The variable exponent De Giorgi class and the variable exponent Hölder continuity
Year: 2024
Type: article
Abstract: For a bounded domain $ \Omega $ in $ \mathbb{R}^{n} $ ($ n \!\ge\! 2 $), we prove that the variable exponent De Giorgi class $ \mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta) $ is a subset of the variable exponent Hölder space $ C^{0, \alpha(x)}_{loc}(\Omega) $. It gives an improvement for the earlier result, which states that $ \mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta)\subset C^{0, \alpha}_{loc}(\Omega) $, where $ \alpha $ is a constant. This result can be applied to get $ C^{0, \alpha(x)}_{loc}(\Omega) $ regularity for certain solutions of variational problems and nonlinear elliptic equations with variable exponents.
DOI: 10.3934/cpaa.2024073
Authors: Ya-Ru Yuan, Zhi-Yuan Cui, Dun Zhao
Institution: Lanzhou University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4402131184