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

I am trying to see what is the analogue for solutions of $-\text{div} A(x)\nabla u=0$ 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?

Crossposted at https://mathoverflow.net/questions/437410/pohozaev-identity-for-linear-equations.



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