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Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  26 January 2019

Begoña Barrios  and
Maria Medina
Begoña Barrios
Affiliation:
Departamento de Análisis Matemático Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, La Laguna 38271, Spain ([email protected])
Maria Medina
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860 Santiago, Chile ([email protected])
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Abstract

We present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical case s = 1 in [23, 24] respectively.

Keywords

Non-local operatorsfractional Laplacianmixed boundary conditionsmaximum principle

MSC classification

Primary: 35B50: Maximum principles
Secondary: 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 475 - 495
Copyright
Copyright © Royal Society of Edinburgh 2019

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