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- W2121106507 abstract "We consider Poisson's equation in an<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$n$><mml:mi>n</mml:mi></mml:math>-dimensional exterior domain<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$G$><mml:mi>G</mml:mi></mml:math><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$(n geq 2)$><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$L^q(G)$><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-spaces there exists a solution in the homogeneous Sobolev space<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S^{2,q}(G)$><mml:msup><mml:mi>S</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, containing functions being local in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$L^q(G)$><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>and having second-order derivatives in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$L^q(G)$><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimension<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$n+1$><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>, independent of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$q$><mml:mi>q</mml:mi></mml:math>." @default.
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- W2121106507 date "2004-01-01" @default.
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- W2121106507 title "The Poisson equation in homogeneous Sobolev spaces" @default.
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