Abstract
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 \( \max \lbrace \frac{2n}{n+2}, 1 \rbrace < p \le 2\).
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1 Introduction
This paper is concerned with the global higher integrability of weak solutions to the following Cauchy-Dirichlet problem with vanishing lateral boundary values:
The prototype is the homogeneous doubly nonlinear equation
which is a special case of the fully doubly nonlinear equation
\(\Omega _T := \Omega \times (0,T)\) is a space-time cylinder consisting of an open, bounded set \(\Omega \subset {\mathbb {R}}^n\) with \(n \ge 1\) and \(T>0\). \(\partial _t\) denotes the derivative with respect to time, while D and \(\mathrm {div}\) denote the spatial gradient and the spatial divergence, respectively.
The vector field \({\mathbf {A}}: {\Omega _T}\times {\mathbb {R}}^N \times {\mathbb {R}}^{Nn} \rightarrow {\mathbb {R}}^{Nn}\) is a Carathéodory function, meaning it is measurable in \({\Omega _T}\) for all \((u,\xi ) \in {\mathbb {R}}^N \times {\mathbb {R}}^{Nn}\) and continuous in \({\mathbb {R}}^N \times {\mathbb {R}}^{Nn}\) for a.e. \((x,t) \in {\Omega _T}\). \({\mathbf {A}}\) is required to satisfy the growth and coercivity conditions
in \({\Omega _T}\) for \(0 < L_1 \le L_2\). The parabolic boundary of \({\Omega _T}\) is denoted by \(\partial _{\mathrm {par}}\Omega _T:= \Omega \times \lbrace 0 \rbrace \cup \partial \Omega \times (0,T)\). In general, p can be a value in \((1,\infty )\). For the right hand side F, one would naturally require that \(F \in L^{p}({\Omega _T})\), while for the initial boundary datum the condition \(u_0 \in W_0^{1,p}(\Omega )\) 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 \(m \ge p-1\), 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 \(Du \in L^p({\Omega _T})\). The aim of this paper is to show that there exists a constant \(\varepsilon >0\) such that \(Du \in L^{p(1+\varepsilon )}({\Omega _T})\). 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 \(\max \lbrace \frac{2n}{n+2},1 \rbrace < p \le 2\). 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 systems, and thus also signed solutions, another approach was used in [5]. Higher integrability for porous medium type systems up to the boundary is shown in [27]. For the singular case indicated by the condition \(m<1\), we refer to [7, 15]. The interior case for the higher integrability for doubly nonlinear systems with \(p=m+1\) in (1.2) is covered in [4].
The approach of using the intrinsic geometry introduced by DiBenedetto [9,10,11] has been used with various deviations for example in the previously mentioned articles related to higher integrability [5, 14, 22, 27] and also in [30]. We continue in similar fashion in the present article.
The structure of this paper is as follows: In Sect. 2, we present the main theorem and a collection of auxiliary material. In Sect. 3, we proof energy estimates in both the lateral and the initial case, in addition to a gluing Lemma. The intrinsic geometry will be introduced in Sect. 4 and used to prove parabolic Sobolev-Poincaré type inequalities. In the following Sect. 5, this will be extended to reverse Hölder inequalities. Finally, in Sect. 6 we construct cylinders on which the previous work will be applied, culminating in the proof of the gradient estimate.
2 Preliminaries
2.1 Setting and main result
Throughout the paper, we will use the notation \(\,\mathrm {d}z = \,\mathrm {d}x \,\mathrm {d}t\). We start by defining the boundary term
for \(u,v \in {\mathbb {R}}^N\). For our main theorem we consider weak solutions, which are defined as follows:
Definition 2.1
Let \({\mathbf {A}}\) fulfil the growth and coercivity conditions (1.3). A measurable function \(u:{\Omega _T}\rightarrow {\mathbb {R}}^N\) with
is called a weak solution to the Cauchy-Dirichlet problem of the doubly nonlinear parabolic system (1.1) if and only if
holds for all \(\varphi \in C_0^\infty ({\Omega _T},{\mathbb {R}}^N)\) and
The condition \(u \in L^{p} \big (0,T; W_0^{1,{p}}(\Omega , {\mathbb {R}}^N) \big )\) also contains the information \(u \in W_0^{1,p}(\Omega ,{\mathbb {R}}^n)\) for a.e. \(t \in (0,T)\), treating the lateral boundary condition, while (2.3) is for the initial boundary. For a center point \(z_0=(x_0,t_0) \in {\mathbb {R}}^n \times {\mathbb {R}}\), a radius \(\varrho >0\) and a scaling parameter \(\mu >0\), we define the respective space-time cylinder by
where
Also, a cylinder which contains only parts with positive time is labelled as
Further, define \(Q_{r,s}(z_0) := B_r(x_0) \times (t_0-s,t_0+s)\). This notation will only be used in Sect. 3. For such cylinders and a function \(v \in L^1({Q_\varrho ^{(\mu )}}, {\mathbb {R}}^N)\), define the mean value of v on \({Q_\varrho ^{(\mu )}}\) as
and the respective mean value of v on a time slice \(t \in {\mathbb {R}}\) as
The latter means exist for all times t if the function v is continuous with respect to time. Furthermore, we sometimes write \(u(t) := u(\cdot ,t)\). The notation \(\varvec{v}^\alpha \) for the power of a vector with \(\alpha >0\) is defined by \(\varvec{v}^\alpha := |v|^{\alpha -1}v\) for \(v \in {\mathbb {R}}^n {\setminus } \lbrace 0 \rbrace \) and \({\varvec{v}}^\alpha =0\) for \(v = 0\).
We will require some form of boundary regularity, more specifically the following property concerning the variational p-capacity \({{\,\mathrm{\mathrm {cap}}\,}}_p\), which will be inspected later on, see Sect. 2.3.
Definition 2.2
Let \(1<p<\infty \). A set \(E \subseteq {\mathbb {R}}^n\) is called uniformly p-thick, if there exist constants \(\nu , \varrho _0 >0\) such that
for all \(x_0 \in E\) and for all \(0< \varrho < \varrho _0\).
The main objective is to prove the following theorem:
Theorem 2.3
Let \(\max \lbrace \frac{2n}{n+2}, 1 \rbrace < p \le 2\) and \(\Omega \subseteq {\mathbb {R}}^n\) such that \({\mathbb {R}}^n {\setminus } \Omega \) is uniformly p-thick with corresponding constants \(\nu , \varrho _0\). Let \({\mathbf {A}}\) be a Carathéodory function fulfilling the growth and coercivity conditions (1.3). Assume that for an \(\varepsilon _2>0\), \(F \in L^{p(1+\varepsilon _2)}({\Omega _T})\) and \(u_0 \in W_0^{1,p(1+\varepsilon _2)}(\Omega )\).
Then there exists \(\varepsilon _1=\varepsilon _1(n,p,N,\nu ,\varrho _0)>0\), such that for any weak solution u to the Cauchy-Dirichlet problem (1.1) there holds
for \(\varepsilon _0 = \min \lbrace \varepsilon _1, \varepsilon _2 \rbrace \). Moreover, for any parabolic cylinder \(Q_{2R}(z_0) \subset {\mathbb {R}}^n \times (-T,T)\) with \(z_0 \in \Omega _T \cup \partial _{\mathrm {par}}\Omega _T\) and any \(\varepsilon \in (0,\varepsilon _0]\) we have
for a constant \(c=c(n,p,N,L_1,L_2,\nu ,\varrho _0)>0\).
2.2 Auxiliary results
The following mollification in time will be used to prove energy estimates.
Definition 2.4
For \(v \in L^1({\Omega _T},{\mathbb {R}}^N)\) 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.
Lemma 2.5
(Mollified system) If u is a weak solution of (1.1) with \(u(0)=0\), then
for any \(\varphi \in L^{p} \big (0,T; W_0^{1,{p}}(\Omega , {\mathbb {R}}^N) \big )\).
The following Lemma collects properties of this mollification, see [24, Lemma 2.9] and [6, Appendix B].
Lemma 2.6
For \(v \in L^1({\Omega _T},{\mathbb {R}}^N)\) and \(p \ge 1\), there holds:
-
1.
If \(v \in L^p(\Omega _T,{\mathbb {R}}^N)\), then \([\![ v ]\!]_{h} \longrightarrow v\) in \(L^p(\Omega _T,{\mathbb {R}}^N)\) as \(h \rightarrow 0\) and
$$\begin{aligned} \Vert [\![ v ]\!]_{h} \Vert _{L^p(\Omega _T,{\mathbb {R}}^N)} \le \Vert v \Vert _{L^p(\Omega _T,{\mathbb {R}}^N)}. \end{aligned}$$ -
2.
If \(v\in L^p(0,T; W^{1,p}(\Omega ,{\mathbb {R}}^N))\), then \([\![ v ]\!]_{h} \longrightarrow v\) in \(L^p(0,T;W^{1,p}(\Omega ,{\mathbb {R}}^N))\) as \(h \rightarrow 0\) and
$$\begin{aligned} \Vert [\![ v ]\!]_{h} \Vert _{L^p(0,T;W^{1,p}(\Omega ,{\mathbb {R}}^N))} \le \Vert v \Vert _{L^p(0,T;W^{1,p}(\Omega ,{\mathbb {R}}^N))}. \end{aligned}$$ -
3.
If \(v \in L^p(0,T; W_0^{1,p}(\Omega ,{\mathbb {R}}^N))\), then \([\![ v ]\!]_{h} \in L^p(0,T;W_0^{1,p}(\Omega ,{\mathbb {R}}^N))\).
-
4.
If \(v \in L^p(0,T; L^p(\Omega ,{\mathbb {R}}^N))\), then \([\![ v ]\!]_{h} \in C([0,T]; L^p(\Omega ,{\mathbb {R}}^N))\).
-
5.
The weak time derivative \(\partial _t [\![ v ]\!]_{h}\) exists in \(\Omega _T\) and can be computed by
$$\begin{aligned} \partial _t [\![ v ]\!]_{h} = \tfrac{1}{h} ( v - [\![ v ]\!]_{h} ). \end{aligned}$$
What follows now is a collection of useful vector inequalities, taken from [4, p.6]
Lemma 2.7
Let \(\alpha >0\) and \(N \in {\mathbb {N}}\). There exists \(c=c(\alpha )>0\) such that for any \(a,b \in {\mathbb {R}}^N\) such that
Further, if \(\alpha \ge 1\), there exists \(c=c(\alpha )>0\) such that for any \(a,b\in {\mathbb {R}}^N\) such that
For the boundary term defined in (2.1), we require the following estimate [4, Lemma 3.4]:
Lemma 2.8
Let \(p \ge 1\) and \(N \in {\mathbb {N}}\). There exists \(c=c(p)>0\) such that for any \(u,v \in {\mathbb {R}}^N\), there holds
An additional fact we use is the quasi-minimality of the mean-value integral. For the proof we refer to [4, Lemma 3.5].
Lemma 2.9
Let \(p \ge 1, \alpha \ge \frac{1}{p}\) and \(k \in {\mathbb {N}}\). There exists \(c=c(\alpha ,p)>0\) such that for \(A \subseteq B \subseteq {\mathbb {R}}^k\) with \(|A|,|B| < \infty \), any \(u \in L^{\alpha p}(B,{\mathbb {R}}^N)\) and any \(a \in {\mathbb {R}}^N\), there holds
Moreover we take the following well known Iteration Lemma from [20, Lemma 6.1]. It will be essential in absorbing certain quantities.
Lemma 2.10
Let \(0< \vartheta < 1\), \(A,C \ge 0\) and \(\alpha >0\). There exsits \(c=c(\alpha ,\vartheta )>0\) such that for any bounded non-negative function \(\phi :[r,\varrho ] \longrightarrow [0,\infty )\) with \(0<r<\varrho \) that satisfies
there holds
2.3 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 \(1<p<\infty \) and a compact set \(D \subset {\mathbb {R}}^n\), the p-capacity of \(C \subset {\mathbb {R}}^n\) is defined by
where the infimum is taken over all functions \(f \in C_0^\infty (D)\) such that \(f \ge 1\) in C. The p-capacity of an open set \(D \subset E\) is defined as the supremum of capacities of compact sets \(C \subset D\). At last, the variational p-capacity of a set \(D \subset {\mathbb {R}}^n\) is defined as the infimum of capacities of open sets \(O \supset D\).
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 p-thickness.
Lemma 2.11
Let \(1<p<\infty \) and \(E \subseteq {\mathbb {R}}^n\) compact and uniformly p-thick. Then E is also uniformly \({\tilde{p}}\)-thick for all \({\tilde{p}} \ge p\).
The uniform thickness admits a self-improving property which was proven in [25] and is as follows:
Theorem 2.12
Let \(1 < p \le n\) and \(E \subseteq {\mathbb {R}}^n\) uniformly p-thick. There exists \(q=q(n,p,\nu ) \in (1,p)\) 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].
Lemma 2.13
Let \(\Omega \subset {\mathbb {R}}^n\) be bounded and open such that \({\mathbb {R}}^n {\setminus } \Omega \) is uniformly p-thick. For a point \(y \in \Omega \) with \(B_{4\varrho /3}(y) {\setminus } \Omega \ne \emptyset \), there exists \({\tilde{\nu }}={\tilde{\nu }}(n,\nu ,\varrho _0,p)>0\) such that
3 Energy estimate
Recall that in this section, the notation \(Q_{r,s}(z_0) := B_r(x_0) \times (t_0-s,t_0+s)\) is used. We start with an energy estimate which is valid even outside of the spatial region \(\Omega \). It thus takes care of the lateral case, but also of cylinders that intersect both the lateral and the initial boundary.
3.1 Lateral boundary
Lemma 3.1
Let u be a weak solution to the Cauchy-Dirichlet problem (1.1), let \(0<R<1\) as well as \(S>0\) and let \({\mathbf {A}}\) satisfy (1.3). There exists \(c=c(n,p,L_1,L_2)>0\), such that for any \(z_0=(x_0,t_0) \in {\mathbb {R}}^{n} \times {\mathbb {R}}\) and \(Q_{r,s}(z_0) \subseteq Q_{R,S}(z_0) \subseteq {\mathbb {R}}^n \times {\mathbb {R}}\) with \(r \in [R/2,R), \; s \in [S/2,S)\), there holds
Proof
We may assume, \(Q_{r,s}(z_0) \cap {\Omega _T}\ne \emptyset \), since otherwise the estimate is obvious. The first step is to define suitable cutoff functions:
-
Let \(\eta \in C_0^1( B_R(x_0), [0,1])\) a cutoff function with \(\eta =1\) on \(B_r(x_0)\) and \(|D\eta | \le \frac{2}{R-r}\).
-
Let \(t_1 \in \Lambda _s(t_0) \cap (0,T)\) and \(\varepsilon >0\) so small such that \(t_1 + \varepsilon < \min \lbrace T,t_0+S \rbrace \). Define \(\psi _\varepsilon \in W^{1,\infty }( (0,t_0+S),[0,1])\) with
$$\begin{aligned} \psi _\varepsilon (t)&:= {\left\{ \begin{array}{ll} 0 &{} t \in (0,\varepsilon ], \\ \frac{t-\varepsilon }{\varepsilon }&{} t \in (\varepsilon ,2\varepsilon ], \\ 1 &{} t \in (2\varepsilon , t_1], \\ 1 - \tfrac{1}{\varepsilon }(t-t_1) &{} t \in (t_1,t_1+\varepsilon ), \\ 0 &{} t \in [t_1+\varepsilon , t_0+S). \end{array}\right. } \end{aligned}$$ -
Let \(\zeta \in W^{1,\infty }(\Lambda _S(t_0),[0,1])\) with
$$\begin{aligned} \zeta (t)&:= {\left\{ \begin{array}{ll} \dfrac{t-t_0+S}{S-s} &{} t \in (t_0-S,t_0-s), \\ 1 &{} t \in [t_0-s,t_0+S). \end{array}\right. } \end{aligned}$$
The method consists of inserting \(\varphi (x,t) := \eta ^p(x) \zeta (t) \psi _\varepsilon (t) (u(x,t)-u_0(x))\) into the mollified system (2.4). Due to the boundary conditions of u, \(\varphi \) does indeed vanish when approaching the boundary of the space-time cylinder \({\Omega _T}\). Thus \(\varphi \) is an admissible test function.
Estimating the parabolic part
With \(\varvec{w}^{p-1}:=[\![ \varvec{u}^{p-1} ]\!]_{h}\), the parabolic term in (2.4) reads
Lemma 2.6 implies \(\partial _t \varvec{w}^{p-1} = \tfrac{1}{h} (\varvec{u}^{p-1} - \varvec{w}^{p-1})\) and thus the second term can be estimated from below by zero. The remaining term can be rewritten due to \((\partial _t \varvec{w}^{p-1}) w = \frac{p-1}{p} \partial _t |w|^p\), \(\partial _t u_0= 0\), \(\partial _t |u_0|^p = 0\) and \(\psi _\varepsilon (0) = 0 = \psi _\varepsilon (T)\), yielding the following estimate:
Now we are able to let \(h \downarrow 0\) by using the convergence properties of the mollification in Lemma 2.6. That way, we obtain
For the first term, there holds
as \(\zeta =1 \) for \(t_1 \in \Lambda _s(t_0) \cap (0,T)\). The initial condition (2.3) took care of the first term when \(\varepsilon \downarrow 0\). The term \(\mathrm {II}_\varepsilon \) on the other hand can be estimated by
which is independent of \(\varepsilon \). We combine the previous estimates and obtain
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 usage of the growth and coercivity conditions (1.3), Young’s inequality and \(|D\eta | \le \frac{1}{R-r}\). The right hand side in (2.4) can be estimated by
with the same arguments as above. The final term on the right hand side of the mollified system (2.4) vanishes when \(h \downarrow 0\) due to \(\varphi (\cdot ,0)=0\). The convergence properties of Lemma 2.6 are also applied.
Combination of the previous estimates
We combine the previous estimates, letting \(\varepsilon \downarrow 0\) and obtain
with \(c=c(n,p,L_1,L_2)>0\). We let \(t_1 \uparrow t_0+s\) for the second term, while taking the supremum over \(t_1 \in \Lambda _s(t_0)\) for the first term and dividing by \(|Q_{R,S}|\). To get the measure of the smaller cylinder on the left hand side, use \(r \in [R/2,R)\), \(s \in [S/2,S)\), also inducing an n-dependency of the constant c. At last, Lemma 2.8 replaces the boundary term \({\mathfrak {b}}[u,u_0]\). \(\square \)
3.2 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 define
Due to the vanishing lateral boundary values, we extend \({\hat{u}}\) outside of \(\Omega \) by zero, for any time t. We do the same for \(u_0 \in W_0^{1,p}(\Omega )\). Also, we recall the notation \(Q_{R,S,+}:= B_R \cap ( \Lambda _S \cap (0,T) )\) 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 \(\Omega \). In this case we have \(Q_{R,S,+} = Q_{R,S} \cap {\Omega _T}\).
Lemma 3.2
Let u be a weak solution to the Cauchy-Dirichlet problem (1.1) and let \({\mathbf {A}}\) satisfy (1.3). There exists \(c=c(n,p,L_1,L_2)>0\), such that for any \(z_0=(x_0,t_0) \in \Omega \times (0,T)\), any \(a \in {\mathbb {R}}^N\) and \(Q_{r,s}(z_0) \subseteq Q_{R,S}(z_0) \subseteq \Omega \times (-T,T)\) with \(r \in [R/2,R), \; s \in [S/2,S)\), there holds
Proof
With the same cutoff functions as in the previous Lemma, \(\varphi = \zeta \psi _\varepsilon \eta ^p(u-a)\) is admissible as a testing function in the mollified equation (2.4), since the ball \(B_R\) lies within \(\Omega \) and \(t_0>0\).
Estimating the parabolic part
With \(\varvec{w}^{p-1}:=[\![ \varvec{u}^{p-1} ]\!]_{h}\), the parabolic term reads
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 \(h \downarrow 0\). These terms will be referred to as \(\mathrm {I}_\varepsilon , \mathrm {II}_\varepsilon \). The first term \(\mathrm {I}_\varepsilon \) converges as follows when \(\varepsilon \downarrow 0\):
For the second term \(\mathrm {II}_\varepsilon \) one can use that the boundary term is non-negative to obtain
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
The last term can be treated in the following way.
If \(t_0-S \ge 0\), then \(\zeta (0)=0\) and the last term vanishes. Otherwise, \(t_0-S<0\) and since \({{\hat{u}}}(x,t) = u_0(x)\) for \(t<0\) there holds
where we refer to the estimate in Lemma 2.8. It remains to prove an estimate for negative times for the spatial integral of \(|\varvec{{{\hat{u}}}}^{\frac{p}{2}} - \varvec{a}^{\frac{p}{2}}|^2\). This is only needed when \(t_0 < s\). Let \(t_2 \in \Lambda _s \cap (-T,0)\). Since for negative times \({{\hat{u}}}=u_0\) and a are constant in time it follows that
As in the proof of the previous Lemma, it follows that
\(\square \)
3.3 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.
Lemma 3.3
Let \(p>1\) and u be a (local) weak solution, so fulfilling merely equation (2.2). Then, on any cylinder \(Q_{R,S}(z_0) \subseteq {\Omega _T}\) with \(R,S >0\) there exists \({{\hat{r}}} \in [ \frac{R}{2}, R)\) such that for all \(t_1,t_2 \in \Lambda _S(t_0)\), we have
where \(c=c(L_2)\).
This can be extended for times \(t<0\) as follows:
Lemma 3.4
(Gluing Lemma) Let \(p>1\) and u be a weak solution to the Cauchy-Dirichlet problem (1.1). Then, on any cylinder \(Q_{R,S}(z_0) \subseteq \Omega \times (-T,T)\) with \(R,S >0\) there exists \({{\hat{r}}} \in [ \frac{R}{2}, R)\) such that for all \(t_1,t_2 \in \Lambda _S(t_0)\), we have
with \(c=c(n,L_2)>0\).
Proof
If \(t_1,t_2 \le 0\), the left hand side vanishes since \({\hat{u}}(x,t_1)={\hat{u}}(x,t_2)=u_0(x)\). If \(t_1,t_2 >0\), the respective result from Lemma 3.3 yields the claim. So let \(t_1 \le 0\) and \(t_2 >0\). Then \(\langle \varvec{{{\hat{u}}}}^{p-1} \rangle (t_1) = \langle \varvec{u_0}^{p-1} \rangle \). Define as testing function \(\varphi (x,t) := \xi (t) \eta (x) e_i\), where \(e_i\) is the i-th canonical basis vector in \({\mathbb {R}}^N\), \(\eta (x)=\eta _\delta (x) := \zeta _\delta (|x-x_0|)\) and
and
for some \(r \in [\frac{R}{2} , R )\). The first term obtained by inserting the test function \(\varphi \) 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
We multiply the weak formulation by \(e_i\), sum over \(i=1,\ldots ,N\) while also replacing \(u_0(\cdot )\) by \({\hat{u}}(\cdot ,t_1)\). This way, we obtain
In the last step, (1.3) was used. We abbreviate \({\mathcal {I}} := \int _{(0,t_0+S)} \big [ L_2|Du|^{p-1} + |F|^{p-1} \big ] \,\mathrm {d}t\). Due to
there must exist a \({{\hat{r}}} \in [R/2,R)\) such that
By choosing \({{\hat{r}}}\) in (3.2) and taking mean values on both sides yields
since \({{\hat{r}}} \ge R/2\) and where \(c=c(n,L_2)>0\). \(\square \)
4 Parabolic Sobolev-Poincaré type inequalities
For convenience, we write \({\mathcal {X}}_{\Omega _T}\) for the characteristic function of the set \({\Omega _T}\), so \({\mathcal {X}}_{\Omega _T}(x,t)=1\) if \((x,t) \in {\Omega _T}\) and \({\mathcal {X}}_{\Omega _T}(x,t) = 0\) otherwise. We consider scaled cylinders \({Q_\varrho ^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^{n+1}\) and when \(B_\varrho (x_0) \subseteq \Omega \), we may write \({Q_{\varrho ,+}^{(\mu )}}(z_0)\) instead of \({Q_\varrho ^{(\mu )}}(z_0) \cap {\Omega _T}\).
4.1 Lateral boundary
If for \(\varrho ,\mu >0\) and some \(K \ge 1\) there holds
then we call such a cylinder \(\mu \)-sub-intrinsic. If on the other hand
holds, then the cylinder is called \(\mu \)-super-intrinsic. The first and second inequality from (4.2) will be individually referenced as (4.2)\(_1\) and (4.2)\(_2\), respectively. A cylinder fulfilling both super- and sub-intrinsic properties is called \(\mu \)-intrinsic. First, we quote [27, Lemma 4.1], which in turn is an adaptation of [8, Lemma 4.2].
Lemma 4.1
Let \(v \in W_0^{1,p}(\Omega )\) for a.e. \(t \in (0,T)\) and \({\mathbb {R}}^n {\setminus } \Omega \) be uniformly p-thick with corresponding constants \(\nu \) and \(\varrho _0\). Consider a cylinder \({Q_\varrho ^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^{n+1}\) with \(B_{\rho /3}(x_0) {\setminus } \Omega \ne \emptyset \). Then there exists a constant \(\gamma =\gamma (n,\nu ) \in (1,p)\) such that for any \(\gamma \le \vartheta \le p\) there holds
for a.e. \(t \in (0,T)\) and also
with \(c=c(n,N,\nu , \varrho _0, \vartheta )>0\).
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 \(\mu \).
Lemma 4.2
Let \(1<p\le 2\). Let u be a weak solution to the Cauchy-Dirichlet problem (1.1) and let \({\mathbb {R}}^n {\setminus } \Omega \) be p-thick with corresponding constants \(\nu \) and \(\varrho _0\). Let \(q = \max \lbrace 2n/(n+2), \gamma , 1 \rbrace \in (1,p)\) with \(\gamma =\gamma (n,\nu )\) from Lemma 4.1. On a cylinder \({Q_\varrho ^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^{n+1}\) with \(\mathrm {dist}(B_{\varrho }(x_0), \partial \Omega ) = 0\) and for any \(\varepsilon \in (0,1]\), there holds
with \(c=c(p,n,N,\nu ,\varrho _0)>0\).
Proof
We omit the center point \(z_0\) and start by calculating
where we use Lemma 2.7 in the second last line. The condition \(\mathrm {dist}(B_{\varrho }(x_0), \partial \Omega )=0\) implies that \(B_{4\varrho /3} {\setminus } \Omega \ne \emptyset \). Thus Lemma 4.1 can be applied for the ball \(B_{4\varrho }\). For the last term we can thus use Sobolev’s inequality (note that \(\frac{p}{q} \ge 1\)), Lemma 4.1 with \(\vartheta = q\) for the ball \(B_{4\varrho }\) as well as Hölder’s inequality with \(\frac{2}{p} \ge 1\). It follows that
with \(c=c(p,n,N,\nu ,\varrho _0)>0\). Inserting this estimate into (4.3) and by applying Young’s inequality with \(\frac{2}{2-q}\) and \(\frac{2}{q}\) we obtain
with \(c=c(p,n,N,\nu ,\varrho _0)>0\). \(\square \)
Lemma 4.3
Let \(1<p\le 2\). Let u be a weak solution to the Cauchy-Dirichlet problem (1.1) and let \({\mathbb {R}}^n {\setminus } \Omega \) be p-thick with corresponding constants \(\nu \) and \(\varrho _0\). On a cylinder \({Q_\varrho ^{(\mu )}}\subseteq {\mathbb {R}}^{n+1}\) satisfying the sub-intrinsic coupling (4.1), there exists \(c=c(K,p,n)>0\) such that for any \(\delta \in (0,1]\), there holds
Proof
We start by using Lemma 2.7 with \(b=\varvec{{\hat{u}}}^{\frac{p}{2}}, a = \varvec{u_0}^{\frac{p}{2}}, \alpha = \frac{2}{p}\) 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 \(\mu \) to the first integral the proof can be finished by applying Young’s inequality. \(\square \)
4.2 Initial boundary
Similar as in the lateral case we call a cylinder \(\mu \)-sub-intrinsic, if for \(\varrho ,\mu >0\) and some \(K \ge 1\) there holds
If on the other hand
holds, then the cylinder is called \(\mu \)-super-intrinsic. As before, a cylinder fulfilling both super- and sub-intrinsic properties is called \(\mu \)-intrinsic. As we consider \(B_\varrho (x_0) \subseteq \Omega \) in the current case, we may write \({Q_{\varrho ,+}^{(\mu )}}(z_0)\) instead of \({Q_\varrho ^{(\mu )}}(z_0) \cap {\Omega _T}\). In this subsection we will follow the strategy as in the local setting [4, Chapter 5].
Lemma 4.4
Let \(1<p \le 2\). Assume that u is a weak solution to the Cauchy-Dirichlet problem (1.1). For \(z_0 \in {\Omega _T}\) and \(\varrho , \mu >0\), consider a cylinder \({Q_\varrho ^{(\mu )}}(z_0) \subseteq \Omega \times (-T,T)\) that fulfils the sub-intrinsic property (4.4). Then there exists \(c=c(p,K)>0\) such that
Proof
Write \({Q_\varrho ^{(\mu )}}={Q_\varrho ^{(\mu )}}(z_0)\). Applying Lemma 2.7 with \(\alpha = \frac{2}{p}\) 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 \(c=c(p,K)>0\). \(\square \)
Lemma 4.5
Let \(1<p \le 2\). Assume that u is a weak solution to the Cauchy-Dirichlet problem (1.1). For \(z_0 \in {\Omega _T}\) and \(\varrho , \mu >0\), consider a cylinder \({Q_\varrho ^{(\mu )}}(z_0) \subseteq \Omega \times (-T,T)\) that fulfils the sub-intrinsic property (4.4). For any \(q \in [1,p]\), there exist \(c=c(n,p,L_2,K)>0\) such that
Proof
Let \({\hat{\varrho }}\) be the radius \({{\hat{r}}}\) from the Gluing Lemma 3.4. By using the quasi-minimality of the mean value for \({{\hat{u}}}\) from Lemma 2.9, it follows that
For the first term, apply the quasi-minimality of the mean value for \(\varvec{{{\hat{u}}}}^{p-1}\) from Lemma 2.9 with the value \(\alpha = \frac{1}{p-1} \ge \frac{1}{q}\) on every time slice. The quotient of measures \(|B_{{\hat{\varrho }}}|/|B_\varrho |\) is compensated by the fact that \({\hat{\varrho }} \in [\frac{\varrho }{2}, \varrho )\). Together with Poincaré’s inequality this implies that
The constant in Poincaré’s inequality depends continuously on q. Since \(q \in [1,p]\), we can thus write the constant as \(c=c(n,p)>0\). Lemma 2.7 with \(\alpha = \frac{1}{p-1} \ge 1\) 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 \(\mu \) cancel each other out. Together with an application of Jensen’s inequality, noting that \(q/(p-1) >1\), it follows that
with \(c=c(p,L_2,K)\). Inserting these inequalities into the initial equation at the start of the proof yields the desired result. \(\square \)
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 \(\lambda >0\). 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:
Lemma 4.6
Let \(\max \lbrace \frac{2n}{n+2},1 \rbrace <p \le 2\). Assume that u is a weak solution to the Cauchy-Dirichlet problem (1.1). For \(z_0 \in \Omega _T\) and \(\varrho , \mu >0\), consider a cylinder \({Q_\varrho ^{(\mu )}}(z_0) \subseteq \Omega \times (-T,T)\) that fulfils both the sub-intrinsic property (4.4) and the super-intrinsic property (4.5). For any \(\varepsilon \in (0,1]\) and for \(q=\max \lbrace \frac{2n}{n+2},1 \rbrace \) there exists \(c=c(n,p,L_2,K)>0\) such that
Proof
Write \(a:= ( {\hat{u}} )_\varrho ^{(\mu )}\). 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
Estimating \(\mathrm {II}\) can be achieved by inspecting \(\mu ^{(2-p)q/2}\) 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 \(\mu ^p \le K\) immediately implies the previous inequality with \(c= K^{\frac{1}{p}}\).
In turn, the following conclusion holds in both cases:
For the first of these terms, use Lemma 2.8 with the exponent \(\alpha = \frac{2}{p} \ge 1\), then Sobolev’s and Hölder’s inequality to conclude that
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 \(\frac{p}{q} >1\) 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 \(\mathrm {II}_2\), use Lemma 2.7 with \(\alpha = \frac{2}{p} \ge 1\) and once again Sobolev’s inequality. This way, we obtain
With Lemma 4.5 it follows that
For \({\tilde{q}} \in \lbrace p,q \rbrace \), we abbreviate
Together with Young’s inequality with the exponents \(\frac{2}{2-q}\) and \(\frac{2}{q}\), 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 \({\mathcal {F}}(p)\). Finally, use Hölder’s inequality for the integral with \(|F|^q\) to obtain the power \(|F|^p\). \(\square \)
5 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.
5.1 Lateral boundary
In the lateral case, we consider cylinders \({Q_\varrho ^{(\mu )}}(z_0) \subset {Q_{8\rho }^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^n \times (-T,T)\) for some \(\varrho , \mu >0\) with \(\mathrm {dist}( B_\varrho (x_0), \partial \Omega ) =0\). For some \(K \ge 1\), we impose the sub-intrinsic condition
and the super-intrinsic condition
These conditions imply the sub- and super-intrinsic conditions (4.1) and (4.2) for every \(s \in [\varrho ,2 \varrho ]\), respectively.
Lemma 5.1
Let \(\max \lbrace 1, \frac{2n}{n+2} \rbrace <p \le 2\) and u be a weak solution to the Cauchy-Dirichlet Problem (1.1). Let \(q = \max \lbrace 2n/(n+2), \gamma , 1 \rbrace \in (1,p)\) with \(\gamma =\gamma (n,\nu )\) from Lemma 4.1. Then, on a cylinder \({Q_{8\varrho }^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^n \times (-T,T)\) with \(\mathrm {dist}\big ( B_{\varrho }(x_0), \partial \Omega \big ) =0\) satisfying the sub-intrinsic coupling (5.1) and the super-intrinsic coupling (5.2), there exists a constant \(c=c(K,p,n,N,L_1,L_2,\nu ,\varrho _0)>0\) such that
Proof
Assume \(\varrho \le r < s \le 2 \varrho \). 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 \({\mathcal {R}}_{r,s} := s/(s-r)\). By using \((s-r)^p \le s^p - r^p\), this leads to
and by Lemma 4.3 to
for any \(\delta \in (0,1]\). Moreover, we use Lemma 4.1 for \({\hat{u}}- u_0\) and \(u_0\), respectively, with \(\vartheta = p\)
and the super-intrinsic coupling (5.2) to show \(\mu ^p \le c(K,n,p)\). Altogether by applying Lemma 4.2 this leads to
for any \(\delta , \varepsilon \in (0,1]\). Now we choose \(\delta \) and \(\varepsilon \) small enough, i.e.
and end up with
with \(\alpha = \left( p+\frac{(2-p)p}{2} \right) \left( 1+\frac{2-q}{q} \right) \). This allows to apply the iteration lemma Lemma 2.10 to get
and finish the proof. \(\square \)
5.2 Initial boundary
In this case, we consider a pair of cylinders \({Q_\varrho ^{(\mu )}}(z_0) \subset {Q_{2\rho }^{(\mu )}}(z_0) \subseteq {\mathbb {R}}^{n+1}\) with \(\varrho , \mu >0\). For some \(K \ge 1\), we impose the sub-intrinsic condition
and the super-intrinsic condition
As we are inspecting the initial boundary case, we replace \({Q_\varrho ^{(\mu )}}(z_0) \cap {\Omega _T}\) with \({Q_{\varrho ,+}^{(\mu )}}(z_0)\) once again.
Lemma 5.2
Let \(\max \lbrace 1, \frac{2n}{n+2} \rbrace < p \le 2\), \(q:= \max \lbrace \frac{2n}{n+2},1 \rbrace \in (1,p)\) and u be a weak solution to the Cauchy-Dirichlet Problem (1.1). Then, on a cylinder \({Q_{2\varrho }^{(\mu )}}(z_0) \subseteq \Omega \times (-T,T)\) with \(z_0 \in \Omega _T\) satisfying the intrinsic coupling (5.3) and (5.4), there exists a constant \(c=c(n,p,L_1,L_2,K)>0\) such that
Proof
Assume \(\varrho \le r < s \le 2 \varrho \). 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
As in the lateral case, set \({\mathcal {R}}_{r,s} := s/(s-r)\). For the first term, we start by applying the quasi-minimality of the mean value integral as in Lemma 2.9 with \(\alpha =1\). Note that we can control the quotient of the size of the occuring cylinders by a power of 2, since \(|{Q_{r,+}^{(\mu )}}| \ge \frac{1}{2} |{Q_{r}^{(\mu )}}| \). Further, since the modified scaling conditions (5.3) and (5.4) imply the sub- and super-intrinsic properties (4.4) and (4.5) for every \(s \in [\varrho ,2\varrho ]\) we can use Lemma 4.4. Lastly, we apply Young’s inequality. This way, we obtain
for every \(\delta \in (0,1]\). For the second term, Lemma 2.9 and \((s-r)^p \le s^p - r^p\) implies
Now, by adding these last two inequalities and applying the parabolic Sobolev-Poincaré type inequality from Lemma 4.6 with \(\varepsilon = \delta ^{\frac{2}{p}}\), it follows that
for the exponent \(q=\max \lbrace \frac{2n}{n+2},1 \rbrace \). Note that for \(t \le 0\) by definition \({\hat{u}}= u_0\) and thus the integral over the part of the cylinder with \(t\le 0\) can be shifted from the \(|D{\hat{u}}|^p\)-term, creating an integral with \(|Du_0|^p\). Putting these estimates together and choosing a sufficiently small \(\delta >0\), i.e. \(\delta = \frac{1}{4c} {\mathcal {R}}_{r,s}^{-p} \) 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
\(\square \)
6 Proof of higher integrability
We consider \(Q_{8R} := Q_{8R,(8R)^p}(\widetilde{z_0}) \subset {\mathbb {R}}^n \times (-T,T)\) with \(R \in (0,1]\) and \(\widetilde{z_0} \in \Omega _T \cup \partial _{\text {par}} \Omega _T\), where we omit the center, since it will be fixed during this chapter. Let
and for \(z_0=(x_0,t_0) \in Q_{2R}\) we define \(d_0 := \frac{1}{2} \mathrm {dist}(x_0,\partial \Omega )\). Moreover we recall the definition of \(Q_\varrho ^{(\mu )}(z_0)\)
and remark that \(Q_\varrho ^{(\mu )}(z_0) \subset Q_\varrho ^{(\kappa )}(z_0)\) whenever \(\kappa \le \mu \).
6.1 Construction of a non-uniform system of cylinder
Following the approach in [4] let \(z_0 \in Q_{2R} \cap {\Omega _T}\) and define for \(\varrho \in (0,R]\)
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 \(\mu \rightarrow \infty \). For better readability we will write \({\widetilde{\mu }}_\rho \) instead of \({{\widetilde{\mu }}_{z_o;\rho }^{(\lambda )}}\) for fixed \(z_0\) and \(\lambda \). Moreover, by the definition of \({Q_{\varrho }^{(\mu )}}(z_0)\) the estimate is equivalent to
respectively, and hence we either have
or
For \(\varrho = R\) this means either \({\widetilde{\mu }}_R = 1\) or with (6.1)
such that in any case
Our first aim is to show that the mapping \((0,R] \ni \varrho \mapsto {\widetilde{\mu }}_\varrho \) is continuous.
Lemma 6.1
The mapping \(\varrho \mapsto {\widetilde{\mu }}_\varrho \) is continuous on \((0,d_0)\) as well as \([d_0,R]\) and moreover
Proof
Consider \(\varepsilon \in (0,R]\) and define \(\mu _{\pm } := {\widetilde{\mu }}_\varrho \pm \varepsilon \). Note first that \({Q_{\varrho }^{(\mu _+)}} (z_0) \subset {Q_{\varrho }^{({\widetilde{\mu }}_\varrho )}} (z_0) \subset {Q_{\varrho }^{(\mu _-)}} (z_0)\) since \(\mu _-< {\widetilde{\mu }}_\varrho < \mu _+\). Hence there exists \(\delta = \delta (\varepsilon ,\varrho ) > 0\) such that
for every r in the same subinterval as \(\varrho \) with \(|r-\varrho | \le \delta \). This can be shown in the following way.
If \({\widetilde{\mu }}_\varrho = 1\), i.e. \(\mu _- \le 1\), then the left hand side of the former inequality holds true trivially. Otherwise, by the definition of \({\widetilde{\mu }}_\varrho \) we have for \(r=\varrho \)
and
Note that both sides are continuous with respect to the radius and hence this implies
In the same manner as above we have
which by the continuity of both sides implies
Altogether this shows the above claim, since otherwise we have a contradiction to the definition of \({\widetilde{\mu }}_\varrho \). Finally the limit value observation is a direct consequence of the definition of \({\widetilde{\mu }}_\varrho \) and \(|{\hat{u}}|^p \le 2^p \left( |{\hat{u}}-u_0|^p + |u_0|^p \right) \). \(\square \)
Unfortunately the mapping \(\varrho \mapsto {\widetilde{\mu }}_\varrho \) might not be monotone or continuous at \(\varrho = d_0\). Therefore we define
where we again omit \(z_0\) and \(\lambda \) if they are fixed. By construction the mapping \((0,R] \ni \varrho \mapsto \mu _\varrho \) is continuous and monotonically decreasing. Moreover cylinders with scaling parameter \(\mu _\varrho \) are sub-intrinsic, which is the topic of the next Lemma.
Lemma 6.2
Cylinders \({Q_{s}^{(\mu _\varrho )}}(z_0)\) are \(\mu \)-subintrinsic with constant \(K=1\) in the sense that for every \(0 < \varrho \le s \le R\)
Proof
By the definition of \(\mu _\varrho \) we have \({\widetilde{\mu }}_s \le \mu _s \le \mu _\varrho \), hence \({Q_{s}^{(\mu _\varrho )}}(z_0) \subset {Q_{s}^{({\widetilde{\mu }}_s)}}(z_0)\) and therefore
in the case \(\varrho < d_0\) as well as
for \(\varrho \ge d_0\). \(\square \)
Next we define
It is easy to see that \(\mu _s = {\widetilde{\mu }}_{{\widetilde{\varrho }}}\) for every \(s \in [\varrho , {\widetilde{\varrho }}]\), especially \(\mu _\varrho = {\widetilde{\mu }}_{{\widetilde{\varrho }}}\).
Our next aim is to show that for every \(s \in (\varrho , R]\) we can estimate
Therefore we first consider the cases \(\mu _\varrho = 1\) as well as \(\mu _\varrho >1\), \(s\in (\varrho , {\widetilde{\varrho }}]\). Then \(\mu _s=1=\mu _\varrho \) or \(\mu _s = {\widetilde{\mu }}_{{\widetilde{\varrho }}} = \mu _\varrho \) respectively and the claim follows directly. The remaining case (\(\mu _\varrho >1\), \(s\in ({\widetilde{\varrho }},R]\)) can be handled by the definition of \({\widetilde{\mu }}_\varrho \), the monotonicity of \(s \mapsto \mu _s\) and Lemma 6.2
and
where we use \(s \ge {\widetilde{\varrho }}\) and \(\mu _s \le \mu _{{\widetilde{\varrho }}}\) to enlarge the area of integration in the penultimate estimate. Therefore (6.5) holds true in any mentioned case. For \(s=R\) we can use the estimate from (6.3) as well as \(\mu _R = {\widetilde{\mu }}_R\) to get an upper bound for \(\mu _\varrho \) by
In what follows we consider a family of cylinders \({Q_{\varrho }^{({\mu _{z_0;\varrho }^{(\lambda )}})}}(z_0)\) with \(\varrho \in (0,R]\), \(z_0 \in Q_{2R}\), which are nested in the sense that
6.2 Covering property
We want to show in this subsection that the cylinders constructed above fulfill a Vitali covering property. More precisely there exists a constant \({\hat{c}} = {\hat{c}}(n,p) \ge 80\) such that for any \(\lambda \ge \lambda _0\) and every collection \({\mathcal {F}}\) of cylinders \({Q_{8r}^{({\mu _{z_0;r}^{(\lambda )}})}}(z)\) with \(r \in (0,\frac{R}{{\hat{c}}}]\) there exists a countable subcollection \({\mathcal {G}} \subset {\mathcal {F}}\) of disjoint cylinders with
where \({\widehat{Q}}:={Q_{{\hat{c}}r}^{({\mu _{z_0;r}^{(\lambda )}})}}(z)\) denotes the \(\frac{1}{8}{\hat{c}}\)-times enlarged cylinder \({Q_{8r}^{({\mu _{z_0;r}^{(\lambda )}})}}(z)\).
Following again the proof of [4, Lemma 7.1] we define for \({\hat{c}}\) to be specified later on
We now select \({\mathcal {G}}_j \subset {\mathcal {F}}_j\) as follows:
-
Let \({\mathcal {G}}_1\) be any maximal disjoint collection of cylinders in \({\mathcal {F}}_1\).
-
When \({\mathcal {G}}_1, {\mathcal {G}}_2, ... , {\mathcal {G}}_{k-1}\) have been selected for some \(k \in {\mathbb {N}}_{\ge 2}\) then let \({\mathcal {G}}_k\) be a maximal disjoint collection in \( \lbrace Q \in {\mathcal {F}}_k :Q \cap {\widetilde{Q}} = \emptyset \text { for any } {\widetilde{Q}} \in \bigcup _{j=1}^{k-1} {\mathcal {G}}_j \rbrace \) .
Note that by the definition of \({\mathcal {F}}_1\) as well as the definition of the cylinders the Lebesgue measure of \(Q \in {\mathcal {G}}_1\) is bounded from below and hence \({\mathcal {G}}_1\) contains finitely many cylinders. Therefore by setting \({\mathcal {G}} := \bigcup _{j=1}^\infty {\mathcal {G}}_j\) we constructed a countable sub-collection of disjoint cylinders in \({\mathcal {F}}\).
It remains to show that for every \(Q={Q_{8r}^{({\mu _{z;r}^{(\lambda )}})}}(z) \in {\mathcal {F}}\) there exists \(Q^*= {Q_{8r_*}^{({\mu _{z_*;r_*}^{(\lambda )}})}}(z_*) \in {\mathcal {G}}\) such that \(Q \cap Q^*\ne \emptyset \) and \(Q \subset \widehat{Q^*}\). For the first statement consider an arbitrary \(Q \in {\mathcal {F}}\), hence \(Q \in {\mathcal {F}}_j\) for some \(j \in {\mathbb {N}}\). If in addition \(Q \in {\mathcal {G}}\) one can choose \(Q^*= Q\). Otherwise \(Q \notin {\mathcal {G}}\) and the maximality of \({\mathcal {G}}_j\) ensures the existence of \(Q^*\in \bigcup _{\ell =1}^{j} {\mathcal {G}}_\ell \) with \(Q \cap Q^*\ne \emptyset \). Furthermore we have \(r \le \frac{2R}{2^j {\hat{c}}} \le 2 r_*\) in both cases.
Next we will present an estimate for \({\mu _{z_*;r_*}^{(\lambda )}}\) in terms of \({\mu _{z;r}^{(\lambda )}}\) which is the main difficulty in showing \(Q \subset \widehat{Q^*}\).
Lemma 6.3
Let \(Q \in {\mathcal {F}}\) and \(Q^*\in {\mathcal {G}}\) as above. Then \({\mu _{z_*;r_*}^{(\lambda )}}\) is bounded from above in terms of \({\mu _{z;r}^{(\lambda )}}\) by \({\mu _{z_*;r_*}^{(\lambda )}} \le (8 \eta )^{\frac{n+2p}{2p-2}} {\mu _{z;r}^{(\lambda )}}\) with \(\eta = 25\).
Proof
To prove the claim, we have to distinguish different cases. Consider first \({\mu _{z_*;r_*}^{(\lambda )}}=1\), then by the definition of \(\mu _\varrho \) and \({\widetilde{\mu }}_\varrho \) it is obvious that \({\mu _{z_*;r_*}^{(\lambda )}}=1\le {\mu _{z;r}^{(\lambda )}}\). For the other case, i.e. \({\mu _{z_*;r_*}^{(\lambda )}} > 1\), let \(\widetilde{r_*}\) be as in (6.4). We know by (6.2) that
and
For \(\widetilde{r_*} > \frac{R}{\eta }\) we can use this inequality to estimate
if \(\varrho < d_0\) as well as for \(\varrho \ge d_0\)
which yields the claim. If otherwise \(\widetilde{r_*} \le \frac{R}{\eta }\) we assume without loss of generality \({\mu _{z;r}^{(\lambda )}} \le {\mu _{z_*;r_*}^{(\lambda )}}\), since otherwise the claim follows immediately. Then the monotonicity of the mapping \(\varrho \mapsto {\mu _{z;\varrho }^{(\lambda )}}\) as well as \(r \le 2r_*\le 2\widetilde{r_*} \le \eta \widetilde{r_*}\) imply
Since \(\widetilde{r_*} \ge r_*\) and \(|x_*- x| \le 8r + 8r_*\le 24r_*\) we know that \(B_{\widetilde{r_*}}(x_*) \subset B_{\eta \widetilde{r_*}}(x)\). Moreover as \(p < 2\)
holds true, therefore \(\Lambda _{\widetilde{r_*}}^{\left( {\mu _{z_*;r_*}^{(\lambda )}} \right) }(t_*) \subset \Lambda _{\eta \widetilde{r_*}}^{\left( {\mu _{z_*;\eta \widetilde{r_*}}^{(\lambda )}} \right) }(t)\) and altogether \({Q_{\widetilde{r_*}}^{({\mu _{z_*;r_*}^{(\lambda )}})}}(z_*) \subset {Q_{\eta \widetilde{r_*}}^{({\mu _{z;\eta \widetilde{r_*}}^{(\lambda )}})}}(z)\). We end up with
or rather
for the different cases of \(\varrho \), where we used Lemma 6.2 in the penultimate line.
\(\square \)
The previous result \({\mu _{z_*;r_*}^{(\lambda )}} \le (8 \eta )^{\frac{n+2p}{2p-2}} {\mu _{z;r}^{(\lambda )}}\) with \(\eta = 25\) implies \(\Lambda _{8r}^{{\mu _{z;r}^{(\lambda )}}}(t) \subset \Lambda _{{\hat{c}} r_*}^{{\mu _{z_*;r_*}^{(\lambda )}}}(t_*)\) for suitable large \({\hat{c}}={\hat{c}}(n,p)\) by
Moreover, if we choose \({\hat{c}} \ge 40\) we also have \(B_{8r}(x) \subset B_{16r + 8r_*}(x_*) \subset B_{40r_*}(x_*) \subset B_{{\hat{c}}r_*}(x_*)\). This shows that \(Q={Q_{8r}^{({\mu _{z;r}^{(\lambda )}})}}(z) \subset {Q_{8r_*}^{({\mu _{z_*;r_*}^{(\lambda )}})}}(z_*) = Q^*\) and finishes the proof of the Vitali type covering property of the constructed cylinders.
6.3 Stopping time argument
First of all we define \(\lambda _0\) by
such that (6.1) is fulfilled. Furthermore we identify with \(E(r,\lambda )\) for \(\lambda \ge \lambda _0\) and \(r \in (0,2R]\) the super-level set for Du, i.e.
where the Lebesgue point has to be understood with respect to the cylinders constructed in Sect. 6.1. For radii \(R \le R_1 < R_2 \le 2R\) we consider concentric cylinders \(Q_R \subset Q_{R_1} \subset Q_{R_2} \subset Q_{2R}\) and fix \(z_0 \in E(R_1,\lambda )\). For \(s \in (0,R]\) we shorten the notation by \(\mu _s = {\mu _{z_0;s}^{(\lambda )}}\). Lebesgue’s differentiation theorem shows
By \({\hat{c}} = {\hat{c}}(n,p)\) we denote the constant from Sect. 6.2 and consider \(\lambda \) satisfying \(\lambda > B\lambda _0\) with \(B:=\left( \frac{8{\hat{c}}R}{R_2-R_1} \right) ^{\frac{n+2}{2p-2}}>1\). By (6.6) and the definition of \(\lambda _0\) we have for every \(\frac{R_2-R_1}{{\hat{c}}} \le s \le R\) and \(1 \le \alpha \le 8\)
By (6.10) it is possible to find small \(0< s < \frac{R_2 - R_1}{{\hat{c}}}\) such that
The continuity of the mapping \(\varrho \mapsto \mu _\varrho \), (6.11) with \(\alpha = 1\) and the absolute continuity of the integral ensure the existence of a maximal radius \(0< \varrho _{z_0} < \frac{R_2-R_1}{{\hat{c}}}\) with
whereby the maximality of \(\varrho _{z_0}\) guarantees that
The monotonicity of \(\varrho \mapsto \mu _\varrho \) together with (6.5) leads to
for every \(\varrho _{z_0} < s \le R\) and hence
for every \(\varrho _{z_0} < s \le R\).
Finally, since \(R_1^p + (R_2-R_1)^p \le R_2^p\), we have \({Q_{{\hat{c}}\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0) \subset Q_{{\hat{c}} \varrho _{z_0}}(z_0) \subset Q_{R_2}\).
6.4 Reverse Hölder inequalities
Consider \(z_0 \in E(R_1,\lambda )\) with \(R_1\) and \(\lambda \) as in Sect. 6.3. By the former construction we have \(0< \varrho _{z_0} < \frac{R_2-R_1}{{\hat{c}}}\) and abbreviate as before \(\mu _{\varrho _{z_0}}:={\mu _{z_0;\varrho _{z_0}}^{(\lambda )}}\). Moreover, according to (6.4) we define \(\tilde{\varrho }_{z_0} \in [\varrho _{z_0},R]\) such that \(\mu _s = \tilde{\mu }_{\tilde{\varrho }_{z_0}}=\mu _{\tilde{\varrho }_{z_0}}\) for every \(s \in [\varrho _{z_0},\tilde{\varrho }_{z_0}]\).
Lateral case \(\varrho _{z_0} \ge d_0\): In this case our aim is to apply Lemma 5.1 on the cylinder \({Q_{2\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\). Therefore we use the (6.14) with \(s=8\varrho _{z_0} \le \frac{R_2-R_1}{5}<R\) and (6.12) to get
for a constant \(c(n,p) > 1\). The second inequality in Lemma 6.2 with \(s=8\varrho _{z_0}\) together with the second part of the former inequality shows
and hence the hypothesis (5.1) is fulfilled for the cylinder \({Q_{2\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) with \(K=c(n,p)2^{n+p}\).
Next, we assume \(\tilde{\varrho }_{z_0} < R\) since otherwise \(\mu _{\varrho _{z_0}}^p = 1\) holds true, satisfying (5.2)\(_2\). We can use (6.2) and apply Lemma 4.1 with \(\vartheta = p\) to the cylinder \({Q_{4\tilde{\varrho }_{z_0}}^{(\mu _{\tilde{\varrho }_{z_0}})}}(z_0)\). Afterwards we can either use the maximality of \(\varrho _{z_0}\) in (6.13) with \(s=4\tilde{\varrho }_{z_0}\) if \(4 \tilde{\varrho }_{z_0} \le R\), or (6.11) with \(s=\tilde{\varrho }_{z_0} \ge \frac{R_2-R_1}{{\hat{c}}}\) and \(\alpha = 4\) to get
with constant \(c=c(n,p,N,\nu ,\varrho _0)\). Note that for the application of (6.13) we have to change the cylinder to \({Q_{4\tilde{\varrho }_{z_0}}^{(\mu _{4\tilde{\varrho }_{z_0}})}}(z_0)\) which is possible by (6.5). This verifies (5.2)\(_2\) for the cylinder \({Q_{2\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) with \(K=c\) in the lateral case. Since moreover \(2\varrho _{z_0} > 2d_0\), hence \(\mathrm {dist}(B_{2\varrho }(x_0),\partial \Omega ) = 0\), we can apply Lemma 5.1 on \({Q_{2\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) and end up with
Initial case \(\varrho _{z_0} < d_0\): Here, we first want to consider the case where also \(\tilde{\varrho }_{z_0} < d_0\). Moreover we distinguish between the non-degenerate case \(\tilde{\varrho }_{z_0} \le 2 \varrho _{z_0}\) and the degenerate case \(2 \varrho _{z_0} < \tilde{\varrho }_{z_0}\). In the former case our aim is to apply Lemma 5.2 to the cylinder \({Q_{{\tilde{\varrho }}_{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\). Since \(2\tilde{\varrho }_{z_0} \le 4\varrho _{z_0}\le \frac{R_2-R_1}{5} < R\) we can apply the first inequality in Lemma 6.2 with \(s=2\tilde{\varrho }_{z_0}\) together with (6.12) to get
This shows that (5.3) is fulfilled. Next, (6.14) with \(2\tilde{\varrho }_{z_0} \le R\) and \(\tilde{\varrho }_{z_0} \le 2\varrho _{z_0}\) leads to
Combining this with (6.2) we achive
and therefore (5.4)\(_1\) is satisfied with \(K=c(n,p)>1\). Since in addition \(B_{2\tilde{\varrho }_{z_0}} \subset \Omega \) we can apply Lemma 5.2 to the cylinder \({Q_{{\tilde{\varrho }}_{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) and end up with
For the second case, i.e. the degenerate one, we want to apply Lemma 5.2 to the cylinder \({Q_{\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\). By the first inequality in Lemma 6.2 with \(s=\varrho _{z_0}\) together with (6.12) we have
showing that (5.3) is satisfied. To verify the super-intrinsic property in this case, note first that the cylinder \({Q_{\tilde{\varrho }_{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) is sub-intrinsic in the sense of Lemma 6.2. Hence we can adapt the proof of Lemma 4.5 with \(q=p\) to these cylinder by replacing the original sub-intrinsic condition (4.4), resulting in an inequality as seen in (4.6). Additionally, we use \(\mu _{\varrho _{z_0}} = \mu _{{\tilde{\varrho }}_{z_0}}\) and apply (6.13) with \(s= {\tilde{\varrho }}_{z_0}\) to estimate
Combining Lemma 2.9, Jensen’s inequality, Lemma 6.2 with \((\varrho ,s)=(\varrho _{z_0},\frac{1}{2} \tilde{\varrho }_{z_0})\) as well as the former inequality leads to
Absorbing the second term on the right hand side we end up with
verifying the super-intrinsic property (5.4)\(_2\) in this case. Since in addition \(B_{2 \varrho _{z_0}} \subset \Omega \) we can apply apply Lemma 5.2 to the cylinder \({Q_{\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) and end up with
Finally we have to consider the remaining case, i.e. \(\varrho _{z_0} < d_0 \le \tilde{\varrho }_{z_0}\). As in the degenerate case we want to apply Lemma 5.2 to the cylinder \({Q_{\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) and by the same argument as above one can see that (5.3) is satisfied in this case. The super-intrinsic property (5.4)\(_2\) can be achived in the same way as in the lateral case, since \(\tilde{\varrho } \ge d_0\). Altogether we end up with (6.16).
Conclusion: Altogether we achieve the estimate
in any case for the exponent \(q=\max \lbrace 2n/(n+2), \gamma , 1 \rbrace \in (1,p)\) with \(\gamma =\gamma (n,\nu ) \in (1,p)\) from Lemma 4.1. The constant c depends on \(n,p,N,L_1,L_2,\nu ,\varrho _0\). Note that we integrate over the whole cylinder on the left hand side, in contrast to the former statements resulting from the Lemmata 5.1 and 5.2. But the remaining piece of the cylinder can be added on both sides of the inequality, as \(|D{\hat{u}}|^p\) is either zero (outside of \(\Omega \)) or equal to \(|Du_0|^p\) (for \(t \le 0\)).
On the right hand side of (6.17), we aim for an integration only on the interior of \({\Omega _T}\). We start by seeing that due to \(u \in W_0^{1,p}(\Omega )\) for a.e. \(t \in (0,T)\), \({\hat{u}}= u_0\) for \(t \le 0\) and \(z_0 \in {\Omega _T}\) it follows that
Thus we can estimate the right hand side of (6.17) further, also using Jensen’s inequality to obtain
6.5 Estimate on super-level set
The aim of this subsection is to show a reverse Hölder inequality on super-level sets. Therefore we define
Moreover we have shown so far, that if \(\lambda \ge B \lambda _0\) with \(B=\big ( \frac{8 {\hat{c}} R}{R_2-R_1} \big )^{\frac{n+2}{2p-2}} > 1\) and \({\hat{c}}(n,p) \ge 80\) then for every \(z_0 \in E(R_1,\lambda )\) there exists a cylinder \({Q_{\varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0)\) such that \({Q_{{\hat{c}} \varrho _{z_0}}^{(\mu _{\varrho _{z_0}})}}(z_0) \subset Q_{R_2}\) and (6.12), (6.14) and (6.18) hold true, where we abbreviate \(\mu _{\varrho _{z_0}} = {\mu _{z_0;\varrho _{z_0}}^{(\lambda )}}\) as before. We introduce the new parameter \(\eta \in (0,1]\), which we specify later. We will apply (6.12), (6.18), Hölder’s inequality and (6.14) with \(s=8\varrho _{z_0}\), \(\alpha =1\). Note that both \(E(R_2,\eta \lambda ), F(R_2,\eta \lambda ) \subset {\Omega _T}\) by definition. This way, it follows that
with \(c=c(n,p,N,\nu ,\varrho _0)\). We now choose \(\eta := \left( \frac{1}{2c} \right) ^p\) to absorb the first term on the right hand side. Therefore we have
Applying (6.14) with \(s={\hat{c}}\varrho _{z_0} \le R_2 \le R\) and \(\alpha = 1\) we can estimate the left hand side from below by
and hence we have
Up to now \(z_0 \in E(R_1,\lambda )\) is arbitrary and hence for \(\lambda > B \lambda _0\) we have a family of cylinders \({\mathcal {F}}=\Big \{ {Q_{8 \varrho _{z_0}}^{({\mu _{z_0;\varrho _{z_0}}^{(\lambda )}})}}(z_0) \Big \}\) covering \(E(R_1,\lambda )\) such that each cylinder is contained in \(Q_{R_2}\) and fulfills (6.19). By the Vitali covering from Sect. 6.2 there exists a countable subfamily \(\Big \{ {Q_{8 \varrho _{z_i}}^{({\mu _{z_i;\varrho _{z_i}}^{(\lambda )}})}}(z_i) \Big \}_{i \in {\mathbb {N}}} \subset {\mathcal {F}}\) of pairwise disjoint cylinders such that
Together with (6.19) we get
Moreover on \(E(R_1,\eta \lambda ) {\setminus } E(R_1,\lambda )\) we have \(\eta \lambda < |Du| \le \lambda \) and hence
Combining the last two inequalities shows
and replacing \(\eta \lambda \) by \(\lambda \) with \(\eta \le 1\) leads to
for any \(\lambda > \eta B \lambda _0 =: \lambda _1\) with constant \(c=c(n,p,N,L_1,L_2,\nu ,\varrho _0)\), which is the required reverse Hölder inequality on super-level sets.
6.6 Proof of the gradient estimate
To finish the gradient estimate of the main result we first consider some \(k > \lambda _1\) und define the truncation of |Du| by \(|Du|_k := \min \{ |Du|,k \}\) and for \(r \in (0,2R]\) the corresponding super-level set by \(E_k(r,\lambda ):=\{ z \in Q_r \cap \Omega _T :|Du|_k > \lambda \}\). Note that \(|Du|_k \le |Du|\) almost everywhere as well as \(E_k(r,\lambda ) = \emptyset \) for \(k \le \lambda \) and \(E_k(r,\lambda ) = E(r,\lambda )\) for \(k > \lambda \). Therefore (6.20) shows
For some \(\varepsilon \in (0,1]\) to be specified later on we multiply the previous inequality with \(\lambda ^{\varepsilon p - 1}\) and integrate the result with respect to \(\lambda \) over \((\lambda _1,\infty )\)
Next we apply Fubini’s theorem to exchange the order of integration in all terms. We start with the left hand side, for which we have
Going to the right hand side of (6.21), the first term reads
Continuing on (6.21), the \(|F|^p\)-term on the right hand side transforms as follows:
Lastly, for the \(|Du_0|^p\)-term in (6.21) there holds
Altogether this yields
Moreover, on \(\left( Q_{R_1} \cap \Omega _T \right) {\setminus } E_k(R_1,\lambda _1)\) we have \(|Du|_k \le \lambda _1\) and hence
and therefore
with constants \(c, c_*\) only depending on \(n,p,N,\nu ,\varrho _0\). Now we choose \(0 < \varepsilon \le \varepsilon _0 := \frac{p-q}{2 p c_*}\). Since \(B \ge 1\), \(\eta < 1\) and \(\varepsilon < 1\) this means \(\lambda _1^\varepsilon = (\eta B \lambda _0)^\varepsilon \le B \lambda _0^\varepsilon \). For every radii \(R \le R_1 < R_2 \le 2R\) we get
Applying the iteration result from Lemma 2.10 shows
with \(c=c(n,p,N,\nu ,\varrho _0)\). Finally we apply Fatou’s lemma to the left hand side and the definition of \(\lambda _0\) to end up with
As the center point \(\widetilde{z_0}\) of \(Q_{8R}\) fulfils \(\widetilde{z_0} \in {\Omega _T}\cup \partial _{\mathrm {par}}\Omega _T\) and \(u_0 \in W_0^{1,p}(\Omega )\) does not depend on time, it follows that
Recalling that \({\hat{u}}(\cdot ,t) =0\) outside of \(\Omega \) for \(t \in (-T,T)\), while \({\hat{u}}(x,t)=u_0(x)\) for \(t \le 0\), we further estimate
At last, we repeat a covering argument as in [27]: We cover the cylinder \(Q_R\) with finitely many cylinders \(Q_{R/8}(z_i)\) and use the previous estimate on these smaller cylinders. The sum of these inequalities finishes the proof of Theorem 2.3.
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Acknowledgements
A. Herán thanks the Deutsche Forschungsgemeinschaft for their support within the Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks. R. Rainer has been supported by the FWF-Project P 31956 “Doubly Nonlinear Evolution Equations”.
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Herán, A., Rainer, R. Global higher integrability for a doubly nonlinear parabolic system. Nonlinear Differ. Equ. Appl. 29, 55 (2022). https://doi.org/10.1007/s00030-022-00787-y
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DOI: https://doi.org/10.1007/s00030-022-00787-y