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}

Now

\displaystyle\begin{gathered} {A_1} = \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {\left[ {{u_{{x_i}}}{\delta _{ij}}{u_{{x_j}}} + {u_{{x_i}}}{x_j}{u_{{x_j}{x_i}}}} \right]dx} } } \hfill \\ \quad= \int_\Omega {|\nabla u{|^2}dx} + \int_\Omega {\sum\limits_{j = 1}^n {{{\left( {\frac{{|\nabla u{|^2}}}{2}} \right)}_{{x_j}}}{x_j}} dx} \hfill \\ \quad= \int_\Omega {|\nabla u{|^2}dx} + \sum\limits_{j = 1}^n {\left[ {\int_\Omega {{{\left( {\frac{{|\nabla u{|^2}}}{2}} \right)}_{{x_j}}}{x_j}dx} } \right]} \hfill \\ \quad= \int_\Omega {|\nabla u{|^2}dx} + \sum\limits_{j = 1}^n {\left[ {\int_{\partial\Omega} {\frac{{|\nabla u{|^2}}}{2}{\nu ^i}{x_j}dx} - \int_\Omega {\frac{{|\nabla u{|^2}}}{2}dx} } \right]} \hfill \\ \quad= \int_\Omega {|\nabla u{|^2}dx} + \sum\limits_{j = 1}^n {\left[ {\int_{\partial\Omega} {\frac{{|\nabla u{|^2}}}{2}{\nu ^i}{x_j}dx} } \right]} - \sum\limits_{j = 1}^n {\left[ {\int_\Omega {\frac{{|\nabla u{|^2}}}{2}dx} } \right]} \hfill \\ \quad= \left( {1 - \frac{n}{2}} \right)\int_\Omega {|\nabla u{|^2}dx} + \int_{\partial\Omega} {\frac{{|\nabla u{|^2}}}{2}(\nu \cdot x)dx} . \hfill \\ \end{gathered}

On the other hand, since u=0 on \partial \Omega, \nabla u is parallel to the normal at each point x \in \partial \Omega. Thus

\nabla u = \pm |\nabla u|\nu .

Using this equality we calculate

\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 } .

Combining all gives

\displaystyle A = \frac{{2 - n}}{2}\int_\Omega {|\nabla u{|^2}dx} - \frac{1}{2}\int_{\partial \Omega } {|\nabla u{|^2}(\nu \cdot x)d\sigma } .

We now evaluate the right hand side.

\displaystyle\begin{gathered} B = \int_\Omega {|u{|^{p - 1}}u\left( {x \cdot \nabla u} \right)} dx \hfill \\ \quad= \sum\limits_{j = 1}^n {\int_\Omega {|u{|^{p - 1}}u{x_j}{u_{{x_j}}}} dx} \hfill \\ \quad= \sum\limits_{j = 1}^n {\int_\Omega {{{\left( {\frac{{|u{|^{p + 1}}}}{{p + 1}}} \right)}_{{x_j}}}{x_j}} dx} \hfill \\ \quad= \sum\limits_{j = 1}^n {\left[ {\int_{\partial \Omega } {\frac{{|u{|^{p + 1}}}}{{p + 1}}{\nu ^j}{x_j}} dx - \int_\Omega {\left( {\frac{{|u{|^{p + 1}}}}{{p + 1}}} \right)} dx} \right]} \hfill \\ \quad= - \sum\limits_{j = 1}^n {\left[ {\int_\Omega {\left( {\frac{{|u{|^{p + 1}}}}{{p + 1}}} \right)} dx} \right]} \hfill \\ \quad= - \frac{n}{{p + 1}}\int_\Omega {|u{|^{p + 1}}dx} . \hfill \\ \end{gathered}

Therefore the identity becomes

\displaystyle\frac{{n - 2}}{2}\int_\Omega {|\nabla u{|^2}dx} + \frac{1}{2}\int_{\partial \Omega } {|\nabla u{|^2}(\nu \cdot x)d\sigma } = \frac{n}{{p + 1}}\int_\Omega {|u{|^{p + 1}}dx} .

Definition. The following identity

\displaystyle\frac{{n - 2}}{2}\int_\Omega {|\nabla u{|^2}dx} +  \frac{1}{2}\int_{\partial \Omega } {|\nabla u{|^2}(\nu \cdot x)d\sigma }  = \frac{n}{{p + 1}}\int_\Omega {|u{|^{p + 1}}dx}

is called the Pohozaev identity for the problem -\Delta u=|u|^{p-1}u.

Note that, if \Omega is the star-shaped domain, we then obtain \nu \cdot x \geqslant 0 on \partial \Omega. Consequently,

\displaystyle\frac{{n - 2}}{2}\int_\Omega {|\nabla u{|^2}dx} \leqslant \frac{n}{{p + 1}}\int_\Omega {|u{|^{p + 1}}dx} .

But once we multiply the PDE by u and integrate by parts, we produce the equality

\displaystyle\int_\Omega {|\nabla u{|^2}dx} = \int_\Omega {|u{|^{p + 1}}dx} .

We thus conclude

\displaystyle\left( {\frac{{n - 2}}{2} - \frac{n}{{p + 1}}} \right)\int_\Omega {|u{|^{p + 1}}dx} \leqslant 0.

Therefore, if u \not \equiv 0, we need p\leqslant \frac{n+2}{n-2}.

8 Comments »

  1. Many Thanks. This entry is nice. I have learned much.

    Comment by HasPas — April 16, 2010 @ 21:49

  2. 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!!

    Comment by Urko — October 31, 2010 @ 7:11

    • 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 }.

      Comment by Ngô Quốc Anh — October 31, 2010 @ 10:39

      • 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?

        Comment by Urko — November 1, 2010 @ 3:56

  3. Right😉

    Comment by Urko — October 31, 2010 @ 21:28

  4. 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.

    Comment by Ngô Quốc Anh — November 1, 2010 @ 12:17

    • 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}

      Comment by Ngô Quốc Anh — November 1, 2010 @ 12:26

    • 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.

      Comment by Urko — November 1, 2010 @ 21:29


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