Metric properties
of the distance in the Carnot groups.
Free boundary
problems.
Relative
isoperimetric inequalities for curvature measures.
Poincaré inequalities and Sobolev
Poincaré.
Regularity of solutions for
elliptic and subelliptic operators.
Metric properties of the distance in the Carnot groups: curvatures
and metric curvatures, geodesics and application to the travel
salesman problems in the Heisenberg group. Free boundary problems:
regularity of the free boundaries in two-phase problems for
elliptic, subelliptic, parabolic and fractional operators,
regularity improvement of the free boundary for linear and
nonlinear operators with variable coefficients, existence of
monotonicity formulas for operators in nondivergence form
(elliptic, parabolic and subelliptic), regularity of level surface
in semilinear problems, existence of solutions of two phase
problems. Relative isoperimetric inequalities for curvature
measures: integral formulas for curvature functions. Poincaré
inequalities and Sobolev Poincaré: solution of semilinear equations
and Poincaré inequalities. Regularity of solutions for elliptic and
subelliptic operators: existence of results like
Aleksandov-Bakelman-Pucci maximum principle.
