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For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says $$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$ Here $u_T$ are the tangential derivative and $u_N$ is the normal derivative.

I am trying to see what is the analogue for solutions of $-\operatorname{div} A(x)\nabla u=0$ where $A(x)$ is symmetric and uniformly elliptic with $\lambda \leq A(x) \leq \Lambda$ (in terms of quadratic forms) and smooth (say $C^{\infty}$) so that weak solutions are also smooth.

How does one go about proving a Pohozaev identity in this setting?

I am more specifically looking for the identity using $\langle x,A(x)\nabla u\rangle$ as a test function. Rather, any other test function is also okay provided all the resulting terms have a specific sign.

Further assumptions: Suppose that all derivatives of $A(x)$ are also uniformly elliptic with some ellipticity constants (in some sense, one can consider the mollification of $A$ as the matrix), then can one get an identity where each of the terms comes with a definite sign?

Clearly, there must be some terms which contain derivatives of $A$ and higher derivatives of $u$, but as long as they are in an expression with a definite sign, then it's okay.

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  • $\begingroup$ You know the test function, and undoubtedly you know that the method is to integrate by parts, twice. It does not work out so nicely with "A" as it does witthout "A", because partial derivatives do not commute with "A". So formulas can be written, but there isn't a canonical clean formula which is better than the others. $\endgroup$
    – username
    Jan 3, 2023 at 9:50
  • $\begingroup$ I am looking for some closed form expression. Writing things out explicitly, things look a bit messy, so maybe there are some extra cancellations that could help with obtaining a clean expression that I am missing. This is essentially the purpose of my question. $\endgroup$
    – Adi
    Jan 3, 2023 at 10:34
  • $\begingroup$ Using $\langle x,\nabla u\rangle$ instead of $\langle x,A(x)\nabla u\rangle$ gives a term that looks like $\int \langle x,\nabla u\rangle\langle x,A(x)\nabla u\rangle$ which does not necessarily have a sign. Hope this adds more context to the purpose of my question. $\endgroup$
    – Adi
    Jan 3, 2023 at 10:37
  • $\begingroup$ How can you expect a specific sign if you allow derivatives of $A$ in the mix ? Or do you mean a specific sign on the highest order term? You said your purpose was to derive smoothness. In that case the Pohozaev-Rellich-Morawetz estimate is read as a way to control the normal derivative by the boundary value and the $H^1$ norm : an inequality suffices, namely $\int_{S_1} |u_N|^2 d \sigma \leq ...$. Is that what you actually want? $\endgroup$
    – username
    Jan 3, 2023 at 21:54
  • $\begingroup$ I am unsure how to answer your question, because I'm not entirely sure what estimate is helpful for what I'm trying. I am optimistically hoping for an identity where each term has a specific sign, could be positive or negative. I assumed everything was smooth in order to not worry about rigorous justification. $\endgroup$
    – Adi
    Jan 3, 2023 at 22:02

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