L^p(0, T;W^1,p(Ω)) is called a (local)parabolic Q-quasiminimizer if it satisfies

(2.2)

− Z

ΩT

u(∂_tφ)dx dt+1 p

Z

suppφ

|∇u|^pdx dt≤ Q p

Z

suppφ

|∇(u−φ)|^pdx dt,

for all test functionsφ∈C_o^∞(Ω_T), where Ω_T denotes the parabolic cylinder Ω×(0, T) and supp denotes the support of the function. In addition, forg∈ L^p(0, T;W^1,p(Ω)), we say that u∈L^p(0, T;W^1,p(Ω)) is a (global)parabolic Q-quasiminimizer with initial and boundary value g ifusatisfies (2.2), (2.3) u(·, t)−g(·, t)∈W_o^1,p(Ω) for almost every t∈(0, T), and

(2.4) lim

h→0

1 h

Z h 0

Z

Ω

|u−g|²dx dt= 0.

Now we are ready to state our main results.

Theorem 2.1. LetT >0andN ≥2. Forp >2N/(N+2), letΩ⊂R^N be a bounded open set such thatR^N\Ωis a uniformlyp-thick. Letpi>2N/(N+2) and Q_i≥1, i= 1,2, . . ., be real numbers such that

(2.5) lim

i→∞p_i=p and lim

i→∞Q_i =Q.

For i ∈ N, let u_i ∈ L^pi(0, T;W^1,pi(Ω)) be a parabolic Q_i-quasiminimizer with initial and boundary value g∈C¹(ΩT) and suppose that there exists a measurable function u such that

(2.6) lim

i→∞u_i(x, t) =u(x, t) for almost every (x, t)∈Ω_T.

Then u∈L^p(0, T;W^1,p(Ω)) andu satisfies

u(·, t)−g(·, t)∈W_o^1,p(Ω) for almost every t∈(0, T), (2.7)

h→0lim 1 h

Z h 0

Z

Ω

|u−g|²dx dt= 0, (2.8)

and moreover (2.9) −

Z

ΩT

u(∂_tφ)dx dt+1 p

Z

suppφ

|∇u|^pdx dt≤ Q p

Z

suppφ

|∇(u−φ)|^pdx dt

for allφ∈C_o^∞(Ω_T).

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. Ifu_i converge to the minimizer asi→ ∞, then we obtain the strong convergence ofui.

Theorem 2.2. Assume the same conditions as in Theorem 2.1. If Q= 1, then

(2.10) ui→u in L^p(0, T;W^1,p(Ω)) as i→ ∞.

3. Preliminary results

In this section we give some preliminary results on properties of quasi- minimizers. 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(Ω), and global higher integrability for the gradients of parabolic quasiminimizers.

3.1. Regularity for parabolic quasiminimizers. We denote byh·,·ithe paring between (L^p(0, T;Wo^1,p(Ω)))^∗ and L^p(0, T;Wo^1,p(Ω)). We say that the functionv ∈(L^p(0, T;Wo^1,p(Ω)))^∗ is the weak derivative of the function u∈L^p(0, T;W^1,p(Ω)) if

hv, ϕi=− Z

ΩT

u(∂_tϕ)dx dt,

for all ϕ∈C_o^∞(ΩT), and v is denoted by ∂tu. 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,Ω⊂R^N be a bounded open set, p >1, T >0, and Q≥1. Let u∈L^p(0, T;W^1,p(Ω)) be a parabolic Q-quasiminimizer. Then

∂tu∈(L^p(0, T;W_o^1,p(Ω)))^∗ and

|h∂_tu, ϕi| ≤ 2^pQ

p kuk^p−1_Lp(0,T;W^1,p(Ω))k∇ϕk_Lp(ΩT), for allϕ∈L^p(0, T;W_o^1,p(Ω)).

Proof. Since the desired inequality holds if kuk_Lp(0,T;W^1,p(Ω)) = 0, we can assume without loss of generality that kuk_Lp(0,T;W^1,p(Ω)) 6= 0. Let ϕ ∈ C_o^∞(Ω_T) be a test function satisfying k∇ϕk_Lp(ΩT) = 1, and set φ = kuk_Lp(0,T;W^1,p(Ω))ϕ. Then, by (2.2) we have

kuk_Lp(0,T;W^1,p(Ω))

Z T 0

Z

Ω

u ∂_tϕ dx dt

≥ −Q p

Z

Ω_T

|∇(u− kuk_Lp(0,T;W^1,p(Ω))ϕ)|^pdx dt

≥ −2^p−1Q p

Z

ΩT

(|∇u|^p+kuk^p_Lp(0,T;W^1,p(Ω))|∇ϕ|^p)dx dt

≥ −2^pQ

p kuk^p_{Lp(0,T;W1,p(Ω))}. (3.1)

On the other hand, replacing φby−φin (2.2), we obtain (3.2) kuk_Lp(0,T;W^1,p(Ω))

Z T 0

Z

Ω

u∂_tϕ dx dt≤ 2^pQ

p kuk^p_{Lp(0,T;W1,p(Ω))}. Since k∇ϕk_Lp(ΩT) = 1, by (3.1) and (3.2) we have

(3.3)

Z T 0

Z

Ω

u ∂_tϕ dx dt

≤ 2^pQ p kuk^p−1

L^p(0,T;W^1,p(Ω))k∇ϕk_Lp(ΩT),

for allϕ∈C_o^∞(Ω_T). SinceC_o^∞(Ω_T) is dense inL^p(0, T;Wo^1,p(Ω)), by (3.3) we obtain∂_tu∈(L^p(0, T;Wo^1,p(Ω)))^∗and the desired inequality. This completes the proof. 2

Next we consider a regularization of quasiminimizers with respect to the time variable. Letφ∈C_o^∞(Ω_T) with suppφ=Ωe×[t₁, t₂]bΩ_T. For 0< ε <

min{t₁, T −t2}, we take a mollifierξε∈C_o^∞(R) satisfying suppξε⊂[−ε, ε], ξ_ε≥0, andkξ_εk_Lp(R) = 1 and denote

(3.4) [φ]ε(x, t) :=

Z ε

−ε

φ(x, t−s)ξε(s)ds in ΩT.

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

Z

ΩT

(∂t[u]ε)φ dx dt+1 p

Z

supp[φ]ε

|∇u|^pdx dt

≤ Q p

Z

supp[φ]ε

|∇(u−[φ]_ε)|^pdx dt, (3.5)

for allφ∈C_o^∞(ΩT). In the above inequality [u]εdenotes the smoothing ofu according to (3.4). By Lemma 2, Corollary 1 and Remark 2 in [16] we know that, for a functionψ ∈L^p_c(0, T;W_o^1,p(Ω)) and for any ε >0, there exists a

functionϕ∈Lip_c(ΩT) and hence also a functionϕ∈C_o^∞(ΩT) such that kψ−ϕk_Lp(0,T;W^1,p(Ω)) < ε, kψ−ϕk_L2(ΩT)< ε,

and |suppϕ\suppψ|< ε.

Using this density result, it is straightforward to show that inequality (3.5) also holds for anyφ∈L^p_c(0, T;W_o^1,p(Ω)).

On the other hand, since kξ_εk_Lp(R^N) = 1, by the H¨older inequality and the Fubini theorem we have

Z _T

0

Z

Ω

[φ]_ε(x, t)

pdx dt≤(2ε)^p−1 Z _T

0

Z

Ω

Z _ε

−ε

φ(x, t−s)

pξ_ε^p(s)ds dx dt

≤ Z ε

−ε

ξ^p_ε(s)hZ T 0

Z

Ω

φ(x, t−s)

pdx dti ds

≤ Z T+ε

−ε

Z

Ω

φ(x, τ)

pdx dτ,

= Z

ΩT

φ(x, t)

pdx dt, (3.6)

and

Z T 0

Z

Ω

|[∇φ]_ε(x, t)|^pdx dt≤ Z

Ω_T

|∇φ(x, t)|^pdx dt

≤ kφk^p_{Lp(0,T;W1,p(Ω))}, (3.7)

for all sufficiently smallε >0. Using these properties, we prove the following lemma.

Lemma 3.2. Let N ≥1, Ω⊂R^N be a bounded open set, p > _N^2N₊₂, T >0 and Q ≥ 1. Let η ∈ W^1,∞(Ω_T) be a function such that η(x, t) = 0 for a.e. (x, t) near t = 0 and t =T. Assume that u ∈ L^p(0, T;W^1,p(Ω)) is a parabolic Q-quasiminimizer. Then ∂_t(ηu)∈(L^p(0, T;Wo^1,p(Ω)))^∗ and

h∂_tv, ηui=−h∂_t(ηu), vi,

for every function v∈L^p(0, T;W_o^1,p(Ω)) with∂_tv∈(L^p(0, T;W_o^1,p(Ω)))^∗. Proof. For simplicity, we set X = L^p(0, T;W^1,p(Ω_T)). Let {η_n}^∞_n=1 ∈ C^∞(Ω_T) be a sequence such that

kη_n−ηk_W1,∞(ΩT)→0 as n→ ∞.

Since ηnϕ∈C_o^∞(ΩT) for all ϕ∈C_o^∞(ΩT), by Lemma 3.1 we have

|h∂_tu, η_nϕi| ≤ 2^pQ

p kuk^p−1_X k∇(η_nϕ)k_Lp(ΩT)

≤ 2^pQ

p kuk^p−1_X kη_nk_W1,∞(ΩT)kϕk_X, (3.8)

for all ϕ∈C_o^∞(ΩT). Moreover, since h∂_tu, ηnϕi=−

Z

ΩT

u(ϕ∂tηn+ηn∂tϕ)dx dt and

Z

ΩT

uϕ(∂tηn)dx dt

≤ k∂_tηnk_L^∞_(ΩT)kuk_L2(ΩT)kϕk_L2(ΩT)

≤C₁k∂_tη_nk_L∞(Ω_T)kuk_Xkϕk_X,

which follows from the continuous embedding W^1,p(Ω) ,→ L²(Ω), as p >

2N/(N + 2), by (3.8) we obtain

Z

ΩT

η_nu(∂_tϕ)dx dt

≤C₁k∂_tη_nk_L∞(ΩT)kuk_Xkϕk_X

+2^pQ

p kuk^p−1_X kη_nk_W1,∞(ΩT)kϕk_X.

Here, C₁ is the constant which depends on N, p and on the domain Ω.

Then, lettingn→ ∞, by the Poincar´e inequality we see that there exists a constantC₂, depending only onp,N, and Qand kηk_W1,∞(ΩT), such that (3.9)

Z

ΩT

ηu(∂_tϕ)dx dt

≤C₂

kuk_X +kuk^p−1_X kϕk_X,

for all ϕ∈C_o^∞(ΩT). This implies that∂t(ηu)∈(L^p(0, T;Wo^1,p(ΩT)))^∗. Let{v_n}^∞_n=1 ⊂C_o^∞(Ω_T) be a sequence such thatkv_n−vk_Lp(0,T;W^1,p(ΩT)) → 0 as n → ∞. Letting ε > 0 be a sufficiently small constant, we have [ηu]_ε ∈ C_o^∞(Ω_T) since η = 0 a.e. near t = 0 and t = T. Then we get by integration by parts that

h∂_tv_n,[ηu]_εi=−hv_n, ∂_t[ηu]_εi, and passing to the limitε→0, by Lemma 3.1 we obtain (3.10) h∂_tvn, ηui=

Z

ΩT

(∂tvn)ηudxdt=−h∂_t(ηu), vni.

Furthermore, by (3.9) we have

|h∂_tv_n−∂_tv, ηui| ≤C₂

kuk_X+kuk^p−1_X

kv_n−vk_X.

This together with Lemma 4.1 and (3.10) yields h∂_tv, ηui=−h∂_t(ηu), vi, and we conclude the proof of Lemma 3.2. 2

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, Ω ⊂ R^N be a bounded open set, and p >

1. Assume that R^N \Ω is uniformly p-thick. Then there exists a positive constant ε such that

W_o^1,p−ε(Ω)∩W^1,p(Ω) =W_o^1,p(Ω).

We also have the following Hardy type estimate, see [1] and [10].

Proposition 3.4. Let N ≥ 1, Ω ⊂ R^N be a bounded domain, and p >

1. Assume that R^N \Ω is uniformly p-thick. Then there exists a positive constant C, depending only on N, p andµ, such that

Z

Ω

|u(x)|

dist(x,R^N \Ω) p

dx≤Ckuk^p_W1,p(Ω),

for allu∈W_o^1,p(Ω).

Furthermore, the following proposition gives a sufficient condition for a function u ∈ W^1,p(Ω) such that u ∈ Wo^1,p(Ω). For the proof of Proposi- tion 3.5, see [4].

Proposition 3.5. Let N ≥ 1, p > 1, and Ω be an open subset of R^N. If u∈W^1,p(Ω) satisfies

Z

Ω

|u(x)|

dist(x,R^N \Ω) p

dx <∞,

thenu∈Wo^1,p(Ω).

3.3. Global higher integrability. One of the fundamental ingredients for our stability proofs is global higher integrability of the gradient∇u on the domain Ω_T, 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 Ω⊂R^N be a bounded open set such that R^N \Ω is uniformly p-thick with parameters µ and r₀. For Q ≥ 1 and g ∈ C¹(Ω_T), let u ∈ L^p(0, T;W^1,p(Ω)) be a parabolic Q- quasiminimizer satisfying (2.3) and (2.4). Then there exists a positive con- stant δ, depending only on N,p,Q, µ, and r₀, such that

u∈L^p+δ(0, T;W^1,p+δ(Ω)).

Furthermore Z

Ω_T

|∇u|^p+δdx dt

is bounded from above by some positive constant depending only onN, p,Q, µ, r₀, δ, g, and k∇uk_Lp(ΩT).

4. Uniform estimate for quasiminizers

In this section we study a uniform estimate of ku_ik_Lpi(0,T;W1,pi(Ω)) 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 >¯ 0, 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∈(0,T)

Z

Ω

|u_i(·, t)−g(·, t)|²dx+ Z

ΩT

|∇u_i|^pidx dt

≤C Z

Ω_T

|∂_tg|^pi/(pi−1)dx dt+ ¯Q Z

Ω_T

|∇g|^pidx dt+ ¯δ Z

Ω_T

|u_i−g|^pidx dt,

for alli∈N. Here Q¯ denotes the upper bound of {Q_i}.

Proof. For 0< s < T and 0< h <min{s/4,(T −s)/2}, we set

χ^h_s(t) :=















0, 0≤t≤h, (t−h)/h, h < t≤2h,

1, 2h < t≤s−2h, (s−t−h)/h, s−2h < t≤s−h,

0, s−h < t≤T.

Then, since the support of the function χ^h_s is compact in (0, T) and by the lateral boundary condition (2.3), we can take the test function

φ^h_ε(x, t) :=χ^h_s([ui]ε−[g]ε)∈Lip_c(0, T;W_o^1,pi(Ω)) in (3.5), and obtain

− Z

ΩT

[u_i]_ε∂_tφ^h_εdx dt+1 p_i

Z

supp[φ^h_ε]ε

|∇u_i|^pidx dt

≤ Q_i pi

Z

supp[φ^h_ε]ε

∇ ui−[φ^h_ε]ε

pi

dx dt, (4.1)

whereε >0 is a sufficiently small constant and [·]_ε is defined by (3.4). Set Z

ΩT

[ui]ε∂tφ^h_εdx dt

= Z

ΩT

([u_i]_ε−[g]_ε)∂_tφ^h_εdx dt+ Z

ΩT

[g]_ε∂_tφ^h_εdx dt

=:I1+I2. (4.2)

By an integration by parts we have I1=

Z

ΩT

|[u_i]ε−[g]ε|²∂tχ^h_s+χ^h_s([ui]ε−[g]ε)∂t([ui]ε−[g]ε) dx dt

= Z

ΩT

|[u_i]_ε−[g]_ε|²∂_tχ^h_sdx dt+1 2

Z

ΩT

χ^h_s∂_t|[u_i]_ε−[g]_ε|²dx dt

= 1 2

Z

ΩT

|[u_i]ε−[g]ε|²∂tχ^h_sdx dt.

Thus, letting ε→0 and thereafter h → 0, we obtain, using also the initial condition (2.4):

I₁−→ −^ε→0 1 2h

Z s−h s−2h

Z

Ω

|u_i(x, t)−g(x, t)|²dx dt + 1

2h Z 2h

h

Z

Ω

|u_i(x, t)−g(x, t)|²dx dt

h→0−→ − Z

Ω

|u_i(·, s)−g(·, s)|²dx, (4.3)

for almost alls∈(0, T).

Furthermore, by integration by parts and the Young inequality, for any δ⁰ >0, we can find a positive constantC_δ⁰, independent of i, such that

I₂=− Z T−h

h

Z

Ω

χ^h_s([u_i]_ε−[g]_ε)∂_t[g]_εdx dt≤ Z

ΩT

|[u_i]_ε−[g]_ε||∂_t[g]_ε|dx dt

≤δ⁰ Z

ΩT

|[u_i]ε−[g]ε|^pidx dt+C_δ⁰ Z

ΩT

|∂_t[g]ε|^pi/(pi−1)dx dt.

Since g∈C¹(ΩT), we obtain (4.4) lim

ε,h→0I2 ≤δ⁰2^p−1¯ Z

ΩT

|u_i−g|^pidx dt+C_δ⁰ Z

ΩT

|∂_tg|^pi/(pi−1)dx dt.

On the other hand, since

ε,h→0lim Z

supp[φ^h_ε]ε

∇ ui−[φ^h_ε]ε

pi

dx dt≤ Z

supp(ui−g)

|∇g|^pi dx dt,

by (4.1)–(4.4) there exists a positive constant C, independent of i and δ⁰, such that

Z

Ω

|u_i(·, s)−g(·, s)|²dx+ Z

ΩT

|∇u_i|^pidx dt

≤p C¯ _δ⁰ Z

ΩT

|∂_tg|^pi/(pi−1)dx dt+C Z

ΩT

|∇g|^pidx dt+Cδ⁰ Z

ΩT

|u_i−g|^pidx dt,

which holds for almost all s∈ (0, T). Here the constant C_δ⁰ depends on ¯p andδ⁰. Therefore, for any ¯δ >0, taking a sufficiently smallδ⁰ >0 satisfying Cδ⁰ <δ¯and passing over to the supremum overs∈(0, T) on the left–hand–

side, we obtain the desired inequality. 2

As a corollary of Lemma 4.1, we obtain a uniform estimate of the norm ku_ik_Lpi(0,T;W1,pi(Ω)).

Corollary 4.2. Assume the same conditions as in Theorem2.1. Then (4.5)

sup

i∈N

sup

t∈(0,T)

Z

Ω

|u_i(·, t)|²dx+ Z

Ω_T

|u_i|^pidx dt+ Z

Ω_T

|∇u_i|^pidx dt

!

<∞.

Furthermore there exists a positive constant δ such that

(4.6) sup

i∈N

Z

ΩT

|u_i|^p+δ+|∇u_i|^p+δ

dx dt <∞.

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 thatg∈C¹(Ω_T). 2

By Corollary 4.2 we can obtain the strong convergence of u_i and weak convergence of the derivatives in L^p+δ(Ω_T), and moreover also in the case p <2 the strong convergence inL².

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(0, T;W^1,p+δ(Ω)) and

u_i →u in L^p+δ(Ω_T)∩L²(Ω_T), (4.7)

∇u_i *∇u weakly in L^p+δ(Ω_T), (4.8)

as i→ ∞. Furthermore ∂tu∈(L^p+δ(0, T;Wo^1,p+δ(Ω)))^∗ and (4.9) ∂tui

* ∂∗ tu in the weak-∗ topology on (L^p+δ(0, T;W_o^1,p+δ(Ω)))^∗ as i→ ∞.

Proof. We can prove (4.7) and (4.8) by the similar argument as in Lemma 3.3 in [9]. Since limi→∞p_i =p and limi→∞Q_i =Q, by Lemma 3.1 there exist positive constantsC andδ, independent of i, such that

|h∂_tu_i, φi| ≤ 2^piQ_i

p_i ku_ik^pi−1

L^pi(0,T;W^1,pi(Ω))k∇φk_Lpi(Ω_T)

≤ 2^piQ_i

p_i ku_ik^pi−1

L^p+δ(0,T;W^1,p+δ(Ω))k∇φk_Lp+δ(ΩT)·(|Ω|T)^{1/pi−1/(p+δ)}

≤Cku_ik^p_Lⁱ_p+δ⁻¹_{(0,T;W1,p+δ(Ω))}k∇φk_Lp+δ(ΩT),

for all φ ∈ C_o^∞(Ω_T) and sufficiently large i ∈ N. This together with (4.6) implies that

(4.10) sup

i∈N

k∂_tuik_(Lp+δ(0,T;W1,p+δ

o (ΩT))^∗<∞.

Then, in view of the Rellich-Kondrachov theorem together with (4.6) and (4.10), by taking a subsequence if necessary we can findue∈L^p+δ(Ω_T) with

∇eu∈L^p+δ(Ω_T) such that

ui →eu in L^p+δ(ΩT),

∇u_i *∇ue weakly in L^p+δ(ΩT),

as i → ∞ (see [20] and [21]). Since ui → u almost everywhere in Ω as i→ ∞, we have ue=u, and obtain (4.7) and (4.8).

Moreover, applying [21, Corollary 8], with the choices X := W^1,p(Ω), B :=L²(Ω) and Y :=W^−1,p0(Ω) = Wo^1,p(Ω)∗

and having the inclusions W^1,p(Ω)⊂L²(Ω)⊂W^−1,p0(Ω),

where the first inclusion is compact, since p > 2N/(N + 2), the uniform bounds (4.6) and (4.10) allow us to conclude also the strong convergence u_i →u inL²(Ω_T) (in fact u_i →u inL^q(0, T;L²(Ω)) for any q <∞).

Next we prove (4.9). Taking a subsequence if necessary, we see that there exists a function v∈(L^p+δ(0, T;Wo^1,p+δ(ΩT))^∗ such that

(4.11) ∂tui

* v∗ in the weak-∗topology on (L^p+δ(0, T;W_o^1,p+δ(ΩT))^∗, asi→ ∞. On the other hand, since it follows from (4.7) that

i→∞limh∂_tui, φi=− lim

i→∞

Z

ΩT

ui∂tφ dx dt=− Z

ΩT

u∂tφ dx dt,

for all φ∈C_o^∞(Ω_T), by (4.11) we have hv, φi= lim

i→∞h∂_tui, φi=− Z

ΩT

u∂tφ dx dt=h∂_tu, φi and

|h∂_tu, φi| ≤ kvk

(L^p+δ(0,T;Wo^1,p+δ(ΩT))^∗k∇φk_Lp+δ(ΩT),

for allφ∈C_o^∞(ΩT). This implies that ∂tu ∈(L^p+δ(0, T;Wo^1,p+δ(ΩT))^∗ and v=∂_tu. Thus we obtain (4.9). 2

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 > 0 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) ∈ W_o^1,p−ε(Ω) for almost all

t∈(0, T), and by Proposition 3.4 we obtain Z

Ω_T

|u_i(x, t)−g(x, t)|

dist(x,R^N \Ω) p−ε

dx dt

≤C₁ Z T

0

ku_i(t)−g(t)k^p−ε_W1,p−ε(Ω)dt

≤C₂ Z T

0

ku_i(t)k^p−ε

W^1,p−ε(Ω)+kgk^p−ε

W^1,p−ε(Ω)

dt.

(5.1)

At this point we use the Caccioppoli type estimate in terms of Lemma 4.1 to control theW^1,p−ε–norm ofui as follows

Z

ΩT

|∇u_i|^p−εdx dt≤ |Ω_T|^{1−(p−ε)/pi} Z

ΩT

|∇u_i|^pidx dt

(p−ε)/p_i

≤ |Ω_T|^{1−(p−ε)/pi}

C Z

Ω_T

|∂_tg|^pi/(pi−1)dx dt+C Z

Ω_T

|∇g|^pidx dt

+ ¯δ Z

ΩT

|u_i−g|^pidx dt

(p−ε)/p_i

≤C|Ω_T|^{1−(p−ε)/pi}

ku_ik^p−ε_Lpi(Ω

T)+k∂_tgk^{(p−ε)/(pi−1)}

L^pi/(pi−1)(ΩT)

+kgk^p−ε

L^pi(0,T;W^1,pi(Ω))

,

for a constant C which depends on N, ¯p and ¯δ and for arbitrary ¯δ > 0.

Combining this with (5.1) and using once again H¨older’s inequality, we arrive at

Z

ΩT

|u_i(x, t)−g(x, t)|

dist(x,R^N \Ω) p−ε

dx dt

≤C

1 +ku_ik^p−ε_Lp+ε(Ω

T)+kgk^p−ε_{Lp+ε(0,T;W1,p+ε(Ω))}+k∂_tgk^{(p−ε)/(pi−1)}

L(p−ε)/(p−ε−1)(ΩT)

. Here we used in the last step also that (p−ε)/(p−ε−1) > p_i/(p_i −1).

The above estimate holds for alli∈N, and C is a positive constant which depends only on N, ¯p, ¯δ and |Ω_T|and blows up as |Ω_T| → ∞. Moreover, the right–hand side of the above inequality is uniformly bounded, sincep_i>

2N/(N + 2)>1 and the uniform bound ofku_ik_Lp+ε(ΩT).

Since u_i →u for almost every (x, t)∈Ω_T, by (4.6), (5.1), and the Fatou lemma we obtain

Z T 0

Z

Ω

|u(x, t)−g(x, t)|

dist(x,R^N \Ω) p−ε

dx dt <∞.

This implies that Z

Ω

|u(x, t)−g(x, t)|

dist(x,R^N\Ω) p−ε

dx <∞,

for almost every t∈ (0, T), and by Proposition 3.5 we obtain u(t)−g(t) ∈ Wo^1,p−ε(Ω) for almost everyt∈(0, T). On the other hand, since R^N \Ω is uniformly p-thick, we can apply Proposition 3.3 to obtain

W_o^1,p−ε0(Ω)∩W^1,p(Ω) =W_o^1,p(Ω),

for some ε⁰ >0. Now, choosing ε small enough, for example ε:= ε⁰/2, we conclude that u(t)−g(t)∈Wo^1,p(Ω) for almost everyt∈(0, T), and obtain (2.7).

In order to prove (2.8), for τ ∈(0, T), we take parametersh, k >0 such that 2h < τ−k < τ and set

χ^h,k_0,τ(t) =















0, 0≤t≤h, (t−h)/h, h < t≤2h,

1, 2h < t≤τ−k, (τ −t)/k, τ−k < t≤τ,

0, τ < t≤T.

Consider the functionχ^h,k_0,τ([u_i]_ε−[g]_ε) in (3.5), and letε→0. Then, by the similar argument as in Lemma 4.1 we obtain

1 k

Z τ τ−k

Z

Ω

|u_i−g|²dx dt− 1 h

Z 2h h

Z

Ω

|u_i−g|²dx dt

≤Csup

i∈N

Z τ o

Z

Ω

|∇u_i|^pi +|u_i−g|^pi+|∂_tg|^pi/(pi−1)+|∇g|^pi dx dt, whereC is a positive constant.

We first pass to the limit h → 0 in the inequality above, and then pass to the limiti→ ∞. Since u_i →u inL^p+δ(Ω_T) and in L²(Ω_T) as i→ ∞ by (4.7) and u_i|_t=0 =g, we obtain

1 k

Z τ τ−k

Z

Ω

|u−g|²dx dt

≤Csup

i∈N

Z τ 0

Z

Ω

|∇u_i|^pi+|u_i−g|^pi+|∂_tg|^pi/(pi−1)+|∇g|^pi dx dt.

(5.2)

Furthermore, since

k→0lim 1 k

Z _τ

τ−k

Z

Ω

|u−g|²dx dt= Z

Ω

|u(x, τ)−g(x, τ)|²dx, by (4.6) and (5.2) we obtain

τ→0lim Z

Ω

|u(x, τ)−g(x, τ)|²dx= 0.

This implies that

h→0lim 1 h

Z h 0

Z

Ω

|u(x, t)−g(x, t)|²dx dt= 0, and we conclude that (2.8) holds.

We next prove (2.9). Fix α >0. Let φ∈C_o^∞(ΩT) and setK = suppφ.

Since K is a compact subset of Ω_T, we can take open sets O₁ and O₂ such thatK bO1bO2 bΩT and

(5.3)

Z

O2\K

|∇u|^p+δdx dt < α,

whereδ >0 is the constant given in Lemma 4.3. Let ε >0 be a sufficiently small constant, and take a test function

ϕi,ε:=φ+η([ui]ε−[u]ε),

where η ∈ C_o^∞(Ω_T) is a cut-off function such that 0 ≤ η ≤ 1, η ≡ 1 in a neighborhood of K, and η ≡0 in Ω_T \O₁. Then, since ϕ_i,ε is a valid test function in (3.5), we obtain

− Z

ΩT

[u_i]_ε∂_t(ϕ_i,ε)dx dt+1 pi

Z

supp[ϕi,ε]ε

|∇u_i|^pidx dt

≤ Q_i pi

Z

supp[ϕi,ε]ε

|∇(u_i−[ϕ_i,ε]_ε)|^pidx dt.

(5.4)

Letβ >0. By (4.8) we have Z

K

|∇u|^p−βdx dt≤lim inf

i→∞

Z

K

|∇u_i|^p−βdx dt

≤lim inf

i→∞

"

Z

K

|∇u_i|^pidx dt

(p−β)/p_i

(|Ω|T)^{1−(p−β)/pi}

#

≤(|Ω|T)^β/p

lim inf

i→∞

Z

K

|∇u_i|^pidx dt

(p−β)/p

.

Since β >0 is arbitrary andK ⊂supp[ϕi,ε]ε for everyi, we obtain (5.5)

Z

K

|∇u|^pdx dt≤lim inf

i→∞

Z

supp[ϕi,ε]ε

|∇u_i|^pidx dt.

Let

I_i,ε:=− Z

ΩT

[u_i]_ε(∂_t(ϕ_i,ε−φ))dx dt=− Z

ΩT

[u_i]_ε∂_t(η[u_i−u]_ε)dx dt.

Then we have

−I_i,ε= Z

ΩT

[u_i−u]_ε∂_t(η[u_i−u]_ε)dx dt+ Z

ΩT

[u]_ε∂_t(η[u_i−u]_ε)dx dt

=:Ji,ε+Ki,ε.

Since η∈C_o^∞(ΩT), we have J_i,ε=

Z

ΩT

(∂_tη)[u_i−u]²_εdx dt+1 2

Z

ΩT

η ∂

∂t[u_i−u]²_εdx dt

= Z

ΩT

(∂tη)[ui−u]²_εdx dt−1 2

Z

ΩT

(∂tη)[ui−u]²_εdx dt

= 1 2

Z

ΩT

(∂_tη)[u_i−u]²_εdx dt, and by (4.7) we obtain

(5.6) lim

i→∞lim

ε→0Ji,ε= 0.

Furthermore, putting Ki,ε=

Z

Ω_T

(∂tη)[u]ε[ui−u]εdx dt+ Z

Ω_T

(∂t[ui−u]ε)η[u]εdx dt

= Z

ΩT

(∂_tη)[u]_ε[u_i−u]_εdx dt

− Z

ΩT

(∂_t[u_i−u]_ε)η(u−[u]_ε)dx dt+ Z

ΩT

(∂_t[u_i−u]_ε)ηu dx dt

=:K_i,ε¹ +K_i,ε² +K_i,ε³ ,

by (3.6), (3.7), and Lemma 3.1 we can find a positive constant C, indepen- dent ofεand i, such that

|K_i,ε¹ | ≤sup

ΩT

|∂_tη| · kuk_L2(ΩT)ku_i−uk_L2(ΩT)

and

|K_i,ε² | ≤Cku_i−uk^p−1_Lp(0,T;W^1,p(Ω))k∇([u]_ε−u)k_Lp(ΩT). By (4.7) we obtain

(5.7) lim

i→∞lim

ε→0 |K_i,ε¹ |+|K_i,ε² |

= 0.

On the other hand, Lemma 3.2 implies that

K_i,ε³ =h∂_t(ηu),[ui−u]εi → h∂_t(ηu), ui−ui=−h∂_t(ui−u), ηui, asε→0 and, consequently, by (4.9) we have

i→∞lim lim

ε→0K_i,ε³ = 0.

This together with (5.6) and (5.7) implies that

(5.8) lim

i→∞lim

ε→0− Z

Ω_T

[ui]ε∂tϕi,εdx dt=− Z

Ω_T

u∂tφ dx dt.

Therefore, lettingε→0 in (5.4), by (5.5) and (5.8) we obtain

− Z

ΩT

u∂tφ dx dt+o(1) + 1 p

Z

K

|∇u|^pdx dt

≤ Q_i pi

Z

O1

|∇(u_i−φ−η(u_i−u))|^pidx dt, (5.9)

for all sufficiently large i∈N. By the definition of η and K we haveη ≡1 in a neighborhood ofK andφ≡0 in ΩT \K, and obtain

Z

O1

|∇(u_i−φ−η(ui−u))|^pidx dt

= Z

K

|∇(u−φ)|^pidx dt+ Z

O1\K

|∇(u_i−η(ui−u))|^pidx dt.

(5.10)

Since

|∇(u_i−η(ui−u))| ≤(1−η)|∇u_i|+|∇η||u_i−u|+η|∇u|

inO₁\K, we have Z

O1\K

|∇(u_i−η(ui−u))|^pidx dt

≤C₁ Z

O1\K

((1−η)^pi|∇u_i|^pi+|∇η|^pi|u_i−u|^pi+η^pi|∇u|^pi) dx dt, (5.11)

where C₁ is a positive constant independent of i. By the H¨older inequality we have

Z

O1\K

η^pi|∇u|^pidx dt≤ Z

O1\K

|∇u|^p+δdx dt

!pi/(p+δ)

|O₁\K|^{1−pi/(p+δ)}.

This together with (5.3) implies that there exists a positive constantC2 such that

(5.12) lim sup

i→∞

Z

O1\K

η^pi|∇u|^pidx dt < C2α^p/(p+δ).

Furthermore, by the H¨older inequality and (4.7) there exists a positive con- stantC such that

(5.13)

Z

O1\K

|∇η|^pi|u_i−u|^pidx dt≤Cku_i−uk^p/(p+δ)

L^p+δ(ΩT) →0,

asi→ ∞. Note that the constant C in the above estimate depends also on

∇η and therefore also onφ, but it is independent of i.

In order to finish the proof of (2.9), it remains to estimate the quantity

i→∞lim Z

O1\K

(1−η)^pi|∇u_i|^pidz.

For this aim we modify the arguments of [7] and [14], and prove the following auxiliary result.