Ngô Quốc Anh

April 11, 2010

The Pohozaev identity: Semilinear elliptic problem with polygonal nonlinearity

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 12:15

Let us start with the Pohozaev identity for semilinear elliptic equation with polygonal nonlinear terms of the form

\displaystyle -\Delta u = |u|^{p-1}u

over an open, star-shaped domain \Omega. We also assume u is identical to zero on the boundary \partial \Omega.

We multiply the PDE by x \cdot \nabla u and integrate over \Omega to find

\displaystyle\int_\Omega {\left( { - \Delta u} \right)\left( {x \cdot \nabla u} \right)} dx = \int_\Omega {|u{|^{p - 1}}u\left( {x \cdot \nabla u} \right)} dx.

The term on the left is just

\displaystyle\begin{gathered} A = - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}{x_i}}}{x_j}{u_{{x_j}}}dx} } } \hfill \\ \quad= - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left[ { - \int_\Omega {{u_{{x_i}}}{{({x_j}u)}_{{x_i}}}dx} + \int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } \right]} } \hfill \\ \quad= \underbrace {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}}}{{({x_j}u_{x_j})}_{{x_i}}}dx} } } }_{{A_1}} - \underbrace {\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 } } } }_{{A_2}}. \hfill \\ \end{gathered}

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