Why is the Cauchy problem severely ill-posed?

Abstract

An answer to the ill-posedness degree issue of the Cauchy problem may be found in the theory of kernel operators. The foundation of the proof is the Steklov–Poincare approach introduced in Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915–36), which consists of reformulating the Cauchy problem as a variational equation, in an appropriate Sobolev scale, and is set on the part of the boundary where data are missing. The linear (Steklov–Poincare) operator involved in that reduced problem turns out to be compact with a non-closed range; hence the ill-posedness. Conducting an accurate spectral analysis of this operator requires characterization of it as a kernel operator, which is obtained through Green's functions of the (Laplace) differential equation. The severe ill-posedness is then settled for smooth domains after showing a fast decaying towards zero of the eigenvalues of that Steklov–Poincare operator. This is achieved by applying the Weyl–Courant min–max principle and some polynomial approximation results. Addressing more general smooth domains with corners, we discuss the regularity of Green's function and we explain why there is a room to extend our analysis to this case and why we are optimistic that it will definitely establish the severe ill-posedness of the Cauchy problem.