We assume that the complement of the domain satisfies a uniform capacity density condition, which is essentially sharp for higher integrability results

PARABOLIC SYSTEMS NEAR THE BOUNDARY

VERENA B ¨OGELEIN^†AND MIKKO PARVIAINEN^∗

ABSTRACT. We establish global regularity results for a wide class of non-linear higher order parabolic systems. The model problem we have in mind is the parabolic p-Laplacian system of order 2m, m≥1,

∂tu+ (−1)^mdiv^m¡

|D^mu|^p−2D^mu¢

=0

with prescribed boundary and initial values. We prove that if the boundary values are sufficiently regular, then D^mu is globally integrable to a better power than the natural p. The method also produces a global estimate.

1. INTRODUCTION

We study the global regularity properties of solutions to a wide class of non- linear higher order parabolic systems. In particular, the parabolic p-Laplacian sys- tem of order 2m, m≥1,

∂tu+ (−1)^mdiv^m¡

|D^mu|^p−2D^mu¢

=0 with the initial and boundary values provides a basic example.

Under suitable conditions on the initial and boundary values, the corresponding initial boundary value problem admits a solution u such that |D^mu|is integrable to the power p. Our aim is to show that D^mu is actually globally integrable to a better power, that is, |D^mu| ∈L^p+ε all the way up to the boundary provided that the boundary values and the domain are sufficiently smooth. We assume that the complement of the domain satisfies a uniform capacity density condition, which is essentially sharp for higher integrability results. Moreover, the method produces an explicit estimate for the L^p+ε-norm of D^mu.

Higher integrability plays an important role in stability and partial regularity results for solutions and gradients in both the elliptic and parabolic cases. For elliptic regularity results with the standard and also non-standard growth condi- tions, see for example Acerbi-Mingione [1, 2, 3, 28]. For recent parabolic appli- cations of higher integrability in the framework of partial regularity and Calder´on- Zygmund type estimates, see for example Acerbi-Mingione [4], Acerbi-Mingione- Seregin [5], B¨ogelein [10], Duzaar-Mingione [16], Duzaar-Mingione-Steffen [17]

and B¨ogelein-Duzaar-Mingione [11].

Date: July 8, 2009.

2000 Mathematics Subject Classification. 35D10, 35G30, 35K50, 35K65.

Key words and phrases. global higher integrability, degenerate parabolic systems, higher order parabolic p-Laplacian.

†Supported by ESF Short Visit Grant.

∗Supported by The Emil Aaltonen Foundation, The Fulbright Center, and The Magnus Ehrnrooth Foundation.

The elliptic higher integrability techniques developed by Gehring [20], and El- crat and Meyers [18] (see also [21]) could not directly be carried over to the par- abolic case. Nevertheless, Giaquinta and Struwe proved a first parabolic analogue for systems with linear growth in [22]. Higher integrability for more general para- bolic systems with non-linear growth conditions remained open for some time: The first positive result for degenerate and singular second order parabolic p-growth systems was obtained by Kinnunen and Lewis [26]. The proof employs the method of intrinsic scaling with respect to the gradient of the solution. The idea to con- sider parabolic cylinders whose scaling depends on the solution itself goes back to DiBenedetto and Friedman [12, 13, 14]. The local higher integrability result was recently extended to higher order parabolic systems in [9], and global higher integrability results for quasiminimizers and second order parabolic systems were obtained in [35, 36, 37].

Our basic strategy follows the guidelines of the local result in [26]. Indeed, we first derive a reverse H¨older inequality on intrinsic cylinders up to the boundary and then use a covering argument to extend the estimates to the whole space-time cylinder. The intuitive idea is to use cylinders whose space-time scaling is roughly speaking comparable to the mean value of |D^mu|^2−p on the same cylinder. In a certain sense this space-time scaling reflects the non-homogeneity of the parabolic system, which is not present in the elliptic case. However, the boundary effects and lower order terms cause extra difficulties: The covering now consists of three kinds of intrinsic cylinders that may lie near the lateral or initial boundary, or inside the domain.

To estimate the lower order terms near the lateral boundary, we employ a bound- ary version of Poincar´e’s inequality iteratively. This step exploits the uniform ca- pacity density condition of the complement. Near the initial boundary, we compare the solution with the mean value polynomial of the initial values. To this end, the oscillation of weighted means of the solution and lower order derivatives between the different time slices needs to be controlled. For the solution itself, we directly exploit the weak formulation of the parabolic system whereas for the derivatives, we utilize the suitable weighted means.

In the singular case, that is when p<2, the quadratic terms on the right-hand side of the Caccioppoli inequality usually cause technical difficulties. Therefore, we employ an iteration method in order to absorb these terms at an early stage (c.f.

Lemma 4.3 and 5.5). In this way, we later avoid additional terms in the scaling which simplifies the proof considerably. Indeed, practically the same proof now runs in both the singular and degenerate cases. This observation is useful even in the local second order higher integrability proof.

2. STATEMENT OF THE RESULT

We consider initial-boundary value problems of the type (2.1)





∂tu+ (−1)^m

∑

|α|=m

D^αA_α(z,D^mu) =0, in ΩT,

u=g, on∂pΩT.

Here, Ω is a bounded domain in Rⁿ and ΩT =Ω×(0,T)⊂Rⁿ⁺¹ stands for a parabolic cylinder. The initial and lateral boundary values g of the solution are prescribed on the parabolic boundary ∂pΩT = (Ω× {0})∪(∂Ω×(0,T))ofΩT.

Moreover, u : ΩT →R^N is a vector valued function and, as usual, we denote by

∂tu=ut the derivative with respect to the time-variable t and by Du, respectively D^ku={D^αu_i}^|_i=1,...,N^α|=k , the derivatives (of order k) with respect to the space-variable x. For convenience of notation, we identify D^mu as a vector in R<sup>`</sup>, `=N¡_n+m−1

m

¢, and similarly for D^ku. Furthermore, we adopt the shorthand notation z= (x,t)∈ Rⁿ⁺¹.

For simplicity of notation, we writeA ={A_α}_|α|=m, whereA_α:ΩT×R^{`</sup>→ R<sup>N</sup>, and thusA :ΩT×R<sup>`}→R^`. We assume thatA is a Carath´eodory function:

z7→A(z,w)is measurable for every w∈R^`, w7→A(z,w)is continuous for almost every z∈ΩT, and satisfies the following p-growth conditions:

hA(z,w),wi ≥ν|w|^p, (2.2)

|A(z,w)| ≤L¡

|w|^p−1+1¢ (2.3) ,

for all z ∈ΩT, w∈ R^` and some constants 0 <ν ≤1 and 1 ≤L <∞ and p>max{1,_n+2m²ⁿ }. Above we have made several simplifications for expository reasons: we could add an inhomogeneity with controlled growth conditions into the right-hand side of (2.1) as well as additional functions to the growth bounds, cf. [9]. Nevertheless, the proofs would remain virtually the same. The restriction p>max{1,_n+2m²ⁿ }is necessary in the parabolic framework because of the Sobolev embedding W^m,n+2m2n ,→L²as we have to deal with the L²-norm of u appearing in Caccioppoli’s inequality.

There will naturally appear several exponents throughout the paper. Set 2∗=max{1,_n+2m²ⁿ }and p∗=max{1,_n+2m^pn },

and observe that when m=1, we simply obtain the usual Sobolev exponents. We will be able to combine the degenerate and singular cases by defining

pb=max{2,p}, pb∗=max{2_∗,p∗}and pb⁰=min{2,_p−1^p }.

Next we define the space V_β^p(0,T ;Ω) for the initial and boundary values. For β ≥0, we denote

V_β^p(0,T ;Ω) =n

ϕ∈L^p+β(0,T ;W^m,p+β(Ω; R^N))∩W^1,bp0+β(0,T ; L^bp0+β(Ω; R^N))

∩C([0,T); L²(Ω; R^N)):ϕ(·,0)∈W^m,bp∗+β(Ω; R^N) o

. The role of the continuity assumption is to fix the right representative. Observe that even smooth boundary values lead to a nontrivial theory. Next we specify the notion of a global solution.

Definition 2.1. Let p>2_∗. A function u∈L^p(0,T ;W^m,p(Ω; R^N))∩C([0,T); L²(Ω;

R^N))is a global (weak) solution to the initial-boundary value problem (2.1) if (2.4)

Z

ΩT

u·ϕt− hA(z,D^mu),D^mϕidz=0 for every test-functionϕ∈C₀^∞(ΩT; R^N)and, moreover,

(2.5) (u−g)(·,t)∈W₀^m,p(Ω; R^N) for almost every t∈(0,T)

and

(2.6) 1

h Z _h

0

Z

Ω|u(x,t)−g(x,0)|²dx dt→0 as h↓0 for a given function g∈V₀^p(0,T ;Ω).

Note that the space L^p(0,T ;W^m,p(Ω; R^N))∩C([0,T); L²(Ω; R^N))seems natural in the light of the existence theorems (see Lions [30] and Showalter [38] Chap- ter III, Proposition 1.2).

We work on the parabolic cylinders of the form

Qz0(ρ,s) =Bx0(ρ)×Λt0(s)⊂Rⁿ⁺¹,

where z₀= (x₀,t₀)∈Rⁿ⁺¹,ρ,s>0 and B_x0(ρ)denotes the open ball in Rⁿwith center x0and radiusρ and

Λt0(s) = (t₀−s,t₀+s)

the interval of length 2s centered at t0. In the case s=ρ^2m, we write Qz0(ρ) = Qz0(ρ,ρ^2m). When no confusion arises, we shall omit the reference points. Fur- thermore, we write

αB(ρ) =B(αρ), αΛ(s) =Λ(α^2ms),andαQ(ρ,s) =Q(αρ,α^2ms), for a ball, interval, and cylinder enlarged by the factorα >0.

Next we state our main theorem. The global higher integrability is achieved under the assumption that the complement of the domainΩsatisfies a uniform ca- pacity density condition. This regularity condition guarantees that there is “enough of complement” near every boundary point. The capacity density condition could be replaced for example by the stronger measure density condition. For the precise formulation of the condition, see Definition 3.1.

Theorem 2.2. Suppose that u is a global solution according to Definition 2.1 with boundary and initial data g∈V_β^p(0,T ;Ω)for someβ >0 and let Rⁿ\Ωbe uni- formly p-thick. Then there existsε=ε(n,N,m,p,L/ν)∈(0,β]such that

u∈L^p+ε(0,T ;W^m,p+ε(Ω; R^N)).

Moreover, for any parabolic cylinder Q0=B0×Λ0=Qz0(R,R²)⊂Rⁿ⁺¹, we have the following boundary estimate

1

|Q₀| Z

1 4Q0∩ΩT

|D^mu|^p+ε dz≤ µ c

|Q₀| Z

Q0∩ΩT

¡|D^mu|^p+G^p¢ dz

¶_(ε+d)/d

+ c

|Q₀| Z

Q0∩ΩT

G^p+εdz+c +

µcδ1

|B₀| Z

B0∩Ω|D^mg(·,0)|^pb∗+ε dx

¶_{(bp+ε)/(bp∗+ε)} . where c=c(n,N,m,p,L/ν)andδ1=1 if 0∈Λ0andδ1=0 otherwise. Here, we have denoted

(2.7) G=¡

|D^mg|^p+|∂τg|^pb0¢_1/p

if B0\Ω6=/0 and G=0 otherwise and

d=

( 2 if p≥2,

p−n(2−p)

2m if 2_∗<p<2.

3. PRELIMINARIES

3.1. Variational p-capacity. Let 1<p<∞andObe an open set. The variational p-capacity of a compact set C⊂O is defined to be

cap_p(C,O) =inf

f

Z

O|∇f|^pdx,

where the infimum is taken over all the functions f ∈C₀^∞(O)such that f=1 in C.

To define the variational p-capacity of an open set U⊂O, we take the supremum over the capacities of the compact sets belonging to U . The variational p-capacity of an arbitrary set E⊂O is defined by taking the infimum over the capacities of the open sets containing E.

For the capacity of a ball, we have

(3.1) cap_p(B(ρ),B(2ρ)) =cρ^n−p.

For further details, see Chapter 4 of Evans-Gariepy [19], Chapter 2 of Heinonen- Kilpel¨ainen-Martio [24], or Chapter 2 of Mal´y-Ziemer [31].

Next we introduce the uniform capacity density condition, which allows us to use a boundary version of a Sobolev-Poincar´e type inequality. This condition is essentially sharp for our main result as shown by Kilpel¨ainen-Koskela [25] in the elliptic case and in [27] for the second order parabolic p-Laplace equation.

Definition 3.1. A set E⊂Rⁿis uniformly p-thick if there exist constantsµ,ρ0>0 such that

cap_p(E∩B_x(ρ),B_x(2ρ))≥µ cap_p(B_x(ρ),B_x(2ρ)), for all x∈E and for all 0<ρ<ρ0.

If p>n, the condition is superfluous since then every set is uniformly p-thick.

The next lemma slightly extends the capacity estimate from the above definition (cf. [36], Lemma 3.8).

Lemma 3.2. LetΩbe a bounded open set in Rⁿ and suppose that Rⁿ\Ωis uni- formly p-thick. Choose y∈Ωsuch that By(4ρ/3)\Ω6= /0. Then there exists a constant ˜µ =µ˜(µ,ρ0,n,p)>0 such that

cap_p¡

By(2ρ)\Ω,By(4ρ)¢

≥µ˜ cap_p¡

By(2ρ),By(4ρ)¢ .

A uniformly q-thick set is also uniformlyϑ-thick for allϑ≥q. This is a conse- quence of H¨older’s and Young’s inequalities.

Lemma 3.3. If a compact set E is uniformly q-thick, then E is uniformlyϑ-thick for allϑ ≥q.

The next theorem states that a uniformly p-thick set has a self-improving prop- erty. This result was shown by Lewis in Theorem 1 of [29]. See also Ancona [7]

and Mikkonen [33].

Theorem 3.4. Let 1<p≤n. If a set E is uniformly p-thick, then there exists γ=γ(n,p,µ)∈(1,p)for which E is uniformlyγ-thick.

Next, we formulate a well-known version of the Sobolev-type inequality. For the proof, see Chapter 10 of Maz’ja’s monograph [32] or Hedberg [23] and also [36]. Later we combine this estimate with the boundary regularity condition and obtain a boundary version of Sobolev’s inequality.

The lemma employs quasicontinuous representatives of Sobolev functions. We call u∈W^1,p(Ω) p-quasicontinuous if for eachε >0 there exists an open set U , U⊂Ω⊂B_R⁰, such that cap_p(U,B_2R⁰)≤ε, and the restriction of u to the setΩ\ U is finite valued and continuous. The p-quasicontinuous functions are closely related to the Sobolev space W^1,p(Ω): For example, if u∈W^1,p(Ω), then u has a

p-quasicontinuous representative.

Adopting the usual notation for the mean value integral Z

B(ρ)|u|^qdx= 1

|B(ρ)|

Z

B(ρ)|u|^qdx.

we have the following

Lemma 3.5. Let B=B(ρ) be a ball in Rⁿ and suppose that u∈W^1,q(B) is q- quasicontinuous. Denote

N_B(ρ/2)(u) ={x∈B(ρ/2): u(x) =0}.

Then there exists a constant c=c(n,q)>0 such that Z

B(ρ)|u|^qdx≤ c

cap_q(N_B(ρ/2)(u),B(ρ)) Z

B(ρ)|Du|^qdx.

3.2. Interpolation estimates. When dealing with higher order problems, inter- polation estimates play an essential role. In several points, particularly when Poincar´e’s inequality cannot be applied they shall help us to treat the intermediate derivatives. First, we provide an interpolation estimate for intermediate derivatives on the annulus, cf. Adams [6], Theorem 4.14 or [8], Lemma B.1. Note that the crucial point here is the right dependence on the width of the annulus.

Lemma 3.6. Let B(r₁)⊂B(r₂)be two concentric balls in Rⁿwith 0<r1<r2≤1 and let u∈W^m,p(B(r₂))with p≥1. Then for any 0≤k≤m−1 and 0<ε≤1 there exists c=c(n,m,p,1/ε), such that

Z

B(r2)\B(r1)

|D^ku|^p

(r₂−r₁)^(m−k)pdx≤ε Z

B(r2)\B(r1)|D^mu|^pdx+c Z

B(r2)\B(r1)

|u|^p (r₂−r₁)^mpdx.

One of the difficulties in proving the main result is the fact that both powers 2 and p play a role in Caccioppoli’s inequality. We now state Gagliardo-Nirenberg- Sobolev’s inequality (see Nirenberg [34]) in a form, which helps us to combine the different powers.

Theorem 3.7. Let B(ρ) be a ball in Rⁿ and u∈W^m,q(B(ρ)), m∈N and 1≤ σ,q,r≤∞andθ ∈(0,1)and 0≤k≤m−1 with k−_σⁿ ≤θ(m−ⁿ_q)−(1−θ)ⁿ_r. Then there exists c=c(n,m,σ)such that

Z

B(ρ)

¯¯

¯ D^ku ρ^m−k

¯¯

¯^σdx≤c

³ m

∑

Z

B(ρ)

¯¯

¯ D^ju ρ^m−j

¯¯

¯^qdx

´_θσ/q³^Z

B(ρ)

¯¯

¯ u ρ^m

¯¯

¯^rdx

´_(1−θ)σ/r . The following lemma will help us to absorb certain integrals into the left-hand side. The proof employs a standard iteration argument, see for instance Giaquinta’s monograph [21], Chapter V, Lemma 3.1.

Lemma 3.8. Let 0<ϑ <1, A,B≥0,α >0 and let f ≥0 be a bounded function satisfying

f(t)≤ϑf(s) +A(s−t)^−α+B for all 0<r≤t<s≤ρ.

Then there exists a constant c_tech=c_tech(α,ϑ), such that f(r)≤ctech

¡A(ρ−r)^−α+B¢ .

3.3. Mean value polynomials. In order to prove a higher integrability result for the m-th derivative of u, we shall approximate the function up to order m−1. For this aim, we shall employ mean value polynomials of order m−1. Let Bx0(r)be a ball in Rⁿand f ∈W^m,1(B_x0(r); R^N). Then its mean value polynomial P_r^(f): Rⁿ→ R^N of degree≤m−1 is defined uniquely by the condition

(3.2) (δPr^(f))_x0;r= (δf)_x0;r,

whereδu= (u,Du, . . . ,D^m−1u)denotes the vector of lower order derivatives and (f)_x0;r=

Z

B_x0(r)f dz

denotes the mean-value of f on Bx0(r). Therefore, (3.2) can be rewritten as (D^kP_r^(f))_x0;r= (D^kf)_x0;r for k=0, . . . ,m−1. The mean value polynomial can be expressed in terms of the mean values of f as

P_r^(f)(x) =

∑

|α|≤m−1

∑

|α+β|≤m−1

b_β

α!(D^α+βf)_x0;r(x−x₀)^α, where

b_β =





1, if|β|=0

−

∑

0<γ≤β

b_β−γ γ!

Z

B_x0(r)(y−x₀)^γdy, if|β| ≥1.

For more details, see for instance Duzaar-Gastel-Grotowski [15].

Due to the defining property of Pr^(f), we can replace in the above representation (D^α+βf)_x0;rby(D^α+βPr^(f))_x0;r. Moreover, we can show that|b_β| ≤c(n,m)r^|β|for all multi-indicesβ with 0≤ |β| ≤m−1. This observation leads us to the estimate (3.3) |P(x)| ≤c(n,m)^m−1

∑

k=0

R^k|(D^kP)_x0;r| for all x∈B_x0(R),

valid for any polynomial P : Rⁿ→R^Nof order≤m−1 and balls B_x0(r), B_x0(R)in Rⁿwith 0<r≤R. See [8], Lemma A.1, for a more detailed proof.

From this estimate, we can deduce a bound for the difference of the mean value polynomials on two different balls. The proof applies the definition of the mean value polynomials together with Poincar´e’s inequality.

Lemma 3.9. Let B_x0(r), B_x0(R) be two balls in Rⁿ with R/2≤r <R and sup- pose that f ∈W^m,1(B_x0(R); R^N). Denote by P_r^(f),P_R^(f): Rⁿ→R^N the mean value polynomials of f of degree≤m−1. Then there exists c=c(n,N,m)such that

|P_r^(f)(x)−P_R^(f)(x)| ≤c R^m Z

B_x0(R)|D^mf|dx for all x∈Bx0(R).

Proof. To estimate the difference of the polynomials, we use (3.3) with (P_r^(f)− P_R^(f))instead of P and exploit the defining property of the polynomial Pr^(f)to find

|(P_r^(f)−P_R^(f))(x)| ≤c

m−1

∑

k=0

R^k

¯¯

¯ Z

B_x0(r)D^k(P_r^(f)−P_R^(f))dy

¯¯

¯

=c

m−1

∑

k=0

R^k

¯¯

¯ Z

B_x0(r)D^k(f−P_R^(f))dy

¯¯

¯.

Next we enlarge the domain of integration in the integrals on the right-hand side and recall that|B_x0(R)|/|B_x0(r)| ≤2ⁿ since R/2≤r. Finally, applying Poincar´e’s inequality m−k times to D^k(f−P_R^(f))which is allowed since(D^k(f−P_R^(f)))_x0,r=0 leads us to

|(P_r^(f)−P_R^(f))(x)| ≤c

m−1

∑

k=0

R^k Z

B_x0(R)|D^k(f−P_R^(f))|dx≤c R^m Z

B_x0(R)|D^mf|dx.

This is the desired estimate. ¤

3.4. Steklov-means. Since weak solutions do not a priori possess any differentia- bility properties with respect to the time variable t, it is standard to use a mollifica- tion in time. Therefore, given a function f ∈L¹(ΩT), we define its Steklov-mean by

f_h(x,t) = [f]_h(x,t) =



 1 h

Z _t+h

t

f(x,s)ds, t∈(0,T−h), 0, t∈(T−h,T),

for 0<h<T and(x,t)∈ΩT. Using Steklov-means, we get for a.e. t∈(0,T)an equivalent system:

(3.4)

Z

Ω∂tuh(·,t)·ϕ+­

[A(·,D^mu)]_h(·,t),D^mϕ® dx=0, for allϕ∈L²(Ω; R^N)∩W₀^m,p(Ω; R^N).

4. ESTIMATES NEAR THE LATERAL BOUNDARY

In this chapter, we derive estimates on parabolic cylinders lying near the lateral boundary∂Ω×(0,T). For notational convenience, in this chapter we will combine the boundary terms and the constant coming from the growth bounds as follows

Ge=¡

|D^mg|^p+|∂τg|^pb0¢_1/p +1.

Since we now are in the lateral boundary situation we haveGe=G+1, where G is from (2.7). As usual, the first step when proving higher integrability is to derive suitable Caccioppoli’s inequality. Although we state it for arbitrary cylinders in Rⁿ⁺¹, it will be needed later only for cylinders intersecting the lateral boundary.

Lemma 4.1. Let u be a global solution according to Definition 2.1. Then there exists cCac = cCac(n,m,p,L/ν) such that for all parabolic cylinders Qz0(r,s), Qz0(R,S)⊂Rⁿ⁺¹ with 0<R/2≤r <R≤1, s=λ^2−pr^2m, S=λ^2−pR^2m ≤1, λ >0 there holds

t∈Λ_t0sup(s)∩(0,T)

Z

B_x0(r)∩Ω|(u−g)(·,t)|²dx+ Z

Q_z0(r,s)∩ΩT

|D^mu|^pdz

≤c_Cac Z

Q_z0(R,S)∩ΩT

λ^p−2¯¯

¯ u−g (R−r)^m

¯¯

¯²+

¯¯

¯ u−g (R−r)^m

¯¯

¯^p+Ge^pdz.

Proof. We choose r ≤r₁<r₂ ≤R and η ∈C^∞₀(B_x0(r₂)), ζ ∈C¹(R) to be two cut-off functions with

(4.1)

( η≡1 in Bx0(r₁),0≤η≤1,|D^kη| ≤_(r ^cη

2−r1)^k for all 0≤k≤m;

ζ ≡0 on(−∞,t0−S),ζ ≡1 on(t₀−s,∞),0≤ζ ≤1, 0≤ζ⁰≤_S−s² . Choosing the test-functionϕh=ηζ²(u_h−gh)in the Steklov-formulation (3.4) and integrating with respect toτover(0,t), we get for t∈(0,T)

(4.2)

Z

Ωt

∂τu_h·ϕh+­

[A(·,D^mu)]_h,D^mϕh

®dz=0,

where we abbreviatedΩt =Ω×(0,t). For the first term on the left-hand side, we find that_Z

Ωt

∂τuh·ϕhdz= Z

Ωt

∂τ(u_h−gh)·ϕh+∂τgh·ϕhdz

→ ¹₂ Z

Ω|(u−g)(·,t)|²ηζ(t)²dx− Z

Ωt

|u−g|²ηζζ⁰dz+ Z

Ωt

∂τg·(u−g)ηζ²dz, as h ↓0. Here we have also taken into account that the initial boundary term vanishes at τ =0 because of the initial condition. The last integral on the right- hand side is now further estimated with the help of Young’s inequality with ex- ponents (2,2) if p<2, respectively (p,p/(p−1))when p≥2. Note also that r2−r1≤R≤1, respectivelyλ^2−p(r₂−r1)^2m≤S≤1. We get

¯¯

¯ Z

Ωt

∂τg·(u−g)ηζ²dz

¯¯

¯≤ Z

Q_z0(R,S)∩ΩT

|∂τg|^bp0+λ^p−2 |u−g|²

(r₂−r₁)^2m+ |u−g|^p (r₂−r₁)^mpdz.

Passing to the limit h↓0 also in the second term on the right side of (4.2), we find Z

Ωt

­[A(·,D^mu)]_h,D^mϕh

®dz

→ Z

Ωt

hA(·,D^mu),D^muiηζ²− hA(·,D^mu),D^mgiηζ²+hA(·,D^mu),LOTiζ²dz, where we abbreviated the lower order terms by

LOT=

m−1

∑

k=0

µm k

¶

D^m−kη¯D^k(u−g).

From the ellipticity (2.2) ofA, we infer for the first term that Z

Ωt

hA(·,D^mu),D^muiηζ²dz≥ν Z

Ωt

|D^mu|^pηζ²dz,

while for the second one, we obtain by the growth bound (2.3) ofA and Young’s inequality forε>0 that

¯¯

¯ Z

Ωt

hA(·,D^mu),D^mgiηζ²dz

¯¯

¯≤ε Z

Ωt

¡|D^mu|^p+1¢

ηζ²dz+c_ε Z

Ωt

|D^mg|^pdz, where c_ε=c_ε(p,L,1/ε). Similarly, for the third term, we get

¯¯

¯ Z

Ωt

hA(·,D^mu),LOTiζ²dz

¯¯

¯≤ε Z

Ωt∩sptη

¡|D^mu|^p+1¢

ζ²dz+c_ε Z

Ωt

|LOT|^pζ²dz, where c_ε =c_ε(p,L,1/ε). To estimate the integral involving the terms of lower or- der, we first note that D^kη=0 on B_x0(r₁)for k≥1. Due to our boundary condition (2.5) we can extend u−g by zero outsideΩT to an L^p−W^m,p function, i.e. for

the extended function we know u−g∈L^p(0,T ;W^m,p(B_x0(r₂); R^N). This allows us to replace the domain of integration by Bx0(r₂)\B_x0(r₁)×(0,t)and then apply the Interpolation-Lemma 3.6 slicewise on the annulus B_x0(r₂)\B_x0(r₁). This yields for 0<ε˜ ≤1 that

Z

Ωt

|LOT|^pζ²dz≤c

m−1

∑

k=0

Z _t

0

Z

B_x0(r2)\B_x0(r1)

|D^k(u−g)|^p (r₂−r₁)^p(m−k)ζ²dz

≤ε˜ Z _t

0

Z

B_x0(r2)\B_x0(r1)|D^m(u−g)|^pζ²dz+cε˜

Z _t

0

Z

B_x0(r2)

|u−g|^p (r₂−r₁)^mpζ²dz, where cε˜=cε˜(n,m,p,1/ε˜).

Combining the previous observations with (4.2), recalling thatη=1 on Bx0(r₁) and choosing ˜ε ¿1 with respect to p,L andε we infer for a.e. t∈(0,T)that

1 2

Z

B_x0(r1)∩Ω|(u−g)(·,t)|²ζ²(t)dx+ν Z _t

0

Z

B_x0(r1)∩Ω|D^mu|^pζ²dx dτ

≤3ε Z _t

0

Z

B_x0(r2)∩Ω|D^mu|^pζ²dz +c

Z

Q_z0(R,S)∩ΩT

λ^p−2 |u−g|²

(r₂−r₁)^2m+ |u−g|^p

(r₂−r₁)^mp+Ge^pdz, where c =c(n,m,p,L,1/ε). Now, we choose ε =ν/6 and multiply with 1/ν. Applying Lemma 3.8, we get rid of the term involving|D^mu|on the right-hand side.

Then, we take the supremum over t ∈Λt0(s)∩(0,T)in the first term on the left- hand side of the resulting inequality and choose t=min{t₀+S,T} in the second term. Finally we recall that ζ ≡1 on Λt0(s) to conclude desired Caccioppoli’s

inequality. ¤

Next, we derive a Poincar´e type inequality for solutions on parabolic cylinders intersecting the lateral boundary∂Ω×(0,T). The capacity density condition and the boundary condition allow us to apply Poincar´e’s inequality slicewise to u−g.

Therefore, in contrast to the local situation, we do not need to compare mean value polynomials between different time slices.

Lemma 4.2. Let u be a global solution according to Definition 2.1 and suppose that Rⁿ\Ωis uniformly p-thick. Furthermore, let Q_z0(ρ,s)⊂Rⁿ⁺¹be a parabolic cylinder such that Bx0(ρ/3)\Ω6= /0. Then there existγ=γ(n,p,µ)∈(1,p)such that for all 0≤k≤m−1 andγ≤ϑ ≤p, we have

Z

Q_z0(ρ,s)∩ΩT

|D^k(u−g)|^ϑdz≤cρ^ϑ(m−k) Z

Q_z0(ρ,s)∩ΩT

|D^mu|^ϑ+G^ϑdz, where c=c(n,m,N,µ,ρ0,ϑ)and G was defined in (2.7).

Proof. Let γ =γ(n,p,µ)∈(1,p) be the constant from Theorem 3.4. Then we know that Rⁿ\Ω is uniformly γ-thick, and therefore also uniformlyϑ-thick by Lemma 3.3. Then we extend u−g by zero outside ofΩT, use the same notation for the extension. We fix k≤ j≤m−1 and t∈Λ(s)∩(0,T)and denote

N_B(^j_ρ/2)={x∈B(ρ/2): D^j(u−g)(x,t) =0}.

From Lemma 3.5, we get (here, we consider for the moment theϑ-quasicontinuous representative of u)

Z

B(ρ)∩Ω|D^j(u−g)(·,t)|^ϑdx= Z

B(ρ)|D^j(u−g)(·,t)|^ϑdx

≤ cρⁿ

cap_ϑ(N_B(^j_ρ/2),B(ρ)) Z

B(ρ)|D^j+1(u−g)(·,t)|^ϑdx, with c=c(n,N,ϑ). Since Rⁿ\Ωis uniformlyϑ-thick, Lemma 3.2 and (3.1) imply

cap_ϑ¡

N_B(^j _ρ/2),B(ρ)¢

≥µ˜ cap_ϑ¡

B(ρ/2),B(ρ)¢

=cρ^n−ϑ.

Note that ˜µ=µ˜(n,µ,ρ0,ϑ). Combining this capacity estimate with the previous one, we conclude

Z

B(ρ)∩Ω|D^j(u−g)(·,t)|^ϑdx≤cρ^ϑ Z

B(ρ)∩Ω|D^j+1(u−g)(·,t)|^ϑdx,

where c=c(n,N,µ,ρ0,ϑ). Integrating with respect to t overΛ(s)∩(0,T)and iter- ating the resulting estimate for j=k, . . . ,m−1, we deduce the following Poincar´e’s inequality

Z

Q(ρ,s)∩ΩT

|D^k(u−g)|^ϑdz≤cρ^ϑ(m−k) Z

Q(ρ,s)∩ΩT

|D^m(u−g)|^ϑdz.

The assertion now follows by Young’s inequality and the definition of G. ¤ Also in the singular case, i.e. when p<2, we will have to estimate the L²- norm of u, since it appears on the right-hand side of Caccioppoli’s inequality in Lemma 4.1. Therefore, we prove a suitable L²-estimate in the following lemma.

This lemma simplifies the proof in the singular case considerably since we absorb the additional terms into the left-hand side. Indeed, due to this lemma, we can apply the same scaling as in the degenerate case.

Lemma 4.3. Letκ≥1, 2_∗<p<2, and u be a global solution according to Defini- tion 2.1. Furthermore, let Q=Qz0(ρ,s)⊂Rⁿ⁺¹with 0<ρ≤1, s=λ^2−pρ^2m≤1, andλ >0 be a parabolic cylinder such that Bx0(2ρ/3)\Ω6=/0. If

1

|2Q|

Z

2Q∩ΩT

¡|D^mu|^p+Ge^p¢

dz≤κ λ^p, (4.3)

then there exists c=c(n,N,m,p,L/ν,µ,ρ0,κ)such that 1

|Q|

Z

Q∩ΩT

|u−g|²dz≤cρ^2mλ^p.

Proof. We first extend u−g by zero outside ofΩT. Next, we choose 1≤α1<

α2 ≤ 2 and denote αiQ =Q_z0(αiρ,α_i^2ms) for i= 1,2. Applying Gagliardo- Nirenberg-Sobolev’s inequality, i.e. Theorem 3.7 with (σ,q,θ,r,k) replaced by (2,p,p/2,2,0)slicewise to(u−g)(·,t), we obtain

Z α1Q∩ΩT

|u−g|²dz= Z

α1Q

|u−g|²dz

≤cρ^mp Z

α1Λ

∑

Z α1B

¯¯

¯D^k(u−g) ρ^m−k

¯¯

¯^pdx

³^Z

α1B

|u−g|²dx

´_(2−p)/2 dt

≤cρ^mp

∑

k=0

Z α1Q

¯¯

¯D^k(u−g) ρ^m−k

¯¯

¯^pdz µ

sup

t∈α1Λ

Z α1B

|(u−g)(·,t)|²dx

¶_(2−p)/2 (4.4) .

To estimate the integrals in the sum on the right-hand side, we recall thatα1Q⊂2Q and u−g=0 on 2Q\ΩT. Therefore, we can replace the domain of integration by 2Q∩ΩT which allows us to apply Poincar´e’s inequality from Lemma 4.2 on 2Q∩ΩT. Finally, using hypothesis (4.3) and the fact that|2Q|=2^n+2m|Q|we infer for 0≤k≤m that

Z α1Q

¯¯

¯D^k(u−g) ρ^m−k

¯¯

¯^pdz≤c Z

2Q∩ΩT

|D^mu|^p+G^pdz≤cλ^p|Q|,

where c=c(n,m,N,µ,ρ0,κ). We now come to the estimate for the sup-term in (4.4). Here, we first apply Caccioppoli’s inequality, i.e. Lemma 4.1. Then we use Young’s inequality, note that p<2, to estimate the second term on the right-hand side and the assumption (4.3) to estimate the term involvingG. Finally, we recalle that|Q|=2ρ^2mλ^2−p|B|. Proceeding this way, we obtain for a.e. t∈α1Λthat

Z α1B

|(u−g)(·,t)|²dx= 1 α₁ⁿ|B|

Z

α1B∩Ω|(u−g)(·,t)|²dx

≤cCac

|B|

Z α2Q∩ΩT

λ^p−2 |u−g|²

(α2ρ−α1ρ)^2m+ |u−g|^p

(α2ρ−α1ρ)^mp+Ge^pdz

≤ c

|B|

Z α2Q∩ΩT

λ^p−2 |u−g|²

(α2ρ−α1ρ)^2m+λ^pdz

= c

|Q|

Z α2Q∩ΩT

|u−g|²

(α2−α1)^2mdz+cρ^2mλ²,

where c=c(n,m,p,L/ν,κ). Joining the previous estimates with (4.4), applying Young’s inequality and recalling that s=λ^2−pρ^2m, we arrive at

Z α1Q∩ΩT

|u−g|²dz≤¹₂ Z

α2Q∩ΩT

|u−g|²dz+ c

(α2−α1)^2m(2−p)/p ρ^2m|Q|λ^p, where c=c(n,N,m,p,L/ν,µ,ρ0,κ). Applying Lemma 3.8, we deduce the desired

estimate. ¤

Now, we have the prerequisites to prove a reverse H¨older type inequality for parabolic cylinders lying near the lateral boundary.

Lemma 4.4. Letκ≥1, and u be a global solution according to Definition 2.1 and suppose that Rⁿ\Ωis uniformly p-thick. Furthermore, let Q=Qz0(ρ,s)⊂Rⁿ⁺¹ with 0<ρ ≤1 and s=λ^2−pρ^2m≤1, λ ≥1 be a parabolic cylinder such that B_x0(4ρ/3)\Ω6=/0. Suppose that

λ^p≤ κ

|Q|

Z

Q∩ΩT

(|D^mu|^p+Ge^p)dz, 1

|8Q|

Z

8Q∩ΩT

(|D^mu|^p+Ge^p)dz≤κ λ^p. (4.5)

and let γ =γ(n,p,µ) be the constant from Lemma 4.2. Then, for any q with max{γ,bp∗} ≤q<p there exists c=c(n,N,m,p,L/ν,µ,ρ0,κ)such that

1

|Q|

Z

Q∩ΩT

|D^mu|^pdz≤ µ c

|4Q|

Z

4Q∩ΩT

|D^mu|^qdz

¶_p/q + c

|4Q|

Z

4Q∩ΩT

Ge^pdz.

Proof. First, we extend u−g by zero outside ofΩT and use the same notation for the extension. From Caccioppoli’s inequality, i.e. Lemma 4.1, we get

1

|Q|

Z

Q∩ΩT

|D^mu|^pdz≤cCac

|Q|

Z

2Q∩ΩT

λ^p−2¯

¯¯u−g ρ^m

¯¯

¯²+

¯¯

¯u−g ρ^m

¯¯

¯^p+Ge^pdz.

(4.6)