{"title":"Soap bubbles and convex cones: optimal quantitative rigidity","authors":"Giorgio Poggesi","doi":"10.1090/tran/9207","DOIUrl":null,"url":null,"abstract":"<p>We consider a class of recent rigidity results in a convex cone <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma subset-of-or-equal-to double-struck upper R Superscript upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Sigma \\subseteq \\mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These include overdetermined Serrin-type problems for a mixed boundary value problem relative to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, Alexandrov’s soap bubble-type results relative to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and Heintze-Karcher’s inequality relative to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma equals double-struck upper R Superscript upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Sigma = \\mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> <p>Stability bounds in terms of the Hausdorff distance are also provided.</p> <p>Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential normal upper Sigma\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂</mml:mi> <mml:mi mathvariant=\"normal\">Σ</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\partial \\Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> <p>We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma subset-of-or-equal-to double-struck upper R Superscript upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Sigma \\subseteq \\mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Sigma equals double-struck upper R Superscript upper N\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Σ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Sigma =\\mathbb {R}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9207","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider a class of recent rigidity results in a convex cone Σ⊆RN\Sigma \subseteq \mathbb {R}^N. These include overdetermined Serrin-type problems for a mixed boundary value problem relative to Σ\Sigma, Alexandrov’s soap bubble-type results relative to Σ\Sigma, and Heintze-Karcher’s inequality relative to Σ\Sigma. Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L2L^2-pseudodistance. In particular, the optimal stability estimate for Heintze-Karcher’s inequality is new even in the classical case Σ=RN\Sigma = \mathbb {R}^N.
Stability bounds in terms of the Hausdorff distance are also provided.
Several new results are established and exploited, including a new Poincaré-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory – relative to the cone Σ\Sigma – for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in ∂Σ\partial \Sigma.
We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone Σ⊆RN\Sigma \subseteq \mathbb {R}^N, which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case Σ=RN\Sigma =\mathbb {R}^N, these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).
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