Given an open bounded subset Ω of ℝn we consider the eigenvalue problem<Equation ID="Equa"> <EquationSource Format="TEX">$\left\{{\matrix{{\Delta u - \left\langle {\nabla u,\nabla V} \right\rangle = - {\lambda _V}u,} & {u > 0\,{\text{in}}\,\Omega,} \cr {u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{on}}\,\partial \Omega,\,\,} \cr}} \right.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>−</mo> <mrow> <mo>⟨</mo> <mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">∇</mi> <mi>V</mi> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <msub> <mi>λ</mi> <mi>V</mi> </msub> </mrow> <mi>u</mi> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>u</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> <mrow> <mtext>in</mtext> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> </mrow> </mtd> <mtd> <mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow> <mtext>on</mtext> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation>where V is a given function defined in Ω and λV is the relevant eigenvalue. We determine sufficient conditions on V such that if Ω is convex, the solution u is log-concave. We also determine sufficient conditions ensuring that λV, as a function of the set Ω, verifies a convexity inequality with respect to the Minkowski addition of sets.

Log-concavity of the first Dirichlet eigenfunction of some elliptic differential operators and convexity inequalities for the relevant eigenvalue

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发表日期：

2025年1月24日

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Andrea Colesanti

摘要

Given an open bounded subset Ω of ℝⁿ we consider the eigenvalue problem

$\left\{{\matrix{{\Delta u - \left\langle {\nabla u,\nabla V} \right\rangle = - {\lambda _V}u,} & {u > 0\,{\text{in}}\,\Omega,} \cr {u = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{on}}\,\partial \Omega,\,\,} \cr}} \right.$ ${\begin{matrix} Δ u - ⟨ \nabla u, \nabla V ⟩ = - λ_{V} u, & u > 0 in Ω, \\ u = 0 & on \partial Ω, \end{matrix}$

where V is a given function defined in Ω and λ_V is the relevant eigenvalue. We determine sufficient conditions on V such that if Ω is convex, the solution u is log-concave. We also determine sufficient conditions ensuring that λ_V, as a function of the set Ω, verifies a convexity inequality with respect to the Minkowski addition of sets.