Porosity of the free boundary in a minimum problem
Yuwei Hu, Jun Ping Zheng · Journal of Nonlinear and Variational Analysis · 2023
Given a bounded domain Ω ⊂ R N (N ≥ 2), a positive constant λ , and functions q, h ∈ L ∞ (Ω), we study geometric properties of non-negative minimizers of the minimum problemover certain class K in the framework of Orlicz-Sobolev spaces, where u + denotes the positive part of u, χ {•} is the standard characteristic function, and the functions A and F satisfy the structural conditions of Lieberman-Tolksdorf's type.In particular, F is allowed to grow with a subcritical exponent.By using the technique of blow-up and the Harnack's inequality, we firstly prove the non-degeneracy of non-negative minimizers near the free boundary Γ + := ∂ {u > 0} ∩ Ω, and then we show that the free boundary Γ + is locally porous.Furthermore, we also prove that {u > 0} has a uniformly positive density.