On weighted Sobolev spaces: Trudinger-Moser and isoperimetric inequalities

Abstract

The main topic of the thesis is the study of Elliptic Partial Differential Equations. The thesis is divided into two Parts: (I) Trudinger-Moser Type inequality on weighted Sobolev spaces; and (II) on existence and nonexistence of isoperimetric inequalities with different monomial weights.

In part I, we establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-\theta} (r^{\alpha}\vert u'(r)\vert^{\beta}u'(r))',$ where $\theta, \beta\geq 0$ and $\alpha>0$, are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted P\'olya-Szeg{\"o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.

In part II, we consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of $\int_{\partial\Omega}x^{A}\mathcal{H}^{N-1}(x)$ among all smooth bounded open sets $\Omega$ in $\mathbb{R}^{N}$ with fixed Lebesgue measure with monomial weight $\int_{\Omega}x^{B}dx$.