Harnack inequalities for nonlocal differential operators and Dirichlet forms (Q2726220)

The author considers nonlocal second order differential operators of Lévy type. The main results of the thesis are Harnack inequalities for weak solutions of both linear and nonlinear equations. The Moser iteration technique shows that the validity of a Harnack inequality implies Hölder continuity of the weak solution. Purely nonlocal and nonlinear equations are treated in detail in extra sections.NEWLINENEWLINENEWLINEThe last part of the thesis is dedicated to applications, where certain nonlinear equations are considered. It is shown that the theorem of Leray-Lions may be used to prove solvability of some approximated problem. A priori estimates for the solution of the approximated problem then imply solvability of the original equation.