Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

By A. L. Bais'on, A. Clop, Raffaella Giova, J. Orobitg, Antonia Passarelli di Napoli. Potential Analysis, 2016

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We study nonlinear elliptic equations in divergence form divA(x,Du)=divG.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\text {div }{\mathcal A}(x,Du)=\text {div } G.$\end{document} When A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal A}$\end{document} has linear growth in Du, and assuming that x↦A(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto {\mathcal A}(x,\xi )$\end{document} enjoys Bnα,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{\alpha }_{\frac {n}\alpha , q}$\end{document} smoothness, local well-posedness is found in Bp,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B^{\alpha }_{p,q}$\end{document} for certain values of p∈[2,nα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in [2,\frac {n}{\alpha })$\end{document} and q∈[1,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\in [1,\infty ]$\end{document}. In the particular case A(x,ξ)=A(x)ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal A}(x,\xi )=A(x)\xi $\end{document}, G = 0 and A∈Bnα,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\in B^{\alpha }_{\frac {n}\alpha ,q}$\end{document}, 1≤q≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq q\leq \infty $\end{document}, we obtain Du∈Bp,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Du\in B^{\alpha }_{p,q}$\end{document} for each p<nα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p<\frac {n}\alpha $\end{document}. Our main tool in the proof is a more general result, that holds also if A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal A}$\end{document} has growth s−1 in Du, 2 ≤ s ≤ n, and asserts local well-posedness in Lq for each q > s, provided that x↦A(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto {\mathcal A}(x,\xi )$\end{document} satisfies a locally uniform VMO condition.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fe85736f837949d69675e1a74d9fa101962bea96

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In Computational Theory and Mathematics

Citations in the corpus

Higher differentiability for solutions to a class of obstacle problems (2018, background): “4) show that the weak differentiability of integer or fractional order of the map A, as function of the x-variable, is a sufficient condition (see [24,25,40–42] for the case of Sobolev space with integer order and [2,15] for the fractional one).”