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arXiv · 2305.11690

Local regularity for nonlocal double phase equations in the Heisenberg group

Yuzhou Fang·Chao Zhang·Junli Zhang
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Abstract

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for nonlinear integro-differential problems in the Heisenberg group $\mathbb{H}^n$. Our proof of the a priori estiamtes bases on the spirit of De Giorgi-Nash-Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev-Poincaré type inequality in local domain, which may be of independent interest and potential applications.

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Yuzhou Fang, Chao Zhang, Junli Zhang. 2023-05-19. Local regularity for nonlocal double phase equations in the Heisenberg group. https://doi.org/10.1017/prm.2024.89

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