6. Unique Continuation [04UC]

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6. Unique Continuation

For our proof of unique continuation we rely on the following classical result:

Theorem 6.1.

Assume that $\Omega\subset\mathbb{R}^{n}$ is a connected open set and $u\in W^{1,2}_{loc}(\Omega)$ is a weak solution to the equation

\partial_{i}(a^{ij}(x)u_{j})+b^{i}(x)u_{i}+c(x)u=0,

where $a^{ij}(x)$ is Lipschitz and uniformly elliptic and $b^{i}(x),c(x)$ are bounded measurable. If $u=0$ on some open subset of $\Omega$ , then $u\equiv 0$ in $\Omega$ .

A proof can be found in Hörmander’s book [H], Theorem $17.2.6$ . In [AS], the authors use the same theorem to prove unique continuation for fully nonlinear uniformly elliptic equations.

We will apply this result to the difference of $u$ and $v$ , which solves a linear equation where $u$ and $v$ are sufficiently regular. Indeed, suppose $u$ and $v$ are $C^{2}$ in a neighborhood of $x$ and let $w_{t}$ be the convex combination $tu+(1-t)v$ . Let $(W_{t})^{ij}$ be the matrix of cofactors for $D^{2}w_{t}$ . Then by expanding $0=\int_{0}^{1}\frac{d}{dt}\det D^{2}w_{t}dt$ we get

a^{ij}(x)(u-v)_{ij}=0,

where

a^{ij}(x)=\int_{0}^{1}(W_{t})^{ij}(x)dt.

The regularity theory of Caffarelli [C2] allows us to use this observation at points of strict convexity for solutions to the Monge-Ampère equation:

Theorem 6.2.

Assume

\det D^{2}u=f\quad\text{ in }\Omega,\quad\quad u|_{\partial\Omega}=0

where $f\in C^{1,\alpha}(\Omega)$ is strictly positive. Then

u\in C^{3,\alpha}(\Omega).

The proof of unique continuation follows easily from these observations and our main theorem.

Proof of Theorem 1.3:.

Let $\Sigma_{u}$ and $\Sigma_{v}$ be the singular sets of $u$ and $v$ respectively, and let $A=\Omega-(\Sigma_{u}\cup\Sigma_{v})$ . Since $A$ is dense in $\Omega$ , it suffices to show that $u=v$ on $A$ .

By Caffarelli’s theory ([C1]), $A$ is an open set, and by Theorem 1.1, $A$ is connected. By Theorem 6.2, the difference $u-v$ satisfies the linear equation

a^{ij}(x)(u-v)_{ij}=0

on $A$ , where $a^{ij}$ are locally uniformly elliptic and $C^{1,\alpha}$ in $A$ . The conclusion follows from Theorem 6.1. ∎