Proof of Courant-Fischer minimax theorem through deformation lemma
In my calculus of variations lecture notes it is claimed that if $Omega$ is an open bounded non-empty subset of $mathbb{R}^N$, then the eigenvalues of the laplacian on the Sobolev space $H^1_0(Omega)$, i.e. the values $lambdainmathbb{R}$ for which there exists $uin H^1_0(Omega)setminus{0}$ such that:
$$forall vin H^1_0(Omega), langle u,vrangle_{H^1_0}=lambdalangle u,vrangle_{L^2}$$
can be obtained through a inf-sup procedure, i.e. the Courant-Fischer method:
$$lambda_k=inf_{Vle H^1_0(Omega)\dim(V)ge k}sup_{uin Vcap S} |u|^2_{H^1_0}$$
where
$kinmathbb{N}$;
$S={uin H^1_0(Omega) | |u|_{L^2}=1}$;- the relation $Vle H^1_0(Omega)$ means that $V$ is a linear subspace of $H^1_0(Omega)$;
$dim(V)$ is the dimension of the linear space $V$.
In the notes, it is also claimed that the Lusternik-Schnirelmann inf-sup procedure is a generalization of the Courant-Fischer procedure and it is suggested that one can prove that the Courant-Fischer procedure produces critical values through a deformation argument, similar to the one given in the Lusternik-Schnirelmann theory: in the Lusternik-Schnirelmann procedure we get the inf-sup values over subsets of the $L^2$-sphere of genus at least $k$ and we use the deformation lemma to prove that these inf-sup values are actually critical values, while in the Courant-Fischer procedure we get the inf-sup values over subsets of the $L^2$-sphere that are the intersection of the $L^2$-sphere with a linear subspace of dimension at least $k$, and we can use a corresponding deformation lemma to prove that these inf-sup values are actually critical values.
How to build such a deformation to prove that the Courant-Fischer procedure produces critical values?
I think that what it's needed is to build a deformation that actually should be linear, in order to get that the deformation of a linear subspace is another linear subspace of at most the same dimension. Also, I think that the deformation should preserve the $L^2(Omega)$ norm, otherwise the deformation of $Vcap S$ where $V$ is a linear subspace of $H^1_0(Omega)$ can't be written in the form $Wcap S$ for some linear subspace of $H^1_0(Omega)$. Also, the deformation should decrease the $H^1_0(Omega)$ norm somehow, like in the deformation lemma used in Lusternik-Schnirelmann theory.
Can someone help providing hints to go on? Also, proofs or references where Courant-Fischer procedure is proved through a deformation argument are welcome.
differential-topology calculus-of-variations
asked Nov 23 at 23:43
Bob
1,4251623
add a comment |
In my calculus of variations lecture notes it is claimed that if $Omega$ is an open bounded non-empty subset of $mathbb{R}^N$, then the eigenvalues of the laplacian on the Sobolev space $H^1_0(Omega)$, i.e. the values $lambdainmathbb{R}$ for which there exists $uin H^1_0(Omega)setminus{0}$ such that:
$$forall vin H^1_0(Omega), langle u,vrangle_{H^1_0}=lambdalangle u,vrangle_{L^2}$$
can be obtained through a inf-sup procedure, i.e. the Courant-Fischer method:
$$lambda_k=inf_{Vle H^1_0(Omega)\dim(V)ge k}sup_{uin Vcap S} |u|^2_{H^1_0}$$
where
$kinmathbb{N}$;
$S={uin H^1_0(Omega) | |u|_{L^2}=1}$;- the relation $Vle H^1_0(Omega)$ means that $V$ is a linear subspace of $H^1_0(Omega)$;
$dim(V)$ is the dimension of the linear space $V$.
In the notes, it is also claimed that the Lusternik-Schnirelmann inf-sup procedure is a generalization of the Courant-Fischer procedure and it is suggested that one can prove that the Courant-Fischer procedure produces critical values through a deformation argument, similar to the one given in the Lusternik-Schnirelmann theory: in the Lusternik-Schnirelmann procedure we get the inf-sup values over subsets of the $L^2$-sphere of genus at least $k$ and we use the deformation lemma to prove that these inf-sup values are actually critical values, while in the Courant-Fischer procedure we get the inf-sup values over subsets of the $L^2$-sphere that are the intersection of the $L^2$-sphere with a linear subspace of dimension at least $k$, and we can use a corresponding deformation lemma to prove that these inf-sup values are actually critical values.
How to build such a deformation to prove that the Courant-Fischer procedure produces critical values?
I think that what it's needed is to build a deformation that actually should be linear, in order to get that the deformation of a linear subspace is another linear subspace of at most the same dimension. Also, I think that the deformation should preserve the $L^2(Omega)$ norm, otherwise the deformation of $Vcap S$ where $V$ is a linear subspace of $H^1_0(Omega)$ can't be written in the form $Wcap S$ for some linear subspace of $H^1_0(Omega)$. Also, the deformation should decrease the $H^1_0(Omega)$ norm somehow, like in the deformation lemma used in Lusternik-Schnirelmann theory.
Can someone help providing hints to go on? Also, proofs or references where Courant-Fischer procedure is proved through a deformation argument are welcome.
differential-topology calculus-of-variations
asked Nov 23 at 23:43
Bob
1,4251623
add a comment |
In my calculus of variations lecture notes it is claimed that if $Omega$ is an open bounded non-empty subset of $mathbb{R}^N$, then the eigenvalues of the laplacian on the Sobolev space $H^1_0(Omega)$, i.e. the values $lambdainmathbb{R}$ for which there exists $uin H^1_0(Omega)setminus{0}$ such that:
$$forall vin H^1_0(Omega), langle u,vrangle_{H^1_0}=lambdalangle u,vrangle_{L^2}$$
can be obtained through a inf-sup procedure, i.e. the Courant-Fischer method:
$$lambda_k=inf_{Vle H^1_0(Omega)\dim(V)ge k}sup_{uin Vcap S} |u|^2_{H^1_0}$$
where
$kinmathbb{N}$;
$S={uin H^1_0(Omega) | |u|_{L^2}=1}$;- the relation $Vle H^1_0(Omega)$ means that $V$ is a linear subspace of $H^1_0(Omega)$;
$dim(V)$ is the dimension of the linear space $V$.
In the notes, it is also claimed that the Lusternik-Schnirelmann inf-sup procedure is a generalization of the Courant-Fischer procedure and it is suggested that one can prove that the Courant-Fischer procedure produces critical values through a deformation argument, similar to the one given in the Lusternik-Schnirelmann theory: in the Lusternik-Schnirelmann procedure we get the inf-sup values over subsets of the $L^2$-sphere of genus at least $k$ and we use the deformation lemma to prove that these inf-sup values are actually critical values, while in the Courant-Fischer procedure we get the inf-sup values over subsets of the $L^2$-sphere that are the intersection of the $L^2$-sphere with a linear subspace of dimension at least $k$, and we can use a corresponding deformation lemma to prove that these inf-sup values are actually critical values.
How to build such a deformation to prove that the Courant-Fischer procedure produces critical values?
I think that what it's needed is to build a deformation that actually should be linear, in order to get that the deformation of a linear subspace is another linear subspace of at most the same dimension. Also, I think that the deformation should preserve the $L^2(Omega)$ norm, otherwise the deformation of $Vcap S$ where $V$ is a linear subspace of $H^1_0(Omega)$ can't be written in the form $Wcap S$ for some linear subspace of $H^1_0(Omega)$. Also, the deformation should decrease the $H^1_0(Omega)$ norm somehow, like in the deformation lemma used in Lusternik-Schnirelmann theory.
Can someone help providing hints to go on? Also, proofs or references where Courant-Fischer procedure is proved through a deformation argument are welcome.
differential-topology calculus-of-variations
asked Nov 23 at 23:43
Bob
1,4251623
In my calculus of variations lecture notes it is claimed that if $Omega$ is an open bounded non-empty subset of $mathbb{R}^N$, then the eigenvalues of the laplacian on the Sobolev space $H^1_0(Omega)$, i.e. the values $lambdainmathbb{R}$ for which there exists $uin H^1_0(Omega)setminus{0}$ such that:
$$forall vin H^1_0(Omega), langle u,vrangle_{H^1_0}=lambdalangle u,vrangle_{L^2}$$
can be obtained through a inf-sup procedure, i.e. the Courant-Fischer method:
$$lambda_k=inf_{Vle H^1_0(Omega)\dim(V)ge k}sup_{uin Vcap S} |u|^2_{H^1_0}$$
where
$kinmathbb{N}$;
$S={uin H^1_0(Omega) | |u|_{L^2}=1}$;- the relation $Vle H^1_0(Omega)$ means that $V$ is a linear subspace of $H^1_0(Omega)$;
$dim(V)$ is the dimension of the linear space $V$.
In the notes, it is also claimed that the Lusternik-Schnirelmann inf-sup procedure is a generalization of the Courant-Fischer procedure and it is suggested that one can prove that the Courant-Fischer procedure produces critical values through a deformation argument, similar to the one given in the Lusternik-Schnirelmann theory: in the Lusternik-Schnirelmann procedure we get the inf-sup values over subsets of the $L^2$-sphere of genus at least $k$ and we use the deformation lemma to prove that these inf-sup values are actually critical values, while in the Courant-Fischer procedure we get the inf-sup values over subsets of the $L^2$-sphere that are the intersection of the $L^2$-sphere with a linear subspace of dimension at least $k$, and we can use a corresponding deformation lemma to prove that these inf-sup values are actually critical values.
How to build such a deformation to prove that the Courant-Fischer procedure produces critical values?
I think that what it's needed is to build a deformation that actually should be linear, in order to get that the deformation of a linear subspace is another linear subspace of at most the same dimension. Also, I think that the deformation should preserve the $L^2(Omega)$ norm, otherwise the deformation of $Vcap S$ where $V$ is a linear subspace of $H^1_0(Omega)$ can't be written in the form $Wcap S$ for some linear subspace of $H^1_0(Omega)$. Also, the deformation should decrease the $H^1_0(Omega)$ norm somehow, like in the deformation lemma used in Lusternik-Schnirelmann theory.
Can someone help providing hints to go on? Also, proofs or references where Courant-Fischer procedure is proved through a deformation argument are welcome.
differential-topology calculus-of-variations
differential-topology calculus-of-variations
asked Nov 23 at 23:43
Bob
1,4251623
asked Nov 23 at 23:43
Bob
1,4251623
edited Nov 24 at 15:17
asked Nov 23 at 23:43
Bob
1,4251623
asked Nov 23 at 23:43
Bob
1,4251623
asked Nov 23 at 23:43
Bob
1,4251623
1,4251623
add a comment |
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