各向异性 $${{textbf {p}}(\cdot )$$ - 拉普拉卡方的弱解和粘性解的等价性和正则性

Pablo Ochoa, Federico Ramos Valverde
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引用次数: 0

摘要

本文阐述了涉及各向异性 \({{textbf {p}}(\cdot )\)-Laplacian 的非均质问题的弱解与粘性解之间的等价性。粘度解是弱解的证明是通过 inf-convolution 技术实现的。然而,由于 \({{textbf {p}}(\cdot )\)-Laplacian 的各向异性,我们调整了 inf-convolution 的定义以适应该算子的非均质性。反之,我们将制定弱解的比较原则。由于局部 Lipschitz 假设是获得粘性弱蕴涵的关键,我们证明了一类有界粘性解确实是局部 Lipschitz 的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{\textbf {p}}}(\cdot )$$ -Laplacian

In this paper, we state the equivalence between weak and viscosity solutions for non-homogeneous problems involving the anisotropic \({{\textbf {p}}}(\cdot )\)-Laplacian. The proof that viscosity solutions are weak solutions is performed by the inf-convolution technique. However, due to the anisotropic nature of the \({{\textbf {p}}}(\cdot )\)-Laplacian we adapt the definition of inf-convolution to the non-homogeneity of this operator. For the converse, we develop comparison principles for weak solutions. Since the locally Lipschitz assumption is crucial to get the viscosity-weak implication, we prove that a class of bounded viscosity solutions are indeed locally Lipschitz.

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