Global higher integrability for a doubly nonlinear parabolic system

In this paper we establish a higher integrability result up to the boundary of weak solutions to doubly nonlinear parabolic systems. We show that the spatial gradient of a weak solution with vanishing lateral boundary values is integrable to a larger power than the natural power p, where the statement holds for parameters in the subquadratic case max{2nn+2,1}<p≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \max \lbrace \frac{2n}{n+2}, 1 \rbrace < p \le 2$$\end{document}.


Introduction
This paper is concerned with the global higher integrability of weak solutions to the following Cauchy-Dirichlet problem with vanishing lateral boundary values: (1.1) The prototype is the homogeneous doubly nonlinear equation which is a special case of the fully doubly nonlinear equation The vector field A : Ω T × R N × R Nn → R Nn is a Carathéodory function, meaning it is measurable in Ω T for all (u, ξ) ∈ R N × R Nn and continuous in R N × R Nn for a.e. (x, t) ∈ Ω T . A is required to satisfy the growth and coercivity conditions A(x, t, u, ξ) · ξ ≥ L 1 |ξ| p , |A(x, t, u, ξ)| ≤ L 2 |ξ| p−1 (1.3) in Ω T for 0 < L 1 ≤ L 2 . The parabolic boundary of Ω T is denoted by ∂ par Ω T := Ω × {0} ∪ ∂Ω × (0, T ). In general, p can be a value in (1, ∞). For the right hand side F , one would naturally require that F ∈ L p (Ω T ), while for the initial boundary datum the condition u 0 ∈ W 1,p 0 (Ω) is appropriate. To obtain our higher integrability result, these conditions must be slightly stronger.
The fully doubly nonlinear equation (1.2) shows different behaviour depending on whether m < p − 1, called the slow diffusion case, or m ≥ p − 1, which is termed the fast diffusion case. The case m = p − 1 thus represents the threshold between these cases.
The spatial gradient of a weak solution to the system (1.1) naturally admits the integrability condition Du ∈ L p (Ω T ). The aim of this paper is to show that there exists a constant ε > 0 such that Du ∈ L p(1+ε) (Ω T ). In particular, we want to show that the self-improving property of integrability holds up to the boundary. We cover the range of exponents given by max{ 2n n+2 , 1} < p ≤ 2. The lower bound is analogous to the higher integrability result for parabolic p-Laplace systems [22].
We continue with a historical overview. In [26], Elcrat & Meyers started by observing this self-improving property in the setting of the elliptic p-Laplace systems, based on Gehring [13]. We also refer to [16,Chapter 11, Theorem 1.2] and [20,Sect. 6.5]. The higher integrability has consequently been used to derive further regularity results, see for example [17,18]. Kilpeläinen and Koskela were able to show in [21] that for equations of p-Laplace type this self-improving property holds up to the boundary.
Giaquinta and Struwe were able to expand onto parabolic systems in [19]. In [22], Kinnunen and Lewis treated more general parabolic systems fulfilling a p-growth condition. They continued to successfully consider very weak solution in [23]. The used approach utilises intrinsic cylinders as introduced by DiBenedetto [9][10][11] to compensate for inhomogeneous behaviour of nonlinear parabolic equations. The higher integrability for the parabolic p-Laplacian was expanded onto the boundary by Parviainen in [28,29]. Higher orders are covered in [2,8] by Bögelein and Parviainen. To obtain the result up to the boundary, it has been shown already in [21] that the natural regularity condition for the domain is the uniform p-thickness of the complement of the domain. This is being reaffirmed in [2,8]. Adimurthi and Byun proved global higher integrability even for very weak solution of parabolic p-Laplace equations in [1].
For the porous medium equation, obtained by setting p = 1 in (1.2), Gianazza and Schwarzacher [14] used the technique of expansion of positivity to obtain a higher integrability result in the interior of the domain. To treat 3) The condition u ∈ L p 0, T ; W 1,p 0 (Ω, R N ) also contains the information u ∈ W 1,p 0 (Ω, R n ) for a.e. t ∈ (0, T ), treating the lateral boundary condition, while (2.3) is for the initial boundary. For a center point z 0 = (x 0 , t 0 ) ∈ R n ×R, 55 Page 4 of 51 A. Herán and R. Rainer NoDEA a radius > 0 and a scaling parameter μ > 0, we define the respective spacetime cylinder by Also, a cylinder which contains only parts with positive time is labelled as . This notation will only be used in Sect. 3. For such cylinders and a function v The latter means exist for all times t if the function v is continuous with respect to time. Furthermore, we sometimes write u(t) := u(·, t). The notation v α for the power of a vector with α > 0 is defined by v α := |v| α−1 v for v ∈ R n \{0} and v α = 0 for v = 0. We will require some form of boundary regularity, more specifically the following property concerning the variational p-capacity cap p , which will be inspected later on, see Sect. 2.3.
for all x 0 ∈ E and for all 0 < < 0 .

Auxiliary results
The following mollification in time will be used to prove energy estimates.

Definition 2.4.
For v ∈ L 1 (Ω T , R N ) and h > 0, define the mollification in time by By inserting a reverse analogue of this mollification into the weak formulation, the following mollified system can be deduced. For reference, see for example [3, p.3293] for a proof in the setting of a porous medium type equation.

The weak time derivative ∂ t [[v]] h exists in Ω T and can be computed by
. What follows now is a collection of useful vector inequalities, taken from [4, p.6] 55 Page 6 of 51 A. Herán and R. Rainer NoDEA Lemma 2.7. Let α > 0 and N ∈ N. There exists c = c(α) > 0 such that for any a, b ∈ R N such that For the boundary term defined in (2.1), we require the following estimate [4, Lemma 3.4]: An additional fact we use is the quasi-minimality of the mean-value integral. For the proof we refer to [4,Lemma 3.5].
Moreover we take the following well known Iteration Lemma from [20, Lemma 6.1]. It will be essential in absorbing certain quantities.

p-capacity
Now, we present a selection of properties of the variational p-capacity. It plays a role in the boundary condition that is the uniform p-thickness in Definition 2.2. We state the definition of the capacity from [27]: For 1 < p < ∞ and a compact set D ⊂ R n , the p-capacity of C ⊂ R n is defined by where the infimum is taken over all functions f ∈ C ∞ 0 (D) such that f ≥ 1 in C. The p-capacity of an open set D ⊂ E is defined as the supremum of NoDEA Global higher integrability for a doubly nonlinear As seen for example in [12,Theorem 4.15(iv)], the capacity of a ball can be computed as with c(n, p) > 0. We continue by observing properties of the uniform pthickness.
Lemma 2.11. Let 1 < p < ∞ and E ⊆ R n compact and uniformly p-thick. Then E is also uniformlyp-thick for allp ≥ p.
The uniform thickness admits a self-improving property which was proven in [25] and is as follows: Theorem 2.12. Let 1 < p ≤ n and E ⊆ R n uniformly p-thick. There exists q = q(n, p, ν) ∈ (1, p) such that E is uniformly q-thick.
In the case p > n, every non-empty set is uniformly p-thick, see [28, p.340]. We conclude this section with the following Lemma, taken from [28, Lemma 3.8].

Energy estimate
Recall that in this section, the notation Q r,s (z 0 ) := B r (x 0 ) × (t 0 − s, t 0 + s) is used. We start with an energy estimate which is valid even outside of the spatial region Ω. It thus takes care of the lateral case, but also of cylinders that intersect both the lateral and the initial boundary.

Lateral boundary
Proof. We may assume, Q r,s (z 0 ) ∩ Ω T = ∅, since otherwise the estimate is obvious. The first step is to define suitable cutoff functions: A. Herán and R. Rainer Due to the boundary conditions of u, ϕ does indeed vanish when approaching the boundary of the space-time cylinder Ω T . Thus ϕ is an admissible test function.

Estimating the parabolic part
and thus the second term can be estimated from below by zero. The remaining term can be rewritten due to , yielding the following estimate: Now we are able to let h ↓ 0 by using the convergence properties of the mollification in Lemma 2.6. That way, we obtain NoDEA Global higher integrability for a doubly nonlinear For the first term, there holds 3) took care of the first term when ε ↓ 0. The term II ε on the other hand can be estimated by which is independent of ε. We combine the previous estimates and obtain lim inf

Estimating the elliptic part and right hand side
The second term of the left hand side in the mollified equation (2.4) can be computed as with the same arguments as above. The final term on the right hand side of the mollified system (2.4) vanishes when h ↓ 0 due to ϕ(·, 0) = 0. The convergence properties of Lemma 2.6 are also applied.

Combination of the previous estimates
We combine the previous estimates, letting ε ↓ 0 and obtain We let t 1 ↑ t 0 +s for the second term, while taking the supremum over t 1 ∈ Λ s (t 0 ) for the first term and dividing by |Q R,S |. To get the measure of the smaller cylinder on the left hand side, use r ∈ [R/2, R), s ∈ [S/2, S), also inducing an n-dependency of the constant c. At last, Lemma 2.8 replaces the boundary term b[u, u 0 ].

Initial boundary
It will be necessary to extend u beyond the initial time t = 0. In [27], this has been done by reflecting the boundary values. Here, we definê Due to the vanishing lateral boundary values, we extendû outside of Ω by zero, for any time t. We do the same for u 0 ∈ W 1,p 0 (Ω). Also, we recall the notation Q R,S,+ := B R ∩ (Λ S ∩ (0, T )) for the part of the cylinder with positive time.
We follow up with an energy estimate valid close to the initial boundary, so restricted inside of the spatial domain Ω. In this case we have Q R,S,+ = Q R,S ∩ Ω T .
Proof. With the same cutoff functions as in the previous Lemma, ϕ = ζψ ε η p (u− a) is admissible as a testing function in the mollified equation (2.4), since the ball B R lies within Ω and t 0 > 0.

Estimating the parabolic part
The properties of the mollification, contained in Lemma 2.6, will be used repeatedly. Once again the second term can be estimated from below by zero, while the first term can be computed as follows: as h ↓ 0. These terms will be referred to as I ε , II ε . The first term I ε converges as follows when ε ↓ 0: For the second term II ε one can use that the boundary term is non-negative to obtain 55 Page 12 of 51 A. Herán and R. Rainer Together, this yields the estimate Estimating the elliptic part and the right hand side Using Du = D(u − a), the second term in the mollified equation (2.4) can be computed similarly as in the previous Lemma: Likewise, for the right hand side in (2.4) there holds the estimate

Combination of the previous estimates
As in to the previous Lemma and by using the estimates for the boundary term in Lemma 2.8, this yields the estimate t1 0 Br The last term can be treated in the following way.
If t 0 − S ≥ 0, then ζ(0) = 0 and the last term vanishes. Otherwise, t 0 − S < 0 and sinceû(x, t) = u 0 (x) for t < 0 there holds NoDEA Global higher integrability for a doubly nonlinear where we refer to the estimate in Lemma 2.8. It remains to prove an estimate for negative times for the spatial integral of |û This is only needed when t 0 < s. Let t 2 ∈ Λ s ∩ (−T, 0). Since for negative timesû = u 0 and a are constant in time it follows that As in the proof of the previous Lemma, it follows that

Gluing Lemma
We quote the Gluing Lemma in the local case, taken from [4,Lemma 4.2]. This result holds true also in our context if both occurring times are positive.
This can be extended for times t < 0 as follows: A. Herán and R. Rainer Proof. If t 1 , t 2 ≤ 0, the left hand side vanishes sinceû(x, t 1 ) =û(x, t 2 ) = u 0 (x). If t 1 , t 2 > 0, the respective result from Lemma 3.3 yields the claim. So let t 1 ≤ 0 and t 2 > 0. Then for some r ∈ [ R 2 , R). The first term obtained by inserting the test function ϕ into the weak formulation (2.2) is given by by the initial boundary condition (2.3). The remaining terms of (2.2) are given by

NoDEA
Global higher integrability for a doubly nonlinear We multiply the weak formulation by e i , sum over i = 1, . . . , N while also replacing u 0 (·) byû(·, t 1 ). This way, we obtain In the last step, (1.3) was used. We abbreviate I : By choosingr in (3.2) and taking mean values on both sides yields sincer ≥ R/2 and where c = c(n, L 2 ) > 0.

Parabolic Sobolev-Poincaré type inequalities
For convenience, we write X ΩT for the characteristic function of the set Ω T , so

Lateral boundary
If for , μ > 0 and some K ≥ 1 there holds holds, then the cylinder is called μ-super-intrinsic. The first and second inequality from (4.2) will be individually referenced as ( for a.e. t ∈ (0, T ) and also Combining this inequality with a slice-wise application of Sobolev's inequality, we are able to derive the following parabolic Sobolev-Poincaré type inequality in the lateral case. We delay the application of the intrinsic property of the cylinder until Lemma 5.1 to obtain a better bound for the scaling parameter μ.  Proof. We omit the center point z 0 and start by calculating with c = c(p, n, N, ν, 0 ) > 0. Inserting this estimate into (4.3) and by applying Young's inequality with 2 2−q and 2 q we obtain with c = c(p, n, N, ν, 0 ) > 0.
Proof. We start by using Lemma 2.7 with b =û p 2 , a = u 0 p 2 , α = 2 p and Hölder's inequality. This way, it follows that The latter integral can be estimated via the sub-intrinsic property (4.1). Hence, with c = c(K, p, n) > 0. After shifting the appearing power of μ to the first integral the proof can be finished by applying Young's inequality.

Initial boundary
Similar as in the lateral case we call a cylinder μ-sub-intrinsic, if for , μ > 0 and some K ≥ 1 there holds If on the other hand holds, then the cylinder is called μ-super-intrinsic. As before, a cylinder fulfilling both super-and sub-intrinsic properties is called μ-intrinsic. As we consider B (x 0 ) ⊆ Ω in the current case, we may write Q (μ) , In this subsection we will follow the strategy as in the local setting [4,Chapter 5]. ⊆ Ω × (−T, T ) that fulfils the sub-intrinsic property (4.4). Then there exists c = c(p, K) > 0 such that . Applying Lemma 2.7 with α = 2 p and Hölder's inequality leads to the estimate where c = c(p) > 0. Due to Jensen's inequality and the sub-intrinsic scaling (4.4) it follows that Together, this yields the desired estimate with c = c(p, K) > 0.
Proof. Letˆ be the radiusr from the Gluing Lemma 3.4. By using the quasiminimality of the mean value forû from Lemma 2.9, it follows that 55 Page 20 of 51 A. Herán and R. Rainer For the first term, apply the quasi-minimality of the mean value forû p−1 from Lemma 2.9 with the value α = 1 p−1 ≥ 1 q on every time slice. The quotient of measures |Bˆ |/|B | is compensated by the fact thatˆ ∈ [ 2 , ). Together with Poincaré's inequality this implies that The constant in Poincaré's inequality depends continuously on q. Since q ∈ [1, p], we can thus write the constant as c = c(n, p) > 0. Lemma 2.7 with α = 1 p−1 ≥ 1 and Hölder's inequality imply that where c = c(p). The Gluing Lemma 3.4 takes care of the supremum term, while the sub-intrinsic scaling (4.4) allows the estimation of the final term. This way, the appearing powers of μ cancel each other out. Together with an application of Jensen's inequality, noting that q/(p − 1) > 1, it follows that Global higher integrability for a doubly nonlinear with c = c(p, L 2 , K). Inserting these inequalities into the initial equation at the start of the proof yields the desired result.
At one point in the final chapter we will work with a cylinder fulfilling an adapted version of the sub-intrinsic property (4.4), namely for some λ > 0. In that instance we will further use the exponent q = p. Note that the sub-intrinsic property is applied merely once in the proof of the previous Lemma. Inserting the adapted version instead results in the following inequality: (4.6) Lemma 4.6. Let max{ 2n n+2 , 1} < p ≤ 2. Assume that u is a weak solution to the Cauchy-Dirichlet problem (1.1). For z 0 ∈ Ω T and , μ > 0, consider a cylinder Q (μ) (z 0 ) ⊆ Ω × (−T, T ) that fulfils both the sub-intrinsic property (4.4) and the super-intrinsic property (4.5). For any ε ∈ (0, 1] and for q = max{ 2n n+2 , 1} there exists c = c(n, p, L 2 , K) > 0 such that Proof. Write a := (û) (μ) . Expanding the left hand side with the powers −q/2+ q/2 and applying Hölder's inequality for the latter part, while taking the supremum over the time slices in the first part yields A. Herán and R. Rainer Estimating II can be achieved by inspecting μ (2−p)q/2 with the help of the super-intrinsic scaling (4.5). This condition consists of two cases. Case (4.5) 1 : First note that by using Lemma 4.5 with q = p one has The super-intrinsic scaling (4.5) 1 then yields Case (4.5) 2 : Here, the condition μ p ≤ K immediately implies the previous inequality with c = K 1 p . In turn, the following conclusion holds in both cases: For the first of these terms, use Lemma 2.8 with the exponent α = 2 p ≥ 1, then Sobolev's and Hölder's inequality to conclude that (4.7)

NoDEA
Global higher integrability for a doubly nonlinear Page 23 of 51 55 By expanding the first of these mean value integrals with the exponents p − 1 and 1 − p, we obtain For the latter integral, use Hölder's inequality with p q > 1 to obtain a right hand side that equals, up to a constant, the right hand side of Lemma 4.5. By inserting this and estimating the second part of the right hand side of (4.7) with Lemma 4.5, it follows that For II 2 , use Lemma 2.7 with α = 2 p ≥ 1 and once again Sobolev's inequality. This way, we obtain With Lemma 4.5 it follows that Forq ∈ {p, q}, we abbreviate |Dû|q + |Du 0 |q + |F |qX ΩT dz. Together with Young's inequality with the exponents 2 2−q and 2 q , the previous estimates imply that Now one can apply Young's inequality twice on the products on the right hand side to gain control over the terms of the form F(p). Finally, use Hölder's inequality for the integral with |F | q to obtain the power |F | p .

Reverse Hölder inequalities
The goal of this section is to obtain revers Hölder type inequalities. Similarly to [27], we must distinguish whether the cylinder is close to the initial or the lateral boundary. Yet, in contrast to the same reference, an intrinsic coupling will be available in any case for our setting.

Lateral boundary
In the lateral case, we consider cylinders and the super-intrinsic condition These conditions imply the sub-and super-intrinsic conditions (4.1) and (4.2) for every s ∈ [ , 2 ], respectively.
Proof. Assume ≤ r < s ≤ 2 . The center point z 0 will be omitted throughout this proof. First note that the energy estimate from Lemma 3.1 thus yields To estimate the first and second term we define R r,s := s/(s − r). By using (s − r) p ≤ s p − r p , this leads to and by Lemma 4.3 to for any δ ∈ (0, 1]. Moreover, we use Lemma 4.1 forû − u 0 and u 0 , respectively, . This allows to apply the iteration lemma Lemma 2.10 to get and finish the proof.

Initial boundary
In this case, we consider a pair of cylinders and the super-intrinsic condition As we are inspecting the initial boundary case, we replace ,+ (z 0 ) once again.

NoDEA
Global higher integrability for a doubly nonlinear Proof. Assume ≤ r < s ≤ 2 . The center point z 0 will be omitted throughout this proof. By inserting mean values as the vector a in the energy estimate from Lemma 3.2, we get Now, by adding these last two inequalities and applying the parabolic Sobolev-Poincaré type inequality from Lemma 4.6 with ε = δ 2 p , it follows that for the exponent q = max{ 2n n+2 , 1}. Note that for t ≤ 0 by definitionû = u 0 and thus the integral over the part of the cylinder with t ≤ 0 can be shifted from the |Dû| p -term, creating an integral with |Du 0 | p . Putting these estimates together and choosing a sufficiently small δ > 0, i.e. δ = 1 4c R −p r,s we end up with  Therefore we are able to apply the Iteration Lemma 2.10 to absorb the supremum term as well as the term involving |Du| p into the left hand side. As a result, we obtain NoDEA Global higher integrability for a doubly nonlinear |Du 0 | p + |F | p χ ΩT dz.

Construction of a non-uniform system of cylinder
Following the approach in [4] let z 0 ∈ Q 2R ∩ Ω T and define for ∈ (0, R] μ (λ) where the set for which the condition of the infimum is satisfied is not empty, since the integrals tends to zero while the right hand sides tends to infinity as μ → ∞. For better readability we will write μ ρ instead of μ (6.2) For = R this means either μ R = 1 or with (6.1) such that in any case Our first aim is to show that the mapping (0, R] → μ is continuous.