# Seminario di Equazioni Differenziali e Applicazioni: “An alternative approach to Hölder continuity of solutions to some elliptic equations”

## Lunedì 21 Maggio 2018, ore 12:15 - Sala Riunioni VII Piano - Fatma Gamze Duzgun

ARGOMENTI: Seminars

Lunedì 21 Maggio 2018 alle ore 12:15 in Sala Riunioni VII Piano, Fatma Gamze Duzgun (Hacettepe University, Ankara, Turchia) terrà un seminario dal titolo “An alternative approach to Hölder continuity of solutions to some elliptic equations”.

Abstract
We give an alternative and self contained proof of the Hölder continuity of locally bounded solutions to some elliptic equations, including the equation associated with the $p-$Laplacian operator. We combine the techniques introduced by De Giorgi and Moser with a 1-dimensional Poincaré nequality.
Particularly we consider the partial differential equation
$$\begin{array}{cr} \mathrm{div}A_p (x, u, Du) = 0 & (1) \end{array}$$
where $\Omega$ is an open set in $\mathbb{R}^n$ for some $n \geq 2$ and $A_p$ is Carathéodory vector field defined in $\Omega\times\mathbb{R}\times\mathbb{R}^n$ ; i.e., $A_p(x,s,\xi)$ is measurable with respect to $x\in\Omega$ and continuous in $(s,\xi)\in\mathbb{R}\times\mathbb{R}^n$ for almost every $x\in\Omega$. We assume that $A_p$ satisfies the following structural conditions (natural growth conditions) for an exponent $p\gt1$ and for some positive constants $m, M$
$$\begin{array}{cr} \left\{\begin{array}{l} A_p(x,s,\xi)\ \xi\geq m\ |\xi|^p\\ |A_p(x,s,\xi)|\ \xi\leq M\ |\xi|^{p-1}. \end{array}\right. & (2)\end{array}$$

References
[1] E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s inequality for degenerate and singular parabolic equation, Springer Monographs in Mathematics. Springer-Verlag, New York, 2012.
[2] F. G. Düzgün, P. Marcellini, V. Vespri, An alternative approach to Hölder continuity of solutions to some elliptic equations, Nonlinear Anal. 94 (2014), 133-141.
[3] F. G. Düzgün, P. Marcellini, V. Vespri, Space expansion for a solution of an anisotropic p-Laplacian equation by using a parabolic approach, Riv. Mat. Univ. Parma, Vol 5 (2014), no.1, 93-111.