Comments on: The Pohozaev identity: Semilinear elliptic problem with polygonal nonlinearity http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/ Học học nữa học mãi Thu, 09 May 2013 22:44:49 +0000 hourly 1 http://wordpress.com/ By: Urko http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-599 Mon, 01 Nov 2010 13:29:13 +0000 http://anhngq.wordpress.com/?p=3068#comment-599 Thanks Ngô,
I supposed that the mistake was in the sign of A_2 but i wasnt sure enough. Im trying to prove a Pohozaev-type identity for another equation (involving fractinal laplacians) and im using this proof as start point ;) .

Thanks again.

]]> By: Ngô Quốc Anh http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-598 Mon, 01 Nov 2010 04:26:39 +0000 http://anhngq.wordpress.com/?p=3068#comment-598 If you have some difficulty, here is my interpretation

\displaystyle\begin{gathered} {A_2} = \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } } \hfill \\ \quad= \int_{\partial \Omega } {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } } \hfill \\ \quad= \int_{\partial \Omega } {\sum\limits_{j = 1}^n {{x_j}{u_{{x_j}}}\left( {\sum\limits_{i = 1}^n {{u_{{x_i}}}{\nu ^i}} } \right)d\sigma } } \hfill \\ \quad= \int_{\partial \Omega } {\sum\limits_{j = 1}^n {{x_j}{u_{{x_j}}}\left( { \pm |\nabla u|\nu \cdot \nu } \right)d\sigma } } \hfill \\ \quad= \int_{\partial \Omega } {\left( { \pm |\nabla u|} \right)\left( {\sum\limits_{j = 1}^n {{x_j}{u_{{x_j}}}} } \right)d\sigma } \hfill \\ \quad= \int_{\partial \Omega } {\left( { \pm |\nabla u|} \right)\left( {\nabla u \cdot x} \right)d\sigma } \hfill \\ \quad= \int_{\partial \Omega } {\left( { \pm |\nabla u|} \right)\left( { \pm |\nabla u|\nu \cdot x} \right)d\sigma } \hfill \\ \quad= \int_{\partial \Omega } {|\nabla u{|^2}(\nu \cdot x)d\sigma }. \hfill \\ \end{gathered}

]]> By: Ngô Quốc Anh http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-597 Mon, 01 Nov 2010 04:17:50 +0000 http://anhngq.wordpress.com/?p=3068#comment-597 Hi Urko,

Thanks again. There is no doubt in both A and A_1. Concerning A_2, there is no minus sign in fact, the correct formula should be

\displaystyle {A_2} = \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } } = \int_{\partial \Omega } {|\nabla u{|^2}(\nu \cdot x)d\sigma }

therefore there is no mistake in the Pohozaev indentity stated in the definition.

Thanks a lot for pointing out the mistake and also for your interest in my blog.

]]> By: Urko http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-596 Sun, 31 Oct 2010 19:56:08 +0000 http://anhngq.wordpress.com/?p=3068#comment-596 In the calculation of A_2 seems to be a mistake with the signs. Since A=A_1-A_2 and the sign of A_2 is “-” then the formula following “Combining all gives” is wrong. Do you agree?

]]> By: Urko http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-595 Sun, 31 Oct 2010 13:28:42 +0000 http://anhngq.wordpress.com/?p=3068#comment-595 Right ;)

]]> By: Ngô Quốc Anh http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-594 Sun, 31 Oct 2010 02:39:22 +0000 http://anhngq.wordpress.com/?p=3068#comment-594 Dear Urko,

Thank you for pointing out the misprint. You are right, the formula is nothing but integration by parts, that is

\displaystyle\int_\Omega {{{({u_{{x_i}}})}_{{x_i}}}({x_j}{u_{{x_j}}})dx} = - \int_\Omega {{u_{{x_i}}}{{({x_j}{u_{{x_j}}})}_{{x_i}}}dx} + \int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma }.

]]> By: Urko http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-592 Sat, 30 Oct 2010 23:11:24 +0000 http://anhngq.wordpress.com/?p=3068#comment-592 Hi there,
First of all, thanks for these posts that help us so much. I was reading this articule and i think that in the third formula, second line (when you are using Green’s Theorem) its u_{x_{1}} instead of u.

Again Thanks!!

]]> By: HasPas http://anhngq.wordpress.com/2010/04/11/the-pohozaev-identity-semilinear-elliptic-problem-with-polygonal-nonlinearity/#comment-415 Fri, 16 Apr 2010 13:49:32 +0000 http://anhngq.wordpress.com/?p=3068#comment-415 Many Thanks. This entry is nice. I have learned much.

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