AIMS OF THIS RESEARCH GROUP

Harnack inequalities play a central role in the theory of elliptic and parabolic differential equations, many of their applications, and to Analysis at large. They are routinely used to study interior and boundary regularity of solutions of a large class of linear, quasi-linear, and fully non-linear equations. They are fundamental in understanding the notion of degeneracy in degenerate and/or singular elliptic - parabolic equations, and the boundary behavior of corresponding solutions. Their role in Analysis began with the classical Liouville Theorem for holomorphic functions, and has continued in producing refined covering Theorems, as well as underscoring the importance of BMO and the DeGiorgi classes. These inequalities are essential in Perron's method of solving the Dirichlet problem for the Laplacian. This method is at the root of the modern theory of viscosity solutions for fully non-linear equations. Thus coming from a rich tradition the Harnack estimate is a key tool in modern developments of Analysis and PDE's.
Harnack inequalities have been established for general linear elliptic and parabolic equations with measurable coefficients, for a large class of quasi-linear singular and degenerate parabolic equations and for fully non-linear equations.
Let us begin by recalling that the classical Harnack inequality for harmonic functions states that if u ≥ 0 is a harmonic function in a ball B_2R(x), then
                                                        sup_BR(x) u ≤ C  inf_BR(x) u
for a constant C>1 independent of the function u and the radius R, and depending only on the ambient space dimension. Note that in particular from the fact that u ≥ 0 in B_2R we conclude that u > 0 in B_R , unless u is identically zero. Thus, the strong maximum principle follows immediately from the Harnack inequality.
There has been great progress in understanding the meaning of Harnack inequalites for non-negative solutions to quasi-linear and fully non-linear elliptic equations. In fact, one can extend the notion of solution considerably and still have a Harnack inequality. For example, let us mention the class of all viscosity solutions to some elliptic equation in non-divergence form with fixed ellipticity introduced by Caffarelli-Cabré.
The inequality stated above holds also for quasi-minima in the Calculus of Variations and for functions in the DeGiorgi classes. Since these are only indirectly related to Partial Differential Equations, it might be that the Harnack inequality is a structural property of some classes rather that PDE's, and it raises the question of identifying such classes.
The first parabolic Harnack-type estimate was established independently by Hadamard and Pini. Consider a non-negative caloric function u in some space-time domain E_T of R^N+1. For (x_o,t_o) in E_T , consider the cylinder
                                                        Q_R(x_o,t_o) = B_R(x_o) × {t_o-R², t_o+R²}
and assume that Q_2R(x_o,t_o) is included in E_T . Then, there exists a constant C, 0<C< 1, depending only upon N, such that
                                                        C sup_BR(xo) u(x,t_o-R²) ≤  u(x_o,t_o) .
Later Moser established the same parabolic Harnack inequality, for non-negative weak solutions of linear parabolic equations in divergence form
                                                        u_t - ∑ _i,j (a_ij(x,t) u_xi)_xj = 0   in  E_T .
Non-linear extensions of the parabolic theory are rather more delicate. For degenerate and/or singular parabolic PDE's, the Harnack inequality needs to be written in term of an intrinsic time scale that reflects the fact that space and time scale differently.
The investigators of this group have worked on versions of the Harnack inequality or its direct applications for many different elliptic and parabolic equations in various different frameworks. In view of the recent advances in various fronts (quasi-linear parabolic equations, the p-Laplacian including p=∞, the Wiener criterion at infinity, and p-harmonic measure), a series of collaborative activities are planned.
This is a cohesive group of investigators, involved on this subject for a number of years, with common way of thinking and vision. They have collaborated and have shared ideas for a long time and some of them have joint contributions. An added component to the group is a rich international connection with investigators involved on the same theme. These connections have resulted in cross-publications and close interactions. The topic is momentous as underscored by the 2005 Summer School in Cortona, Italy, the recent 2007 Summer Schools in Saariselka, Finland, and the coming 2009 Summer School in Cetraro, Italy (see in Future scientific events).