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Global higher integrability for parabolic quasiminimizers in nonsmooth domains. (English) Zbl 1173.35036

The paper is concerned with the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. A function \(u\in L_{\text{loc}}^{2}(0,T;W^{1,2}(\Omega))\) is a parabolic quasiminimizer if
\[ -\int_{\operatorname{spt} \varphi} u\frac{\partial\varphi}{\partial t}\, dx\,dt +\int_{\operatorname{spt}\phi} F(x,t,\nabla u)\, dx\,dt \leq K\int_{\operatorname{spt}\varphi}\, F\left(x,t,\nabla(u-\varphi)\right)\, dx\,dt \]
for every \(\varphi\in C_{0}^{\infty}(\Omega\times(0,T))\) where \(\Omega\) is a bounded open set in \(\mathbb{R}^{n}\), \(n\geq 2\) \(u :\Omega\times(0,T)\rightarrow\mathbb{R}\), \(K\geq 1\) and \(F :\Omega\times(0,T)\times\mathbb{R}^{n}\rightarrow\mathbb{R}\) satisfies the following assumptions: (a) \(x\,\rightarrow\, F(x,t,\xi)\) and \(t\,\rightarrow\, F(x,t,\xi)\) are measurable for every \(\xi\), (b) \(\xi\,\rightarrow\, F(x,t,\xi)\) is continuous for almost every \((x,t)\), (c) there exist \(0<\alpha\leq\beta<\infty\) such that for every \(\xi\) and almost every \((x,t)\), we have
\[ \alpha| \xi| ^{2}\leq F(x,t,\xi)\leq\beta| \xi|^2. \]
The author shows that the parabolic quasiminimizer globally belongs to a higher Sobolev space than assumed a a priori and, in particular, that the gradient of a quasiminimizer satisfies a reverse Hölder inequality. In this case, the regularity of the boundary and the regularity of boundary value play a role. The proofs are based on Caccioppoli and Poincarè type inequalities and a self-improving property of a reverse Hölder inequality.

MSC:

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
49N60 Regularity of solutions in optimal control
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