$W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909665097416704 |
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| author | Valette, Guillaume |
| author_facet | Valette, Guillaume |
| contents | We prove a priori estimates for solutions of order $2$ linear elliptic PDEs in divergence form on subanalytic domains. More precisely, we study the solutions of a strongly elliptic equation $Lu=f$, with $f\in L^2(\mathcalΩ)$ and $Lu=div (A(x) \nabla u)$, and, given a bounded subanalytic domain $\mathcalΩ$, possibly admitting non metrically conical singularities within its boundary, we provide explicit conditions on the tangent cone of the singularities of the boundary which ensure that $||u||_{ W^{1,p}(\mathcalΩ)}\le C||f||_{L^2(\mathcalΩ)}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $δ\mathcalΩ$, but not on $u$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_22913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains Valette, Guillaume Analysis of PDEs 35B45, 35A21, 32B20, 14P10 We prove a priori estimates for solutions of order $2$ linear elliptic PDEs in divergence form on subanalytic domains. More precisely, we study the solutions of a strongly elliptic equation $Lu=f$, with $f\in L^2(\mathcalΩ)$ and $Lu=div (A(x) \nabla u)$, and, given a bounded subanalytic domain $\mathcalΩ$, possibly admitting non metrically conical singularities within its boundary, we provide explicit conditions on the tangent cone of the singularities of the boundary which ensure that $||u||_{ W^{1,p}(\mathcalΩ)}\le C||f||_{L^2(\mathcalΩ)}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $δ\mathcalΩ$, but not on $u$. |
| title | $W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains |
| topic | Analysis of PDEs 35B45, 35A21, 32B20, 14P10 |
| url | https://arxiv.org/abs/2506.22913 |