Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916269806059520 |
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| author | Poggio, Andrea |
| author_facet | Poggio, Andrea |
| contents | Let $Ω\subseteq\mathbb{R}^{d}$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $Ω$ with $L^{\infty}$ coefficients, $b$ and $c$ two $d$-dimensional vector-valued functions on $Ω$ with $L^{\infty}$ coefficients and $V$ a locally integrable nonegative function on $Ω$. Consider the operator ${\mathscr L}^{A,b,c,V}=-{\rm div}\,(A\nabla) + \left\langle \nabla , b \right\rangle - {\rm div}\,(c \, \cdot) + V $ with mixed boundary conditions on $Ω$. We extend the bilinear inequality that Carbonaro and Dragičević proved in the special cases when $b=c = 0$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A,b,c,V}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(Ω)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that Carbonaro and Dragičević introduced in arXiv:1611.00653 and $b,c,V$ satisfy another condition that we introduce in this paper. Roughly speaking, $V$ has to be ``big'' with respect to $b$ and $c$. We do not impose any conditions on $Ω$, in particular, we do not assume any regularity of $\partialΩ$, nor the existence of a Sobolev embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains Poggio, Andrea Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J15, 47D06, 42B25 Let $Ω\subseteq\mathbb{R}^{d}$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $Ω$ with $L^{\infty}$ coefficients, $b$ and $c$ two $d$-dimensional vector-valued functions on $Ω$ with $L^{\infty}$ coefficients and $V$ a locally integrable nonegative function on $Ω$. Consider the operator ${\mathscr L}^{A,b,c,V}=-{\rm div}\,(A\nabla) + \left\langle \nabla , b \right\rangle - {\rm div}\,(c \, \cdot) + V $ with mixed boundary conditions on $Ω$. We extend the bilinear inequality that Carbonaro and Dragičević proved in the special cases when $b=c = 0$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A,b,c,V}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(Ω)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that Carbonaro and Dragičević introduced in arXiv:1611.00653 and $b,c,V$ satisfy another condition that we introduce in this paper. Roughly speaking, $V$ has to be ``big'' with respect to $b$ and $c$. We do not impose any conditions on $Ω$, in particular, we do not assume any regularity of $\partialΩ$, nor the existence of a Sobolev embedding. |
| title | Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J15, 47D06, 42B25 |
| url | https://arxiv.org/abs/2406.00774 |