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
F(u)=N∑i=11pi∫Ω|∂iu|pidx,Ω⊂⊂RN,
known to be the prototype of orthotropic non-standard growth functionals 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.