Uniqueness and Stability in the Cauchy Problem

In this chapter we formulate and in many cases prove results on uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carleman-type estimates. In Section 3.1 we describe the results for a simplest problem of this kind (the backward parabolic equation), where a choice of the weight function in Carleman estimates is obvious, and the method is equivalent to that of logarithmic convexity. In Section 3.2 we formulate general conditional Carleman estimates, and we apply the results to the general Cauchy problem. We also formulate a global version of Holmgren’s theorem and the recent result of Tataru on nonanalytic equations. In Section 3.3 we consider elliptic and parabolic equations of second order, construct for them pseudoconvex weight functions and obtain complete and general uniqueness and stability results for the Cauchy problem. Section 3.4 is devoted to substantially less understood hyperbolic equations and Schrödinger-type equations. Here, for some particular but interesting domains we also give appropriate weight functions and obtain a quite explicit description of uniqueness domains for lateral Cauchy problems. Additional information to Sections 3.2–3.4 can be found in the book of Zuily [Z].