Bilinear embedding for divergence-form operators with first-order terms and negative potentials
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| Format: | Preprint |
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2026
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| _version_ | 1866918501596266496 |
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| author | Morelato, Lorenzo Luciano Poggio, Andrea |
| author_facet | Morelato, Lorenzo Luciano Poggio, Andrea |
| contents | This article establishes a bilinear embedding for second-order divergence-form operators with complex coefficients, characterized by the simultaneous presence of first-order terms and negative potentials. This work provides a further development of the theory initiated by Carbonaro and Dragičević for the homogeneous case, and recently extended by the second author to cases where first-order terms or negative potentials were treated in isolation.
We work in the general setting of arbitrary open subsets of $\mathbb{R}^d$ under Dirichlet, Neumann, or mixed boundary conditions. Our main contribution is the introduction of a unified notion of generalized $p$-ellipticity that extends all its predecessors and serves as the natural condition for the bilinear inequality. Methodologically, we overcome the rigidity of the Bellman-heat method on arbitrary open subsets by introducing a novel sequence-based approach that unifies and simplifies the previous techniques.
As fundamental applications, we prove the boundedness of the $H^\infty$-calculus on $L^p$ and establish $L^p$-maximal regularity. Moreover, we show that this generalized $p$-ellipticity provides a sufficient condition for the $L^p$-contractivity and $L^p$-analyticity of the generated semigroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14699 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bilinear embedding for divergence-form operators with first-order terms and negative potentials Morelato, Lorenzo Luciano Poggio, Andrea Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J15, 47D06, 42B25, 47A60 This article establishes a bilinear embedding for second-order divergence-form operators with complex coefficients, characterized by the simultaneous presence of first-order terms and negative potentials. This work provides a further development of the theory initiated by Carbonaro and Dragičević for the homogeneous case, and recently extended by the second author to cases where first-order terms or negative potentials were treated in isolation. We work in the general setting of arbitrary open subsets of $\mathbb{R}^d$ under Dirichlet, Neumann, or mixed boundary conditions. Our main contribution is the introduction of a unified notion of generalized $p$-ellipticity that extends all its predecessors and serves as the natural condition for the bilinear inequality. Methodologically, we overcome the rigidity of the Bellman-heat method on arbitrary open subsets by introducing a novel sequence-based approach that unifies and simplifies the previous techniques. As fundamental applications, we prove the boundedness of the $H^\infty$-calculus on $L^p$ and establish $L^p$-maximal regularity. Moreover, we show that this generalized $p$-ellipticity provides a sufficient condition for the $L^p$-contractivity and $L^p$-analyticity of the generated semigroup. |
| title | Bilinear embedding for divergence-form operators with first-order terms and negative potentials |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35J15, 47D06, 42B25, 47A60 |
| url | https://arxiv.org/abs/2605.14699 |