On s-harmonic functions on cones
| Autor: | Giorgio Tortone, Susanna Terracini, Stefano Vita |
|---|---|
| Přispěvatelé: | Terracini, S, Tortone, G, Vita, S |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Boundary (topology)
Monotonic function 01 natural sciences Combinatorics Fractional Laplacian 35R11 35B45 35B08 Conic function Mathematics - Analysis of PDEs FOS: Mathematics 0101 mathematics Numerical Analysi Eigenvalues and eigenvectors Mathematics Conic functions Numerical Analysis Applied Mathematics 010102 general mathematics Zero (complex analysis) Analysi 35B08 35B45 Asymptotic behavior Vertex (geometry) Martin kernel Analysis 010101 applied mathematics 35R11 Harmonic function Cone (topology) Conic section Analysis of PDEs (math.AP) |
| Zdroj: | Anal. PDE 11, no. 7 (2018), 1653-1691 |
| Popis: | We deal with non negative functions satisfying \[ \left\{ \begin{array}{ll} (-��)^s u_s=0 & \mathrm{in}\quad C, u_s=0 & \mathrm{in}\quad \mathbb{R}^n\setminus C, \end{array}\right. \] where $s\in(0,1)$ and $C$ is a given cone on $\mathbb R^n$ with vertex at zero. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half sphere. These conic functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt-Caffarelli-Friedman monotonicity formula to the case of fractional diffusions. 37 pages, 3 figures |
| Databáze: | OpenAIRE |
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