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• W4306178462 abstract "In this article we characterize the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper L Superscript normal infinity> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathrm {L}^infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> eigenvalue problem associated to the Rayleigh quotient <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar nabla u double-vertical-bar Subscript normal upper L Sub Superscript normal infinity Baseline slash double-vertical-bar u double-vertical-bar Subscript normal infinity> <mml:semantics> <mml:mrow> <mml:mo fence=true stretchy=true symmetric=true /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-CLOSE> </mml:mrow> <mml:mo stretchy=true>/</mml:mo> <mml:mrow class=MJX-TeXAtom-OPEN> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> <mml:mo fence=true stretchy=true symmetric=true /> </mml:mrow> <mml:annotation encoding=application/x-tex>left .{|nabla u|_{mathrm {L}^infty }}middle /{|u|_infty }right .</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and relate it to a divergence-form PDE, similarly to what is known for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper L Superscript p> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathrm {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> eigenvalue problems and the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Laplacian for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Contrary to existing methods, which study <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper L Superscript normal infinity> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathrm {L}^infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-problems as limits of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper L Superscript p> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathrm {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-problems for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p right-arrow normal infinity> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>pto infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u right-arrow from bar double-vertical-bar nabla u double-vertical-bar Subscript normal upper L Sub Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>↦<!-- ↦ --></mml:mo> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>L</mml:mi> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>umapsto |nabla u|_{mathrm {L}^infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the eigenvalue problem takes the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda nu u equals minus d i v left-parenthesis tau nabla Subscript tau Baseline u right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>ν<!-- ν --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>div</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:msub> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>τ<!-- τ --></mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda nu u =-operatorname {div}(tau nabla _tau u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=nu> <mml:semantics> <mml:mi>ν<!-- ν --></mml:mi> <mml:annotation encoding=application/x-tex>nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding=application/x-tex>tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are non-negative measures concentrated where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue u EndAbsoluteValue> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>|u|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> respectively <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue nabla u EndAbsoluteValue> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>|nabla u|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are maximal, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=nabla Subscript tau Baseline u> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>∇<!-- ∇ --></mml:mi> <mml:mi>τ<!-- τ --></mml:mi> </mml:msub> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>nabla _tau u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the tangential gradient of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding=application/x-tex>u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tau> <mml:semantics> <mml:mi>τ<!-- τ --></mml:mi> <mml:annotation encoding=application/x-tex>tau</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Lastly, we investigate a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich–Rubinstein norm. Our results apply to all stationary points of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature." @default.
• W4306178462 created "2022-10-14" @default.
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• W4306178462 date "2022-10-14" @default.
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• W4306178462 title "Eigenvalue problems in 𝐿^{∞}: optimality conditions, duality, and relations with optimal transport" @default.
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• W4306178462 doi "https://doi.org/10.1090/cams/11" @default.
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