Local regularity for nonlocal equations with variable exponentsopen access
- Authors
- Chaker, Jamil; Kim, Minhyun
- Issue Date
- Sep-2023
- Publisher
- John Wiley and Sons Inc
- Keywords
- Caccioppoli estimate; De Giorgi iteration; fractional Sobolev space; Holder regularity; local boundedness; nonlocal equation; variable exponent; weak solution
- Citation
- Mathematische Nachrichten, v.296, no.9, pp.4463 - 4489
- Indexed
- SCIE
SCOPUS
- Journal Title
- Mathematische Nachrichten
- Volume
- 296
- Number
- 9
- Start Page
- 4463
- End Page
- 4489
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/192295
- DOI
- 10.1002/mana.202100521
- ISSN
- 0025-584X
- Abstract
- In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler–Lagrange equations. We show that weak solutions are locally bounded when the variable exponent p is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on p. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder–type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on p outside the domain.
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