Regularity of the solutions for PDE and perturbation problem

A natural question in the theory of partial differential equation of second order with analytic coefficients is the regularity of the solutions. A strongly related problem concerns the conditions in order that such an operator preserves its regularity properties of the solutions if it is perturbed by adding an operator of smaller order. The project is devoted to investigate such problems. 1967 Hörmander characterized the smoothness in the case of real coefficients. Later on it was shown how the Hörmander condition is not enough to ensure that the solutions are analytic functions. The progresses done in the study of the regularity of the solutions have shown that there is a strong link between the geometrical properties associated to the operator and the regularity of the solutions, even if there is not a general theory to explain this connection.
The first goal of the project aspires to understand the problem of the regularity in relation to the geometrical properties of the operator. We will start studying a case with a prescribed geometrical structure to obtain the canonical form of the associated operator and consequently the regularity. Moreover we will study general models to detect a general scheme. One of the main tool will be the use of suitable a priori estimates. To obtain the results we also want to develop the knowledges concerning the so called Green operator related to the operator, subject already studied in literature but whose understanding is still limited. A most comprehension of the properties of the Green operator will also allow us to solve the perturbation problem. As was shown in a recent work by Cordaro and the applicant, the Green operator plays an important role to understand the regularity and the perturbation problem. The a priori estimates appear to be effective tools but are not always sufficient if not supported by a better understanding of the Green operator. Our studies are also devoted to approach the problem of the global regularity for operators defined on the torus as their underlying geometrical structure. The starting point will be the study of a class of operators with prescribed properties.
In the case when the coefficients of the operators are complex-valued the situation changes radically. Indeed the Hörmander theory cannot be extended to this setting as was recently shown. To obtain new insights in this direction we will start focusing on a particular class of second order partial differential operators, which admits a reach and interesting structure. To do this we will take advantage of the spectral property of the operator to construct, via linear algebra, a sort of ``inverse” of the operator. Also in this case the purpose is to develop strategies to obtain results which can be used to investigate more general clases.