1992 | OriginalPaper | Buchkapitel
A Collocation Method for a Screen Problem in ℝ3
verfasst von : Martin Costabel, Frank Penzel, Reinhold Schneider
Erschienen in: Symposium “Analysis on Manifolds with Singularities”, Breitenbrunn 1990
Verlag: Vieweg+Teubner Verlag
Enthalten in: Professional Book Archive
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Let Ω: = (−1,1) × (−1,1) ⊂ ℝ2. The first kind integral equation on Ω, 1.1<m:math display='block'> <m:semantics> <m:mrow> <m:mi>V</m:mi><m:mi>u</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>:</m:mo><m:mo>=</m:mo><m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>4</m:mn><m:mi>π</m:mi> </m:mrow> </m:mfrac> <m:mstyle displaystyle='true'> <m:mrow> <m:munder> <m:mo>∫</m:mo> <m:mi>Ω</m:mi> </m:munder> <m:mrow> <m:mi>u</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>y</m:mi> <m:mo>)</m:mo></m:mrow><m:mfrac> <m:mrow> <m:mi>d</m:mi><m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mrow><m:mo>|</m:mo> <m:mrow> <m:mi>x</m:mi><m:mo>−</m:mo><m:mi>y</m:mi> </m:mrow> <m:mo>|</m:mo></m:mrow> </m:mrow> </m:mfrac> <m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mtext> </m:mtext><m:mrow><m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi><m:mo>∈</m:mo><m:mi>Ω</m:mi> </m:mrow> <m:mo>)</m:mo></m:mrow> </m:mrow> </m:mrow> </m:mstyle> </m:mrow> <m:annotation encoding='MathType-MTEF'> </m:annotation> </m:semantics> </m:math>$$ Vu\left( x \right): = \frac{1} {{4\pi }}\int\limits_\Omega {u\left( y \right)\frac{{dy}} {{\left| {x - y} \right|}} = f\left( x \right)\quad \left( {x \in \Omega } \right)} $$ gives the solution of the Dirichlet problem for the Laplace equation in ℝ3\Ω̄ with Dirichlet data f given on Ω, (“screen problem”) (see [6], [16]).