Regularity results for solutions of mixed local and nonlocal elliptic equations

By Xifeng Su, Enrico Valdinoci, Yuanhong Wei, Jiwen Zhang. Mathematische Zeitschrift, 2022

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We consider the mixed local-nonlocal semi-linear elliptic equations driven by the superposition of Brownian and Lévy processes -Δu+(-Δ)su=g(x,u)inΩ,u=0inRn\Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} - \Delta u + (-\Delta )^s u = g(x,u) &{} \text {in } \Omega , \\ u=0 &{} \text {in } \mathbb {R}^n\backslash \Omega . \\ \end{array} \right. \end{aligned}$$\end{document}Under mild assumptions on the nonlinear term g, we show the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} boundedness of any weak solution (either not changing sign or sign-changing) by the Moser iteration method. Moreover, when s∈(0,12]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0, \frac{1}{2}]$$\end{document}, we obtain that the solution is unique and actually belongs to C1,α(Ω¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }(\overline{\Omega })$$\end{document} for any α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/7e91607b5cb3a1f8470c8bbd2b60356945a17a40