Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry
| Autor: | Gareth Speight, Nageswari Shanmugalingam, Gianmarco Giovannardi, Sylvester Eriksson-Bique, Riikka Korte |
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| Přispěvatelé: | University of Oulu, Università degli Studi di Trento, Department of Mathematics and Systems Analysis, University of Cincinnati, Aalto-yliopisto, Aalto University |
| Rok vydání: | 2022 |
| Předmět: |
Primary: 31E05
Secondary: 35A15 50C25 35J70 Hölder condition Metric measure space 01 natural sciences Fractional Laplacian Combinatorics 010104 statistics & probability Mathematics - Analysis of PDEs Mathematics - Metric Geometry Traces and extensions FOS: Mathematics Uniqueness 0101 mathematics Besov space Existence and uniqueness for Dirichlet problem Mathematics Dirichlet problem Applied Mathematics 010102 general mathematics Metric Geometry (math.MG) Dirichlet's energy Metric space Bounded function Laplace operator Analysis Strong maximum principle Analysis of PDEs (math.AP) |
| Zdroj: | Journal of Differential Equations. 306:590-632 |
| ISSN: | 0022-0396 |
| Popis: | We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups. Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction |
| Databáze: | OpenAIRE |
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