Remarks on Calder'on Projections for Elliptic Equations with Parameter

L. R. Volevich, A. R. Shirikyan


Abstract: The paper is devoted to investigating the Cauchy problem for scalar elliptic equations with parameter. The main result is a new proof of the existence of Calder'on projections. In contrast to the classical Calder'on-Seeley approach using the Green formula, the suggested method is based on the unique solvability of the Dirichlet problem and admits a generalization to the case of nonlinear elliptic equations. In particular, it is proved that the subspace of Cauchy's data of solutions to the equation in question that belong to Sobolev spaces can be represented as a graph of a pseudo-differential operator (acting in the phase space) whose symbol can be calculated explicitly in terms of the symbol for the original equation.