Second order regularity of solutions of elliptic equations in divergence form with Sobolev coefficients
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914240510558208 |
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| author | Perelmuter, M. A. |
| author_facet | Perelmuter, M. A. |
| contents | We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $Ω\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(Ω) \bigcap L^s(Ω)$ $$\|Δu\|_{2} \leq \begin{cases} c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_q^2\|f\|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\\ c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_4^2\|f\|_s, & \text{if } s > d/2 \end{cases}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second order regularity of solutions of elliptic equations in divergence form with Sobolev coefficients Perelmuter, M. A. Analysis of PDEs 35B45 35J15 42B37 46E35 We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $Ω\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(Ω) \bigcap L^s(Ω)$ $$\|Δu\|_{2} \leq \begin{cases} c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_q^2\|f\|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\\ c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_4^2\|f\|_s, & \text{if } s > d/2 \end{cases}.$$ |
| title | Second order regularity of solutions of elliptic equations in divergence form with Sobolev coefficients |
| topic | Analysis of PDEs 35B45 35J15 42B37 46E35 |
| url | https://arxiv.org/abs/2411.09378 |