A strong unique continuation theorem for parabolic equations

Abstract

The unique continuation is best understood for second order elliptic operators. The classical paper by Carleman [7] established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients. This was remarkable since it indicates that even a non-analytic solution of an elliptic equation can behave somewhat in an “analytic” manner. The powerful technique he used, the so-called “Carleman weighted inequality” has played a central role in later developments. In 1950’s, Aronszajn [4] and Cordes [11] generalized his result to higher dimensions. In recent years this classical subject caught new attention from a great number of people. Much efforts have been made to relax the smoothness hypotheses on the coefficients (see [6, 17, 43, 1, 22]). A seminal paper by Jerison and Kenig [26] has shown the unique continuation property for operators of the form ∆ + c(x ) with c ∈ L /2 loc where N ≥ 3 is the space dimension. Further improvements have been carried out in considering other classes of coefficients (Fefferman-Phong class and Kato class) [42, 8, 12, 29], in extending the result to operators with first derivative terms and variable leading coefficients [46, 47, 51, 52], and etc. (See [25, 15, 16, 28, 5, 27] and the references therein.) For second order linear parabolic operators with time-independent coefficients, the strong unique continuation property along with detailed estimates on the Hausdorff measures of nodal sets was reduced in [32, 33] to the previously established elliptic counterparts. The reduction from time-independent parabolic