**Simone Ciani**

Technische Universität Darmstadt

Fachbereich Mathematik, Schlossgartenstrasse 7, 64289 Germany.

In this brief talk we will introduce an equation in divergence form whose principal part is the Euler-Lagrange equation of the energy integral

$$\mathcal{F}(u) = \sum_{i=1}^N \frac{1}{p_i} \int_{\Omega} |\partial_i u|^{p_i}\, dx , \quad \Omega \subset \subset \mathbb{R}^N,$$

known to be the prototype of orthotropic non-standard growth functionals $\mathcal{F}$ in Calculus of Variations. We will focus on the similarities and differences of the theory of regularity in comparison with the $p$-Laplacian one, and describe a problem which is still active and widely open after more than fifty years.

We will show why a new approach to the theory of regularity concerning these general functionals is required, and present some possible new methods, adapted from parabolic differential equations.

Key Words: Anisotropic $p$-Laplacian, degenerate and singular equations, parabolic and elliptic equations.