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3 results on '"Existence and uniqueness for Dirichlet problem"'

1. Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry.

Author

Eriksson-Bique, Sylvester, Giovannardi, Gianmarco, Korte, Riikka, Shanmugalingam, Nageswari, and Speight, Gareth

Subjects
*METRIC spaces, *DIRICHLET problem, *HOLDER spaces, *GEOMETRY, *BESOV spaces, *MAXIMUM principles (Mathematics)
Abstract

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 , μ X) satisfying a 2-Poincaré inequality. Given a bounded domain Ω ⊂ X with μ X (X ∖ Ω) > 0 , and a function f in the Besov class B 2 , 2 θ (X) ∩ L 2 (X) , we study the problem of finding a function u ∈ B 2 , 2 θ (X) such that u = f in X ∖ Ω and E θ (u , u) ≤ E θ (h , h) whenever h ∈ B 2 , 2 θ (X) with h = f in X ∖ Ω. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on Ω, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups. [ABSTRACT FROM AUTHOR]

Published
2022
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2. Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

Author

Gareth Speight, Nageswari Shanmugalingam, Gianmarco Giovannardi, Sylvester Eriksson-Bique, Riikka Korte, University of Oulu, Università degli Studi di Trento, Department of Mathematics and Systems Analysis, University of Cincinnati, Aalto-yliopisto, and Aalto University

Subjects
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)
Abstract

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

Published
2022

3. Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

Author

Eriksson-Bique, S. (Sylvester), Giovannardi, G. (Gianmarco), Korte, R. (Riikka), Shanmugalingam, N. (Nageswari), Speight, G. (Gareth), Eriksson-Bique, S. (Sylvester), Giovannardi, G. (Gianmarco), Korte, R. (Riikka), Shanmugalingam, N. (Nageswari), and Speight, G. (Gareth)

Abstract

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-Poincaré 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 \(\mathcal{E}_{\theta}(u,u) \leq \mathcal{E}_{\theta}(h,h)\) 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 Hölder continuous on \(\Omega\), and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

Published
2022

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