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Weak and viscosity solutions of the fractional Laplace equation

Raffaella Servadei and Enrico Valdinoci

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Abstract

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation $$ \begin{cases} (-\Delta)^s u=f &\text{in }\Omega\\ u=g &\text{in }\mathbb R^n\setminus\Omega \end{cases} $$ are also continuous solutions (up to the boundary) of this problem in the viscosity sense.

Here $s\in(0,1)$ is a fixed parameter, $\Omega$ is a bounded, open subset of $\mathbb R^n$ ($n\geqslant1$) with $C^2$-boundary, and $(-\Delta)^s$ is the fractional Laplacian operator, that may be defined as $$(-\Delta)^su(x):=c(n,s)\int\limits_{\mathbb R^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy,$$ for a suitable positive normalizing constant $c(n,s)$, depending only on $n$ and $s$.

In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem.

As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of $(-\Delta)^s$ is strictly positive in $\Omega$.

Article information

Source
Publ. Mat., Volume 58, Number 1 (2014), 133-154.

Dates
First available in Project Euclid: 20 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.pm/1387570393

Mathematical Reviews number (MathSciNet)
MR3161511

Zentralblatt MATH identifier
1292.35315

Subjects
Primary: 35R09: Integro-partial differential equations [See also 45Kxx] 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 49N60: Regularity of solutions 35D30: Weak solutions

Keywords
Integrodifferential operators fractional Laplacian weak solutions viscosity solutions regularity theory

Citation

Servadei, Raffaella; Valdinoci, Enrico. Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58 (2014), no. 1, 133--154. https://projecteuclid.org/euclid.pm/1387570393


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