关于混合局部与非局部加权拟线性椭圆方程的正则性理论
On the regularity theory for mixed local and nonlocal weighted quasilinear elliptic equations
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中文总结 AI 辅助
该研究针对带Muckenhoupt权的混合局部与非局部退化p-拉普拉斯方程，建立了首个系统的局部正则性理论，扩展了相关现有理论框架。
中文摘要 AI 辅助
我们研究一类带有一般右端项的混合局部与非局部退化p-拉普拉斯方程，其退化由Muckenhoupt A_p-权控制，形成局部与非局部算子可同时退化的高度非均匀椭圆框架。我们建立了全面的局部正则性理论，包括弱下解的局部有界性与下半连续性、弱上解的弱哈纳克不等式、哈纳克不等式以及弱解的局部赫尔德连续性。我们的方法结合了加权分析技术与适配混合局部-非局部框架的De Giorgi–Nash–Moser迭代法。据我们所知，这是首个针对带有Muckenhoupt权的混合局部与非局部方程的系统正则性理论，其结果甚至对自然假设w∈A₂下的齐次线性方程（p=2）也是新的，因此将现有的混合局部-非局部方程正则性理论实质性扩展至带有一般右端项的退化加权框架。
英文摘要
We investigate a broad class of mixed local and nonlocal degenerate $p$-Laplace equations with general right-hand sides. The degeneracy is governed by Muckenhoupt $A_p$-weights, yielding a highly nonuniform elliptic framework in which both the local and nonlocal operators may degenerate simultaneously. We establish a comprehensive local regularity theory, including local boundedness and lower semicontinuity of weak subsolutions, weak Harnack inequalities for weak supersolutions, Harnack inequalities, and local Hölder continuity of weak solutions. Our approach combines weighted analytic techniques with the De Giorgi--Nash--Moser iteration method, adapted to the mixed local--nonlocal setting. To the best of our knowledge, this is the first systematic regularity theory for mixed local and nonlocal equations with Muckenhoupt weights. In particular, our results are new even for homogeneous linear equations ($p=2$) under the natural assumption $w\in A_2$, and therefore substantially extend the existing regularity theory for mixed local--nonlocal equations to a degenerate weighted framework with general right-hand sides.
