Connection between Rayleigh's formula and solution of the Steklov eigenvalue problem The Next CEO of Stack OverflowFirst Weighted Eigenvalue of the LaplacianWeak formulation for nonhomogeneous problem $-Delta u = 0$rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)Does the solution of my variational problem satisfy a specific Neumann boundary condition?Calculation of $fracdlambdadt$ under volume preserving mean curvature flowSecond Steklov operator and Dirichet-to-Neumann operator for a diskEnergy functional of Poisson equationProof that Poisson formula solves the Neumann Problem for Laplace Equation in Unit DiskLaplace equation with the Robin's boundary problemAbout positivity of a solution to a sub-critical semilinear elliptic problem
Can I equip Skullclamp on a creature I am sacrificing?
How does the Z80 determine which peripheral sent an interrupt?
Why am I allowed to create multiple unique pointers from a single object?
How did people program for Consoles with multiple CPUs?
Why do professional authors make "consistency" mistakes? And how to avoid them?
How do I go from 300 unfinished/half written blog posts, to published posts?
Interfacing a button to MCU (and PC) with 50m long cable
What's the best way to handle refactoring a big file?
Why don't programming languages automatically manage the synchronous/asynchronous problem?
Novel about a guy who is possessed by the divine essence and the world ends?
Is there an analogue of projective spaces for proper schemes?
Is it my responsibility to learn a new technology in my own time my employer wants to implement?
Is it ever safe to open a suspicious html file (e.g. email attachment)?
Is it professional to write unrelated content in an almost-empty email?
Is micro rebar a better way to reinforce concrete than rebar?
Why does the UK parliament need a vote on the political declaration?
If the heap is initialized for security, then why is the stack uninitialized?
How did the Bene Gesserit know how to make a Kwisatz Haderach?
Make solar eclipses exceedingly rare, but still have new moons
Help understanding this unsettling image of Titan, Epimetheus, and Saturn's rings?
Why has the US not been more assertive in confronting Russia in recent years?
Calculus II Question
In excess I'm lethal
What benefits would be gained by using human laborers instead of drones in deep sea mining?
Connection between Rayleigh's formula and solution of the Steklov eigenvalue problem
The Next CEO of Stack OverflowFirst Weighted Eigenvalue of the LaplacianWeak formulation for nonhomogeneous problem $-Delta u = 0$rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)Does the solution of my variational problem satisfy a specific Neumann boundary condition?Calculation of $fracdlambdadt$ under volume preserving mean curvature flowSecond Steklov operator and Dirichet-to-Neumann operator for a diskEnergy functional of Poisson equationProof that Poisson formula solves the Neumann Problem for Laplace Equation in Unit DiskLaplace equation with the Robin's boundary problemAbout positivity of a solution to a sub-critical semilinear elliptic problem
$begingroup$
Given the Steklov eigenvalue problem
$$
beginalign*
Delta u &= 0 ~~~textin B_r \
partial_nu u &= lambda u ~~~textin partial B_r.
endalign*
$$
Then we can express the first positiv eigenvalue $lambda_1$ of this problem via Rayleigh's formula
$$
lambda_1 :=min_uin C^infty(B_R) leftlbrace fracint_B_R int_partial B_R u^2~~|~~int_partial B_R u = 0 rightrbrace.
$$
Say now that this minimum is obtained at $u^*$.
My Question now is, if $u^*$ also solves the Steklov Prolem?
functional-analysis analysis pde
asked 2 days ago
BaraBara
6510
$endgroup$
add a comment |
$begingroup$
Given the Steklov eigenvalue problem
$$
beginalign*
Delta u &= 0 ~~~textin B_r \
partial_nu u &= lambda u ~~~textin partial B_r.
endalign*
$$
Then we can express the first positiv eigenvalue $lambda_1$ of this problem via Rayleigh's formula
$$
lambda_1 :=min_uin C^infty(B_R) leftlbrace fracint_B_R int_partial B_R u^2~~|~~int_partial B_R u = 0 rightrbrace.
$$
Say now that this minimum is obtained at $u^*$.
My Question now is, if $u^*$ also solves the Steklov Prolem?
functional-analysis analysis pde
asked 2 days ago
BaraBara
6510
$endgroup$
1
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago
add a comment |
$begingroup$
Given the Steklov eigenvalue problem
$$
beginalign*
Delta u &= 0 ~~~textin B_r \
partial_nu u &= lambda u ~~~textin partial B_r.
endalign*
$$
Then we can express the first positiv eigenvalue $lambda_1$ of this problem via Rayleigh's formula
$$
lambda_1 :=min_uin C^infty(B_R) leftlbrace fracint_B_R int_partial B_R u^2~~|~~int_partial B_R u = 0 rightrbrace.
$$
Say now that this minimum is obtained at $u^*$.
My Question now is, if $u^*$ also solves the Steklov Prolem?
functional-analysis analysis pde
asked 2 days ago
BaraBara
6510
$endgroup$
Given the Steklov eigenvalue problem
$$
beginalign*
Delta u &= 0 ~~~textin B_r \
partial_nu u &= lambda u ~~~textin partial B_r.
endalign*
$$
Then we can express the first positiv eigenvalue $lambda_1$ of this problem via Rayleigh's formula
$$
lambda_1 :=min_uin C^infty(B_R) leftlbrace fracint_B_R int_partial B_R u^2~~|~~int_partial B_R u = 0 rightrbrace.
$$
Say now that this minimum is obtained at $u^*$.
My Question now is, if $u^*$ also solves the Steklov Prolem?
functional-analysis analysis pde
functional-analysis analysis pde
asked 2 days ago
BaraBara
6510
asked 2 days ago
BaraBara
6510
asked 2 days ago
BaraBara
6510
asked 2 days ago
BaraBara
6510
asked 2 days ago
BaraBara
6510
6510
1
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago
add a comment |
1
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago
1
1
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3164693%2fconnection-between-rayleighs-formula-and-solution-of-the-steklov-eigenvalue-pro%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3164693%2fconnection-between-rayleighs-formula-and-solution-of-the-steklov-eigenvalue-pro%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
What you have there is the lowest eigenvalue of the Laplacian $-Delta$ with Dirichlet boundary conditions. I don't see how this is related to your problem.
$endgroup$
– amsmath
2 days ago
$begingroup$
The Dirichlet eigenvalue problem should pop up if the denominator would be an integral over $B_R$ not $partial B_R$ or am I mistaken there?
$endgroup$
– Bara
2 days ago
$begingroup$
Oh, correct. Missed that, sorry. But how do you get this minimum? What is your operator?
$endgroup$
– amsmath
2 days ago
$begingroup$
Just read your actual question. If $A$ is a self-adjoint operator which is bounded below by $lambda_1$ and $(Au,u) = lambda_1|u|^2$, then $Au = lambda_1 u$.
$endgroup$
– amsmath
2 days ago