Bilinear embedding for divergence-form operators with negative potentials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911672546885632 |
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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, and $V$ a locally integrable function on $Ω$ whose negative part is subcritical. We consider the operator $\mathscr{L} = -\mathrm{div}(A\nabla) + V$ with mixed boundary conditions on $Ω$. We extend the bilinear inequality of Carbonaro and Dragičević [15], originally established for nonnegative potentials, by introducing a novel condition on the coefficients that reduces to standard $p$-ellipticity when $V$ is nonnegative. As a consequence, we show that the solution to the parabolic problem $u'(t) + \mathscr{L} u(t) = f(t)$ with $u(0)=0$ has maximal regularity on $L^p(Ω)$, in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by $-\mathscr{L}$ under this new condition, thereby extending classical results for the Schrödinger operator $-Δ+ V$ on $\mathbb{R}^d$ [8,47]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bilinear embedding for divergence-form operators with negative potentials Poggio, Andrea Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J10, 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, and $V$ a locally integrable function on $Ω$ whose negative part is subcritical. We consider the operator $\mathscr{L} = -\mathrm{div}(A\nabla) + V$ with mixed boundary conditions on $Ω$. We extend the bilinear inequality of Carbonaro and Dragičević [15], originally established for nonnegative potentials, by introducing a novel condition on the coefficients that reduces to standard $p$-ellipticity when $V$ is nonnegative. As a consequence, we show that the solution to the parabolic problem $u'(t) + \mathscr{L} u(t) = f(t)$ with $u(0)=0$ has maximal regularity on $L^p(Ω)$, in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by $-\mathscr{L}$ under this new condition, thereby extending classical results for the Schrödinger operator $-Δ+ V$ on $\mathbb{R}^d$ [8,47]. |
| title | Bilinear embedding for divergence-form operators with negative potentials |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J10, 35J15, 47D06, 42B25 |
| url | https://arxiv.org/abs/2510.05714 |