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Bilinear embedding for divergence-form operators with negative potentials

Andrea Poggio

TL;DR

The paper develops a dimension-free bilinear embedding theory for divergence-form operators with potentials that may be negative by introducing a perturbed $p$-ellipticity condition and a Bellman-function heat-flow approach. It proves $L^p$ contractivity and analyticity of the associated semigroups under these new structural assumptions, leading to maximal parabolic regularity and a bounded holomorphic functional calculus for the Schrödinger-type operators on general open sets. The analysis unifies and extends classical results for nonnegative potentials, Schrödinger operators, and complex-coefficient systems, providing extrapolation ranges and off-diagonal estimates through complex-time arguments. It also develops a robust framework for handling strongly subcritical potentials via Hardy-type inequalities and domain geometry, enabling a broad class of $V$ to be accommodated in the bilinear embedding and maximal-regularity theory.

Abstract

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].

Bilinear embedding for divergence-form operators with negative potentials

TL;DR

The paper develops a dimension-free bilinear embedding theory for divergence-form operators with potentials that may be negative by introducing a perturbed -ellipticity condition and a Bellman-function heat-flow approach. It proves contractivity and analyticity of the associated semigroups under these new structural assumptions, leading to maximal parabolic regularity and a bounded holomorphic functional calculus for the Schrödinger-type operators on general open sets. The analysis unifies and extends classical results for nonnegative potentials, Schrödinger operators, and complex-coefficient systems, providing extrapolation ranges and off-diagonal estimates through complex-time arguments. It also develops a robust framework for handling strongly subcritical potentials via Hardy-type inequalities and domain geometry, enabling a broad class of to be accommodated in the bilinear embedding and maximal-regularity theory.

Abstract

Let be open, a complex uniformly strictly accretive matrix-valued function on with coefficients, and a locally integrable function on whose negative part is subcritical. We consider the operator with mixed boundary conditions on . 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 -ellipticity when is nonnegative. As a consequence, we show that the solution to the parabolic problem with has maximal regularity on , in the same spirit as [13]. Moreover, we study mapping properties of the semigroup generated by under this new condition, thereby extending classical results for the Schrödinger operator on [8,47].

Paper Structure

This paper contains 39 sections, 44 theorems, 291 equations, 1 table.

Table of Contents

  1. Introduction and statement of the main results
  2. Strongly subcritical potentials
  3. The operator $-{\rm div}(A\nabla u) + Vu$
  4. Boundary conditions
  5. The $p$-ellipticity condition
  6. The perturbed $p$-ellipticity
  7. Semigroup properties on $L^p$
  8. Bilinear embeddings for nonegative potentials
  9. Maximal regularity and functional calculus
  10. Notation
  11. Organization of the paper
  12. The domain of the form
  13. Heat-flow monotonicity and generalized convexity
  14. Real form of complex operators
  15. Convexity with respect to complex matrices
  16. ...and 24 more sections

Key Result

Theorem 1.2

Suppose that ${\mathscr V}$ satisfies eq: inv P and eq: inv N. Choose $p>1$, $(A,V) \in {\mathcal{A}}{\mathcal{P}}(\Omega,{\mathscr V})$ and $\phi \in \mathbb{R}$ such that $|\phi| <\pi/2 - \vartheta_0$ and $(e^{i\phi}A, (\cos\phi)V) \in \widetilde{{\mathcal{A}}{\mathcal{P}}}_p(\Omega,{\mathscr V}) extends to a strongly continuous semigroup of contractions on $L^p(\Omega)$. If $V_-=0$, the same c

Theorems & Definitions (85)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • ...and 75 more