Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents (Q1733226)

This paper deals with classes of quasilinear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded domains. The corresponding equations contain the $\vec p(\cdot)$-Laplace operator, namely \[\Delta_{\vec p(\cdot)} u= \sum^N_1\partial_{x_i}(|\partial_{x_i}|^{p_i(\cdot)- 2}\partial_{x_i}),\quad x\in\Omega\subset\mathbb{R}^N.\] $\Omega$ can be a domain of non-Lipschitz type as some domain with fractal boundary. The considerations are done in the anisotropic Sobolev space with variable exponent $W^{1,\vec p}$, where $\vec p$ is a $L^\infty$ vector function from $\overline\Omega$ into $[1,\infty]^N$. Solvability and global regularity results for the elliptic and parabolic Robin problems are proved in Theorem 1 and Theorem 3.\par In the elliptic case the weak solution $u$ turns out to be Hölder contiuous in $\overline\Omega$. An a-priori estimate for $u$ is given by formula (1.7). The unique strong solution of the well-posed over $C(\overline\Omega)$ parabolic problem is written in explicit form by the Markovian $C_0$-semigroup generated by $-\Delta_{\vec p(\cdot)}$ acting on the Cauchy data. In Theorem 3 it is shown that, if in the elliptic case the right-hand sides of the Robin problem are nonnegative, then the weak solution is nonnegative in $\overline\Omega$. The difference of two strong solutions of the parabolic problem provides the ultracontractivity property (Theorem 4).