A unique continuation theorem for second order parabolic differential operators
Denote by \((x,t)\) a generic point in \({\mathbb{R}}^ n\times {\mathbb{R}}\), \(n\geq 2\), and let \(X\subset {\mathbb{R}}^{n+1}\) be an open connected set. Let \(L=D_ t+P(t,x,D_ x)\) be a differential operator of parabolic type, where P is a second order elliptic operator with bounded coefficients, whose principal part is real and \(C^{\infty}.\) The author establishes the following result: Suppose that \(Lu\in L^ 2_{loc}(X)\) for some \(u\in {\mathcal D}'(X)\) and that \[ (*)\;| Lu| \leq | Vu| \text{ in } X \] for some \(V\in L_{loc}^{(n+2)/2}(X)\). Then the normal set of the support of u is non-characteristic for L. As a corollary, a unique continuation theorem is obtained: Let \(X=\Omega \times [-T,T]\), where \(\Omega \subset {\mathbb{R}}^ n\) is an open connected set and \(T>0\). Let L and V be as above and suppose that (*) holds. Assume that u vanishes in an open subset \(G\subset X\). Then u must vanish identically in the horizontal component of G, i.e., \(u(x,\tau)\in G\) for some \(\xi\). The proof of these results relies on a Carleman type inequality, the derivation of which constitutes the main part of the paper.
scientific article; zbMATH DE number 4154968
scientific article