| Abstract: |
| I will present some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of
rearrangement invariant spaces (e.g. $L^{p}$, Lorentz, Orlicz, etc...).
The monomial weights are defined by
$\begin{equation*}
d\mu (x):=x^{A}dx=|x_{1}|^{A_{1}}\cdots |x_{n}|^{A_{n}}dx, \label{mu}
\end{equation*}$
where $A=(A_{1},A_{2},\dots ,A_{n})$ is a vector in $\mathbb{R}^{n}$ with $%
A_{i}\geq 0$ for $i=1,\dots ,n$.
Some applications to study local boundedness of minimizers of a class of non uniformly elliptic integral functionals are also given. |
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