Why is the Cauchy problem severely ill-posed?

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.