Speaker
Antonella Nastasi
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 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).
References
[1] A. Björn, J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17, European Mathematical Society (EMS), Zurich (2011).
[2] V. Bögelein, F. Duzaar, P. Marcellini, C. Scheven, A variational approach to doubly nonlinear equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 739–772.
[3] J. Kinnunen, N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math., 105 (2001), 401–423.
[4] P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl., 501 (2021), https://doi.org/10.1016/j.jmaa.2020.124408.
[5] A. Nastasi, C. Pacchiano Camacho, Regularity properties for quasiminimizers of a (p, q)- Dirichlet integral, Calc. Var., 227 (60) (2021).