加权广泛退化问题的二阶正则性，显式依赖于$u$

中文摘要 AI 辅助

我们考虑如下类型的广泛退化椭圆偏微分方程的局部弱解：\begin{equation} \label{equazione mia} \mathrm{div}\Biggl(|x|^\beta(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=\frac{|u|^{q-2}u}{|x|^\alpha} \\ \\ \text{ in }\Omega, \end{equation} 其中$2\leq p<n$，$\alpha,\beta>0$为固定指数，$\Omega$是$\mathbb{R}^n$中包含原点的开子集，$n>2$，$(\\ \cdot\\ )_+$表示正部。我们在对数据适当的假设下，建立了梯度与一个在单位球内消失的合适函数的复合的高可微性结果。与以往关于该主题的论文相比，这里的新颖之处在于右端项显式依赖于解$u$，并且左端和右端的权重之间存在不匹配。

英文摘要

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(|x|^β(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=\frac{|u|^{q-2}u}{|x|^α} \ \ \text{ in }Ω, \end{equation} where $2\leq p<n$, $α,β>0$ are fixed exponents, $Ω$ is an open subset of $\mathbb{R}^n,$ that contains the origin, $n>2,$ and $( \ \cdot \ )_+$ stands for the positive part. We establish a higher differentiability result for the composition of the gradient with a suitable function that vanishes in the unit ball for the gradient, under appropriate assumptions on the datum. The novelty here with respect to previous papers on the subject is that the right-hand side explicitly depends on the solution $u$ and we have a mismatch between the weight on the left-hand side and the right-hand side.

发表机构

• Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II"（那不勒斯费德里科二世大学应用数学系）

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