# Antonella Nastasi: Regularity for quasiminimizers of an anisotropic problem

Time: Tue 2022-09-20 14.00 - 15.00

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Antonella Nastasi (University of Palermo)

Abstract:

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 Hölder 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 Hölder 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).