Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets

Jae-Hwan Choi; Kyeong-Hun Kim; Junhee Ryu

doi:10.3934/dcds.2023050

Article Contents

2023, Volume 43, Issue 9: 3338-3377. Doi: 10.3934/dcds.2023050

This issue Previous Article Hautus–Yamamoto criteria for approximate and exact controllability of linear difference delay equations Next Article Asymptotic stability and large time behavior of some three dimension magnetohydrodynamic equations

Sobolev regularity theory for the non-local elliptic and parabolic equations on $ C^{1,1} $ open sets

1.
Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea
2.
(Current address) Department of Mathematical Sciences, KAIST, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

^*Corresponding author: Junhee Ryu
^*Corresponding author: Junhee Ryu

Received: November 2022

Revised: April 2023

Early access: May 16, 2023

Published online: May 16, 2023

The authors were supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324).

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Abstract

We study the zero exterior problem for the elliptic equation

$ \Delta^{\alpha/2}u-\lambda u = f, \quad x\in D\, ; \quad u|_{D^c} = 0 $

as well as for the parabolic equation

$ u_t = \Delta^{\alpha/2}u+f, \quad t>0, \, x\in D \, ; \quad u(0, \cdot)|_D = u_0, \, u|_{[0, T]\times D^c} = 0. $

Here, $ \alpha\in (0, 2) $, $ \lambda \geq 0 $ and $ D $ is a $ C^{1, 1} $ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.

Keywords:

Non-local elliptic and parabolic equations,
fractional Laplacian,
Dirichlet problem,
Sobolev regularity theory,
Hölder estimates.

Mathematics Subject Classification: Primary: 35S16, 35B65, 45K05; Secondary: 46E35.

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