Sharp forms of Moser-Trudinger inequalities and Laplace operators of self-similar measures on

Abstract

We show a sharp estimate constant and extremal function for the modified Moser-Trudinger inequality on the Lebesgue norm in two dimensions. We consider the eigenvalues and eigenfunctions of the fractal Laplacians on the unit interval. These eigenfunctions can be considered fractal analogs of the classical Fourier sine and cosine functions. We construct the iterated function systems with overlaps and show the behavior of the Laplace operators that related to self-similar measures on the Euclidian space. We study the Trudinger type inequal-ties in the Euclidian space and find the structure of their best exponents. We also show the Moser-Trudinger inequalities of vector bundles over a compact Riemannian manifold of dimension two for unbounded domains and higher order derivatives. We investigate a weighted Moser-Trudinger inequality and its relation to the Caffrelli - Cohn- Nirenberg inequalities in two dimensions and other higher dimensions. inequalities in bounded domains. The best constants and blowing up analysis is considered. We give some applications of Pohozaev identity and show discussions of the extremal functions for Moser-Trudinger inequality.

In chapter 5, we show the maximizers sequences and blowing up analysis with test functions. We also show that for the entire space the supremum of the Trudinger- Moser inequality over such functions is attained and the proof is based on a blow up procedure.

In chapter 6, the inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy- Sobolev inequality, as established in two dimensions. In fact, for suitable sets of parameters, asymptotically sharp, we show symmetry or symmetry breaking by means of a blow- up method. In this way, the weighted Moser- Trudinger inequality appears as a limit case of the Hardy- Sobolev inequality.