Ph.D Thesis


Ph.D StudentVersano Idan
SubjectOn Positive Solutions and Families of Optimal
Hardy-Weights for Elliptic Equations
DepartmentDepartment of Mathematics
Supervisor PROFESSOR EMERITUS Yehuda Pinchover


Abstract

In this thesis, we study positive solutions  of elliptic operators and the associated optimal Hardy-weights.


The first main goal of the thesis is to study  positive solutions of a second-order linear elliptic operator in divergence form defined on a domain, and satisfy an oblique boundary condition on a  Robin boundary portion .In particular, we discuss  the generalized maximum principle,  existence of a principal eigenvalue,  and the existence of a positive minimal Green function in bounded domains. Then, we establish a criticality theory for positive solutions in a general domain with no boundary condition on the Dirichlet part and no growth condition at infinity. Our results generalize and extend the results obtained by Pinchover and Saadon (2002) for classical solutions, where stronger regularity assumptions on the coefficients of the operator and boundary   are assumed.


The second main goal of the thesis  is to utilize the criticality theory developed  in the first part of the thesis to construct families of optimal Hardy-weights for a subcritical linear second-order linear elliptic operator. First, we characterize all optimal Hardy-weights of a general Sturm-Liouville operator defined on an interval. Then, we present a new approach to construct families of optimal Hardy-weights in n dimensions using a one-dimensional reduction. Finally, we show how our results improve several familiar Hardy-type inequalities.


The last part of the thesis is devoted to  Hardy inequalities for half-linear operators.

First, we obtain new Hardy-inequalities for subcritical  Schr\"odinger-type (quasi-linear) operators  with  nontrivial potentials of a constant sign. Then, we  construct families of optimal Hardy-weights to the (p,A)-Laplacian in a general domain.