Seminar

# Regularity for quasiminimizers of an anisotropic problem

#### Geometric Aspects of Nonlinear Partial Differential Equations

#### 20 September 14:00 - 15:00

#### Antonella Nastasi (Online) - University of Palermo

"We study quasiminimizers of the following anisotropic energy (p, q) - Dirichlet integral \int_\Omega ag_u^p dmu + \int_\omega bg_u^q dmu in metric measure spaces, with g_u the minimal q-weak upper gradient of u. Here, \Omega\in X is an open bounded set, where (X, d, µ) is a complete metric measure space with metric d and a doubling Borel regular measure µ, supporting a weak (1, p)-Poincar´e inequality for 1 < p < q. We consider some coefficient functions a and b to be measurable and satisfying 0<\alpha\leq a, b \leq \beta, for some positive constants \alpha,\beta. Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary. We extend local properties of quasiminimizers of the p-energy integral on metric spaces studied by Kinnunen and Shanmugalingam [3] to an anisotropic case. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H¨older continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H¨older continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points. This is a joint work [5] with Cintia Pacchiano Camacho (Aalto University)."

Organizers

Panagiota Daskalopoulos

Columbia University

Alessio Figalli

ETH Zürich

Erik Lindgren

Uppsala University

Henrik Shahgholian

KTH Royal Institute of Technology

Susanna Terracini,

University of Turin