Regularity for a strongly degenerate equation with explicit $u$-dependence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908624728621056 |
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| author | Piccirillo, Miriam |
| author_facet | Piccirillo, Miriam |
| contents | We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation}
\label{equazione mia}
\mathrm{div}\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ \text{ in }Ω,
\end{equation} where $2\leq p<\infty,\textbf{ } Ω$ is an open subset of $\mathbb{R}^n,n>2,$ 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 suitable assumptions on the datum $b(x,u)$ and the coefficient $a(x).$ The novelty here with respect to previous papers on the subject is that the right hand side explicitly depends on the solution $u.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00976 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity for a strongly degenerate equation with explicit $u$-dependence Piccirillo, Miriam Analysis of PDEs 35J70, 35J75 We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ \text{ in }Ω, \end{equation} where $2\leq p<\infty,\textbf{ } Ω$ is an open subset of $\mathbb{R}^n,n>2,$ 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 suitable assumptions on the datum $b(x,u)$ and the coefficient $a(x).$ The novelty here with respect to previous papers on the subject is that the right hand side explicitly depends on the solution $u.$ |
| title | Regularity for a strongly degenerate equation with explicit $u$-dependence |
| topic | Analysis of PDEs 35J70, 35J75 |
| url | https://arxiv.org/abs/2511.00976 |