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On a closed manifold $M$, let $P(f)=0$ be a linear overdetermined elliptic system of PDE of 2nd order with smooth coefficients. By overdetermined ellipticity, I mean the principal sympbol is injective. For simplicity, let's assume it is for a real-valued function $f$. If a solution $f$ is zero on an open subset, then is $f$ zero on the whole of $M$? If this is not true in general, what would be conditions for it to be true? Also please recommend some references. Thanks a lot!

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Is there any reason you cannot use Holmgren's theorem? –  Michael Renardy Aug 2 '13 at 10:49
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What do you mean exactly by overdetermined ? Is $P$ vector (or complex) valued ? –  BS. Aug 2 '13 at 12:23
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You need to specify what you mean by overdetermined. Many but not all classes of eliptic systems satisfy the unique continuation property. –  Liviu Nicolaescu Aug 2 '13 at 12:51
    
So you have two elliptic PDEs, or one elliptic and one something else, or more than two? I think that if you already have one elliptic PDE, you should be able to apply: Aronszajn, N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. –  Ben McKay Aug 2 '13 at 13:11
    
I cannot use Holmgren's theorem, because it requires real-analytic coefficients. It is a system of PDE for one real-valued function, but each one of those equations is alone not elliptic. –  user38064 Aug 3 '13 at 4:03
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1 Answer

(1) The most classical result, due to Aronszajn, and later to Calder\'{o}n and (independently) to H\"ormander. Take a second order elliptic operator $P$ with Lipschitz and real coefficients in the principal part. Then if $Pu=0$ on an open set $\Omega$ and u vanishes on an open set $\omega$, this implies that $u$ vanishes on the connected component of $\omega$.

(2) The same conclusion holds if $Pu=0$ on an open set $\Omega$ and $u$ is flat at a point $m$, i.e. $$ \forall \alpha,\ (\partial^\alpha u)(m)=0\quad\text{or if $u$ is not smooth, $\int_{\vert x-m\vert\le R}\vert u(x)\vert dx=O(R^\infty)$}. $$ (3) You cannot use Holmgren to get this since no analyticity of the coefficients is required and also the result is true if you replace the equation $Pu=0$ by an inequality $$ \vert (Pu)(x)\vert\le \vert V(x) u(x)\vert+\vert W(x) \nabla u(x)\vert, $$ with $V\in L^{n/2}, W\in L^{n+\epsilon}$. The latter result is due to Jerison \& Kenig without $W$ and to Koch \& Tataru in general.

(4) The Lipschitz continuity assumption should be taken seriously because of counterexamples due to Plis, Miller, Filonov.

(5) Also the real coefficients assumption is important since there are counterexamples due to Alinhac for a second order elliptic operator with non-conjugate roots.

You will find many references in Chapter 28 of the fourth volume of H\"ormander's ALPDO.

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