Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subseteq \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> be open, A a complex uniformly strictly accretive <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d\times d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> matrix-valued function on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> coefficients, and V a locally integrable function whose negative part is subcritical. We consider the operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${{\mathscr {L}}}= -\textrm{div}(A\nabla ) + V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>=</mo> <mo>-</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> with mixed boundary conditions on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. We extend the bilinear embedding of Carbonaro and Dragičević (Publ Mat 69(1):83–108, 2025), established for nonnegative potentials under the p-ellipticity of the matrices, by introducing a novel condition on the coefficients that reduces to p-ellipticity when V is nonnegative. As a consequence, the solution to the parabolic problem <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$u^\prime (t) + {{\mathscr {L}}}u(t) = f(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi mathvariant="script">L</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$u(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> has maximal regularity on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^p(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, under this new condition, we study mapping properties of the semigroup generated by <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$-{{\mathscr {L}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation>, thereby extending classical results for the Schrödinger operator <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$-\Delta + V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>.

Bilinear embedding for divergence-form operators with negative potentials

期刊论文
发表日期：

2026年5月18日

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Andrea Poggio

摘要

Let $\Omega \subseteq \mathbb {R}^d$ $Ω \subseteq R^{d}$ be open, A a complex uniformly strictly accretive $d\times d$ $d \times d$ matrix-valued function on $\Omega $ $Ω$ with $L^\infty $ $L^{\infty}$ coefficients, and V a locally integrable function whose negative part is subcritical. We consider the operator ${{\mathscr {L}}}= -\textrm{div}(A\nabla ) + V$ $L = - div (A \nabla) + V$ with mixed boundary conditions on $\Omega $ $Ω$ . We extend the bilinear embedding of Carbonaro and Dragičević (Publ Mat 69(1):83–108, 2025), established for nonnegative potentials under the p-ellipticity of the matrices, by introducing a novel condition on the coefficients that reduces to p-ellipticity when V is nonnegative. As a consequence, the solution to the parabolic problem $u^\prime (t) + {{\mathscr {L}}}u(t) = f(t)$ $u^{'} (t) + L u (t) = f (t)$ with $u(0)=0$ $u (0) = 0$ has maximal regularity on $L^p(\Omega )$ $L^{p} (Ω)$ . Moreover, under this new condition, we study mapping properties of the semigroup generated by $-{{\mathscr {L}}}$ $- L$ , thereby extending classical results for the Schrödinger operator $-\Delta + V$ $- Δ + V$ on $\mathbb {R}^d$ $R^{d}$ .