Unique continuation for some evolution equations

Abstract

This work is concerned with unique continuation results for some evolution equations, essentially of parabolic type. To be more precise, let L be an evolution operator acting on functions defined on some connected open set B of R”+’ = R; x R,. L is said to have the unique continuation property if every solution u of Lu = 0 which vanishes one some nonempty open set U.I of fl vanishes in the horizontal component of o in 0, i.e., in {(x, t) E 0; 3X,) (X,) t) E co}. The first general result of this type was established by Mizohata [6] in the case where L is a second order parabolic operator with smooth coefficients. However, Mizohata’s proof is somewhat delicate (it is based in particular on an extended class of Calderon-Zygmund operators), and does not lead obviously to a weakening of the smoothness of the coefficients. On the other hand, it is important for the applications to deal with nonsmooth coefficients since L is often obtained as the linearization of some nonlinear operator at a (nonnecessarily smooth) solution. We refer to Lions [4], and Kernevez and Lions [3], for applications to control theory and to [ 1 l] for applications to transversality techniques. Let us now describe the content of this work. In the first section we prove a unique continuation theorem when L is a second order parabolic equation. Our proof is simple and based on the derivation of a Carleman estimate which is reminiscent of the classical Carleman estimates for second order elliptic operators (see [ 121). This Carleman inequality allows the weakening of the 118 0022-0396187 $3.00