On the second-order regularity of solutions to widely singular or degenerate elliptic equations
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| Format: | Preprint |
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2024
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| author | Ambrosio, Pasquale Grimaldi, Antonio Giuseppe di Napoli, Antonia Passarelli |
| author_facet | Ambrosio, Pasquale Grimaldi, Antonio Giuseppe di Napoli, Antonia Passarelli |
| contents | We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,Ω, \] where $1<p<\infty$, $Ω$ is an open subset of $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a positive constant and $(\,\cdot\,)_{+}$ stands for the positive part. Equations of this form are widely degenerate for $p\ge 2$ and widely singular for $1<p<2$. We establish higher differentiability results for a suitable nonlinear function of the gradient $Du$ of the local weak solutions, assuming that $f$ belongs to the local Besov space $B^{(p-2)/p}_{p',1,loc}(Ω)$ when $p>2$, and that $f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(Ω)$ if $1<p\leq2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $λ=0$, our results give back those contained in [12,28]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the second-order regularity of solutions to widely singular or degenerate elliptic equations Ambrosio, Pasquale Grimaldi, Antonio Giuseppe di Napoli, Antonia Passarelli Analysis of PDEs 35J70, 35J75, 35J92, 49K20 We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,Ω, \] where $1<p<\infty$, $Ω$ is an open subset of $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a positive constant and $(\,\cdot\,)_{+}$ stands for the positive part. Equations of this form are widely degenerate for $p\ge 2$ and widely singular for $1<p<2$. We establish higher differentiability results for a suitable nonlinear function of the gradient $Du$ of the local weak solutions, assuming that $f$ belongs to the local Besov space $B^{(p-2)/p}_{p',1,loc}(Ω)$ when $p>2$, and that $f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(Ω)$ if $1<p\leq2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $λ=0$, our results give back those contained in [12,28]. |
| title | On the second-order regularity of solutions to widely singular or degenerate elliptic equations |
| topic | Analysis of PDEs 35J70, 35J75, 35J92, 49K20 |
| url | https://arxiv.org/abs/2401.13116 |