PDE Seminar
Abstract: In this talk, I will review Harnack inequalities for second order elliptic operators in the Euclidean setting and their extension to the Riemannian manifolds. Harnack inequalities play a central role in the theory of elliptic and parabolic differential equations, many of their applications, and to analysis at large. 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 De Giorgi 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 nonlinear equations. Thus coming from a rich tradition the Harnack estimate is a key tool in modern developments of analysis and PDEs.