Regularity results for functionals with general growth

Regularity results for functionals with general growth

Mon, 15/02/2010
17:00 		Bianca Stroffolini (University of Naples) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Regularity results for functionals with general growth
In this talk I will present some results on functionals with general growth, obtained in collaboration with L. Diening and A. Verde. Let $ \phi $ be a convex, $ C^1 $-function and consider the functional:
$$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$
where $ \Omega\subset \mathbb{R}^n $ is a bounded open set and $ \bf u: \Omega \to \mathbb{R}^N $. The associated Euler Lagrange system is
$$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$
In a fundamental paper K.~Uhlenbeck proved everywhere $ C^{1,\alpha} $-regularity for local minimizers of the $ p $-growth functional with $ p\ge 2 $. Later on a large number of generalizations have been made. The case $ 1 {\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$ -Hölder continuous for some $\alpha>0$ . We present a unified approach to the superquadratic and subquadratic $p$ -growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.