Elliptic PDEs on unbounded sets
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I recently tried to apply the classical methods of Sobolev Spaces and the Lax-Milgram-theorem to a PDE on an unbounded domain. This did not work, since the Rellich-Kondrachov-Theorem can not be used and there is no Poincaré-inequality.
Then I read online that it is possible to define weighted Sobolev Spaces - for example $H^{1}_{0}(Omega)$ equipped with the norm
$|u|:=(int_{Omega}frac{u^{2}}{(1+|x|)^{2}}+(nabla{u})^{2})^{frac{1}{2}}$
in order to establish the desired inequality of the form:
$int_{Omega}frac{u^{2}}{(1+|x|)^{2}}le C int_{Omega}(nabla u)^{2}$
I suspect one could use the Hardy-inequality in order to prove it, but I couldn´t find any references.
Does anyone have any literature suggestions on weighted spaces for PDEs on unbounded sets?
And is it possible to prove that any solution in such a space decays pointwise at infinity?
sobolev-spaces
asked Dec 17 '18 at 20:35
Falc14Falc14
164
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add a comment |
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I recently tried to apply the classical methods of Sobolev Spaces and the Lax-Milgram-theorem to a PDE on an unbounded domain. This did not work, since the Rellich-Kondrachov-Theorem can not be used and there is no Poincaré-inequality.
Then I read online that it is possible to define weighted Sobolev Spaces - for example $H^{1}_{0}(Omega)$ equipped with the norm
$|u|:=(int_{Omega}frac{u^{2}}{(1+|x|)^{2}}+(nabla{u})^{2})^{frac{1}{2}}$
in order to establish the desired inequality of the form:
$int_{Omega}frac{u^{2}}{(1+|x|)^{2}}le C int_{Omega}(nabla u)^{2}$
I suspect one could use the Hardy-inequality in order to prove it, but I couldn´t find any references.
Does anyone have any literature suggestions on weighted spaces for PDEs on unbounded sets?
And is it possible to prove that any solution in such a space decays pointwise at infinity?
sobolev-spaces
asked Dec 17 '18 at 20:35
Falc14Falc14
164
$endgroup$
add a comment |
$begingroup$
I recently tried to apply the classical methods of Sobolev Spaces and the Lax-Milgram-theorem to a PDE on an unbounded domain. This did not work, since the Rellich-Kondrachov-Theorem can not be used and there is no Poincaré-inequality.
Then I read online that it is possible to define weighted Sobolev Spaces - for example $H^{1}_{0}(Omega)$ equipped with the norm
$|u|:=(int_{Omega}frac{u^{2}}{(1+|x|)^{2}}+(nabla{u})^{2})^{frac{1}{2}}$
in order to establish the desired inequality of the form:
$int_{Omega}frac{u^{2}}{(1+|x|)^{2}}le C int_{Omega}(nabla u)^{2}$
I suspect one could use the Hardy-inequality in order to prove it, but I couldn´t find any references.
Does anyone have any literature suggestions on weighted spaces for PDEs on unbounded sets?
And is it possible to prove that any solution in such a space decays pointwise at infinity?
sobolev-spaces
asked Dec 17 '18 at 20:35
Falc14Falc14
164
$endgroup$
I recently tried to apply the classical methods of Sobolev Spaces and the Lax-Milgram-theorem to a PDE on an unbounded domain. This did not work, since the Rellich-Kondrachov-Theorem can not be used and there is no Poincaré-inequality.
Then I read online that it is possible to define weighted Sobolev Spaces - for example $H^{1}_{0}(Omega)$ equipped with the norm
$|u|:=(int_{Omega}frac{u^{2}}{(1+|x|)^{2}}+(nabla{u})^{2})^{frac{1}{2}}$
in order to establish the desired inequality of the form:
$int_{Omega}frac{u^{2}}{(1+|x|)^{2}}le C int_{Omega}(nabla u)^{2}$
I suspect one could use the Hardy-inequality in order to prove it, but I couldn´t find any references.
Does anyone have any literature suggestions on weighted spaces for PDEs on unbounded sets?
And is it possible to prove that any solution in such a space decays pointwise at infinity?
sobolev-spaces
sobolev-spaces
asked Dec 17 '18 at 20:35
Falc14Falc14
164
asked Dec 17 '18 at 20:35
Falc14Falc14
164
asked Dec 17 '18 at 20:35
Falc14Falc14
164
asked Dec 17 '18 at 20:35
Falc14Falc14
164
asked Dec 17 '18 at 20:35
Falc14Falc14
164
164
add a comment |
add a comment |
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