Interior regularity results for inhomogeneous anisotropic quasilinear equations

By C. Antonini, Giulio Ciraolo, Alberto Farina. Mathematische Annalen, 2021

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We consider inhomogeneous p-Laplace type equations of the form -diva(∇u)=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\textrm{div}\left( a(\nabla u)\right) =f$$\end{document} in a possibly anisotropic setting. Under general assumptions on the source term f, we obtain quantitative Sobolev regularity results for the stress field a(∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(\nabla u)$$\end{document} and weighted L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} estimates for the Hessian of the solution. As far as we know, our results are new or refine the ones available in literature also when restricted to the Euclidean setting.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5be11022dd66d3c4d96acd377e0b52e64c09f0e4