Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real analytic coefficients in the interior $\Omega$. $u$ and $f$ are real analytic in the interior $\Omega$ and both are smooth up to the boundary, with zero Dirichlet boundary data $u|_{\partial \Omega} = f|_{\partial \Omega} = 0$. Suppose in addition we assume that $u$ satisfies zero Neumann data $ \frac{\partial u}{\partial \eta}|_{\partial \Omega} = 0$.

I am looking for references which give unique continuation results like $u = f = 0$. I am not sure if we need additional restrictions on $L$ to ensure such a conclusion (I am happy with references that say something about $L = \Delta$). Any pointers will be highly appreciated.

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    $\begingroup$ Aronszajn, N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36 (1957), 235–249. $\endgroup$ – Liviu Nicolaescu Nov 8 '18 at 18:54
  • $\begingroup$ @LiviuNicolaescu I cannot seem to find the Aronszajn paper anywhere online. Is it a particular result there that you are referring to, or is it a general technique that I can may be locate somewhere else? Thanks a lot ! $\endgroup$ – user131136 Nov 9 '18 at 12:35
  • $\begingroup$ I have access only to hardcopies of the journal. Read the MathSciNet review. It describes precisely Aronszajn's result. mathscinet.ams.org/mathscinet/pdf/… $\endgroup$ – Liviu Nicolaescu Nov 9 '18 at 14:00

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