Degenerate partial differential equations
Research focus
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1. Dirichlet forms. One of our aim is to extend the methods of potential theory to operator algebras and investigate the geometry underlying the notion of energy (cf. [B2]). We focus on studying problems in the perspective of Banach algebras (cf. [J14]). This provides a common framework both for the classical theory on locally compact metric spaces as well as for studying the notions of energy on operator algebras associated to singular quotient spaces (foliations), graphs and discrete groups, systems in quantum statistical mechanics (cf. [J16]) and to introduce notions of positivity for spinors or exterior forms on riemannian manifolds (cf. [J17], [J18]). Efforts are directed in developing a differential calculus and tools of differential topology in Dirichlet spaces (cf. [J16]). Our second aim is to investigate the notions of nonlinear Markov semigroups and nonlinear Dirichlet forms and extend to the nonlinear setting the criteria characterizing the order-preserving and markovian properties of semigroups in terms of corresponding properties of Dirichlet forms (cf. [J15]). The research is motivated by the study of non-linear Dirichlet problems of geometric type (cf. [J15]), the investigations of regularity properties of harmonics of homogeneous Dirichlet forms with various boundary conditions on perforated domains and homogenization problems for nonlinear weighted subelliptic operators (cf. [J3], [J5], [J10]). 2. Hörmander type operators. The study of degenerate PDEs with nonnegative characteristic form has been developed, classically, under different points of view. According to one of these, introduced by L. Hörmander in 1967, the equation is written as a sum of second order derivatives with respect to vector fields, which are strictly less than the dimension of the space, but on the other hand satisfy a suitable rank condition (known as “Hörmander’s condition”). In recent years new classes of second order PDOs, which generalize nonvariational elliptic and parabolic operators when the spacial derivatives are replaced by “Hörmander's vector fields”, have been introduced and studied. Motivations for this study come from geometric theory of several complex variables, as well as from some models of human vision. The systematic study of this new theory of stationary or evolutionary nonvariational operators structured on Hörmander’s vector fields is the main focus of this line of research, see [J11], [J12], [J13], and references therein. Related to this central issue as useful tools are both the study of singular and fractional integrals in the abstract context of spaces of homogeneous type, in the sense of Coifman-Weiss, and the study of the properties of metrics induced by systems of vector fields (widely known as “Carnot-Carathéodory metrics”). 3. Mixed type PDE’s. We study boundary value problems for linear and nonlinear equations of mixed elliptic hyperbolic type that arise in applications to transonic gas dynamics and in geometric problems where curvature changes sign. The main difficulty lies in the mixture of qualitative types that we seek to treat in a global manner in order to obtain robust results with respect to boundary geometry and to give clear explanations as to which problems are well posed and why. In particular, we are interested in nonlinear problems which have a variational structure and linear problems which represent possible two dimensional transonic potential flow patterns in the linearized hodograph plane. Among other typical nonlinear results a critical exponent phenomenon of power type has been exhibited for a wide class of mixed and degenerate type equations with various classical boundary conditions (cf. [J6]).
Dipartimento di afferenza
Docenti afferenti
Full Professors
Marco Biroli
Daniela Lupo
Associate Professors
Marco Bramanti
Fabio Cipriani