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  • Weak and viscosity solutions of the fractional Laplace equation
    Servadei, Raffaella, 1973- ; Valdinoci, Enrico, 1974-
    Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ▫$$\begin{cases} (-\Delta)^s u = f & \text{in} \quad \Omega,\\ u = g & \text{in} \quad \mathbb{R}^n ... \setminus \Omega \end{cases}$$▫ are also continuous solutions of this problem in the viscosity sense. Here ▫$s\in(0,1)$▫ is a fixed parameter, ▫$\Omega$▫ is a bounded, open subset of ▫$\mathbb{R}^n$▫, ▫ $n \geqslant 1$▫, 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 ▫$\Omega$▫.
    Vir: Publicacions matemàtiques. - ISSN 0214-1493 (Vol. 58, no. 1, 2014, str. 133-154)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2014
    Jezik - angleški
    COBISS.SI-ID - 17084249

    Povezava(-e):

    http://dx.doi.org/10.5565/PUBLMAT_58114_06

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vir: Publicacions matemàtiques. - ISSN 0214-1493 (Vol. 58, no. 1, 2014, str. 133-154)
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