| Abstract: |
| In this talk we consider bounded local weak solutions to the following class of anisotropic elliptic equations
\begin{equation}
\label{first eq}
\sum_{i=1}^{N-1} \frac {\partial } {\partial x_i} A_{q,i}(x,u, Du) + \frac {\partial} {\partial x_N }A_p( x,u, Du) = 0 \quad
\mbox{in} \ \ \Omega,
\end{equation}
with $\Omega$ a regular domain in $\mathbb{R}^N. $
The functions
$A_{q,i}(x,u,D u)$ and
$A_p(x,u,D u) : \Omega \times \mathbb{R}^{N+1} \rightarrow \mathbb{R}^N$ are assumed to be measurable and
satisfying the structure conditions.\
We prove the H\older regularity of solutions to \ref{first eq} when $p>q>1$ (singular case).
A first step to face this problem was made by Liskevich and Skrypnik (\cite {LS}) in 2009.
Then Duzgum, Marcellini and Vespri (\cite {DMV}) in 2014 extended the result proved in \cite {LS} to the case when in the equation \ref{first eq} it is assumed $q>p>1$ (degenerate case).
The results have been established in a joint research with Francesco Ragnedda (Cagliari) e Vincenzo Vespri (Firenze).\
\begin{thebibliography}{99}
\bibitem{DMV} F.G. D\uzg\un, P. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic $p$-Laplacian equation by using a parabolic approach, {\it Riv. Math. Univ. Parma, {\bf 5}, (2014), 93 --111}.
\bibitem{LS} V. Liskevich and I.I. Skrypnik, H\older continuity of solutions to an anisotropic elliptic equation, {\it Nonlinear Anal., {\bf 71}, (2009), 1699--1708}.
\end{thebibliography} |
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