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arXiv · 2402.06747

Boundary value problems for holomorphic functions on Lipschitz planar domains

William Gryc·Loredana Lanzani·Jue Xiong·Yuan Zhang
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Abstract

We study the $\bar\partial$ equation subject to various boundary value conditions on bounded simply connected Lipschitz domains $D\subset\mathbb C$: for the Dirichlet problem with datum in $L^p(bD, \sigma)$, this is simply a restatement of the fact that members of the holomorphic Hardy spaces are uniquely and completely determined by their boundary values. Here we identify the maximal data spaces and obtain estimates in the maximal $p$-range for the Dirichlet, Regularity-for-Dirichlet, Neumann, and Robin boundary conditions for $\bar\partial$.

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William Gryc, Loredana Lanzani, Jue Xiong, Yuan Zhang. 2024-02-09. Boundary value problems for holomorphic functions on Lipschitz planar domains. https://arxiv.org/abs/2402.06747

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