arXiv subjects

Andrea Poggio

Publications and source records attributed to Andrea Poggio.

3 recordsLinked to original sources

arXiv2026

Bilinear embedding for divergence-form operators with first-order terms and negative potentials

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.

math.AP↗

arXiv2026

Bilinear embedding for divergence-form operators with negative potentials

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].

math.AP↗

arXiv2024

Bilinear embedding for perturbed divergence-form operator with complex coefficients on irregular domains

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.

math.AP↗