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• W2036766039 abstract "The Dirichlet problem <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta u equals lamda f left-parenthesis u right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mspace width=thinmathspace /> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Delta u = lambda ,f(u)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega comma u equals 1> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega ,,u = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is considered with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis t right-parenthesis equals 0> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>f(t) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t less-than-or-equal-to 0 comma f left-parenthesis t right-parenthesis greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>t leq 0,,f(t) > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t greater-than 0 comma f left-parenthesis t right-parenthesis tilde t Superscript p> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>∼<!-- ∼ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>t > 0,,f(t) sim {t^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t down-arrow 0 comma 0 greater-than p greater-than 1 semicolon f left-parenthesis t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy=false>↓<!-- ↓ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>;</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>t downarrow 0,0 > p > 1;f(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not monotone in general. The set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace u equals 0 right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{ u = 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the “free boundary” <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential left-brace u equals 0 right-brace> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>partial { u = 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are studied. Sharp asymptotic estimates are established as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda right-arrow normal infinity> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda to infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For suitable <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, under the assumption that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a two-dimensional convex domain, it is shown that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace u equals 0 right-brace> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{ u = 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a convex set. Analogous results are established also in the case where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential u slash partial-differential v plus mu left-parenthesis u minus 1 right-parenthesis equals 0> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>v</mml:mi> <mml:mo>+</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>partial u/partial v + mu (u - 1) = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2036766039 created "2016-06-24" @default.
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• W2036766039 date "1984-01-01" @default.
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• W2036766039 title "The free boundary of a semilinear elliptic equation" @default.
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