Higher-order boundary regularity estimates for nonlocal parabolic\n equations

2017/11/06 by Xavier Ros‐Oton, Ros-Oton, Xavier, Hernán Vivas +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1711.02075

openalex publication_date 2017/11/06 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We establish sharp higher-order H "older regularity estimates up to the\nboundary for solutions to equations of the form \∂_t u-Lu=f(t,x) in\nI\×\Ω where I\⊂\ℝ, \Ω\⊂\ℝⁿ and f\nis H "older continuous. The nonlocal operators L considered are those arising\nin stochastic processes with jumps such as the fractional Laplacian\n(-\Δ)^s, s\∈(0,1).\n Our main result establishes that, if f is C^\γ is space and\nC^\γ/2s in time, and \Ω is a C^2,\γ domain, then u/d^s\nis C^s+\γ up to the boundary in space and u is C^1+\γ/2s up\nthe boundary in time, where d is the distance to \∂\Ω. This is\nthe first higher order boundary regularity estimate for nonlocal parabolic\nequations, and is new even for the fractional Laplacian in C^\∞ domains.\n

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