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- W4283702920 abstract "Abstract In this paper, we study the Γ-limit, as <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>p</m:mi> <m:mo>→</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {pto 1} , of the functional <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>J</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>u</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo></m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mi>u</m:mi> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mi>u</m:mi> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:mfrac> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> J_{p}(u)=frac{int_{Omega}lvertnabla urvert^{p}+betaint_{partialOmega% }lvert urvert^{p}}{int_{Omega}lvert urvert^{p}}, where Ω is a smooth bounded open set in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> {mathbb{R}^{N}} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {p>1} and β is a real number. Among our results, for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>β</m:mi> <m:mo>></m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {beta>-1} , we derive an isoperimetric inequality for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi mathvariant=normal>Λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:munder> <m:mo movablelimits=false>inf</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>BV</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace=4.2pt>,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>≢</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> </m:munder> <m:mo></m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mrow> <m:mi>min</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>β</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo></m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mi>u</m:mi> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:mi mathvariant=normal>Ω</m:mi> </m:msub> <m:mrow> <m:mo fence=true stretchy=false>|</m:mo> <m:mi>u</m:mi> <m:mo fence=true stretchy=false>|</m:mo> </m:mrow> </m:mrow> </m:mfrac> </m:mrow> </m:mrow> </m:math> Lambda(Omega,beta)=inf_{uinoperatorname{BV}(Omega),,unotequiv 0}% frac{lvert Durvert(Omega)+min(beta,1)int_{partialOmega}lvert urvert% }{int_{Omega}lvert urvert} which is the limit as <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>p</m:mi> <m:mo>→</m:mo> <m:msup> <m:mn>1</m:mn> <m:mo>+</m:mo> </m:msup> </m:mrow> </m:math> {pto 1^{+}} of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>min</m:mi> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:msub> <m:mo></m:mo> <m:msub> <m:mi>J</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>u</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {lambda(Omega,p,beta)=min_{uin W^{1,p}(Omega)}J_{p}(u)} . We show that among all bounded and smooth open sets with given volume, the ball maximizes <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=normal>Λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {Lambda(Omega,beta)} when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {betain(-1,0)} and minimizes <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi mathvariant=normal>Λ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi mathvariant=normal>Ω</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {Lambda(Omega,beta)} when <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=false>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>∞</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {betain[0,infty)} ." @default.
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- W4283702920 date "2022-06-29" @default.
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- W4283702920 title "On the behavior of the first eigenvalue of the <i>p</i>-Laplacian with Robin boundary conditions as <i>p</i> goes to 1" @default.
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