Unique continuation for parabolic operators
Unique continuation results for elliptic and parabolic operators have been proved by many authors under various assumptions and using a variety of techniques. Here the authors consider this problem for parabolic operators of the form \[ Pu = \sum_{i,j=1}^n \partial_i \left( a^{ij}(x,t)\partial_ju \right) + \partial_tu \tag{1} \] where \(a(x,t)=[a^{ij}(x,t)]\) is symmetric with eigenvalues bounded between \(\lambda\) and \(1/\lambda\) for some \(\lambda\in(0,1]\). The main result is that if there are constants \(M>0\) and \(\beta\in(0,1)\) such that either \[ |a(x,t)-a(y,s)|\leq M(|x-y|^2+|t-s|)^{1/2} \] or \[ \begin{aligned}|a(x,t)-a(y,s)|&\leq M(|x-y|^2+|t-s|)^{\beta/2},\\ |\nabla a(x,t)|&\leq M|x|^{\beta-1}, \\ |\partial_t a(x,t)|&\leq Mt^{\beta/2-1}, \end{aligned} \] for all \((x,t),(y,s)\in{\mathbb{R}}^{n+1}\), and \(u\) satisfies \[ |Pu|\leq M(|\nabla u|+|u|) \quad\text{in}\quad B_2\times[0,2] \] and \[ |u(x,t)|\leq C_k(|x|+\sqrt{t})^k \] for all nonnegative integers \(k\), then \(u(x,0)=0\) for all \(x\in B_2\). Furthermore, \(u\) satisfies the estimate \[ |u(x,t)|\leq Ne^{-1/Nt} \|u\|_{L^\infty}(B_2\times(0,2)) \quad\text{for}\quad (x,t)\in B_1\times(0,1) \] where \(N\) depends only on \(\lambda,\beta,n\) and \(M\). The results are also proved up to the boundary under suitable Dirichlet or Neumann boundary conditions, extending earlier work of \textit{L.~Escauriaza} and \textit{V.~Adolfsson } in [Commun. Pure Appl. Math. 50, 935-969 (1997; Zbl 0899.31004)] on parabolic operators with time independent coefficients. Examples of \textit{K.~Miller} in [Arch. Ration. Mech. Anal. 54, 105-117 (1963; Zbl 0289.35046)] and \textit{A.~Plis} in [Bull. Acad. Polon. Sci., Math. Astron. Phys. 11, 95-100 (1963; Zbl 0107.07901)] show that the Lipschitz regularity of the coefficients in the space variables is necessary, but is is not known whether the \(1/2\)-Hölder regularity with respect to time is optimal. The result is derived from a Carleman type inequality. A similar approach to unique continuation theorems for elliptic equations was used by \textit{N.~Aronszajn}, \textit{A.~Krzywicki} and \textit{J.~Szarski} in [Ark. Mat. 4, 417-453 (1962; Zbl 0107.07803)] and by \textit{L.~Hörmander} in [Commun. Partial Differ. Equ. 8, 21-64 (1983; Zbl 0546.35023)].
scientific article; zbMATH DE number 2017825