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- W1978392779 abstract "We study the Poisson problem <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta u equals f> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta u = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Helmholtz problem <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta u plus lamda u equals f> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta u + lambda u = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in bounded domains with angular corners in the plane and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u equals 0> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>u=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but not in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H squared> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in general, even though <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> may be very regular. We formulate these as variational problems in weighted Sobolev spaces and prove existence and uniqueness of solutions in what would be weighted counterparts of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H squared intersection upper H 0 Superscript 1> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding=application/x-tex>H^2 cap H^1_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The specific forms of our variational formulations are motivated by, and are particularly suited to, applying a finite element scheme for solving the time-dependent Navier-Stokes equations of fluid mechanics." @default.
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- W1978392779 date "2010-05-19" @default.
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- W1978392779 title "Variational problems in weighted Sobolev spaces on non-smooth domains" @default.
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