# Unique continuation from the boundary for inhomogeneous elliptic pde

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.

• 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. – Liviu Nicolaescu Nov 8 '18 at 18:54
• @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 ! – user131136 Nov 9 '18 at 12:35
• I have access only to hardcopies of the journal. Read the MathSciNet review. It describes precisely Aronszajn's result. mathscinet.ams.org/mathscinet/pdf/… – Liviu Nicolaescu Nov 9 '18 at 14:00