Ph.D Thesis | |

Ph.D Student | Versano Idan |
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Subject | On Positive Solutions and Families of Optimal Hardy-Weights for Elliptic Equations |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Yehuda Pinchover |

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.