Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

Abstract

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type ($1< p< \infty$) with strong absorption condition:

$$\begin{aligned} -\mathrm {div}(\Phi (x, u, \nabla u)) + \lambda _0(x) u_{+}^q(x) = 0 \quad \hbox {in} \quad \Omega \subset \mathbb {R}^N, \end{aligned}$$

where $\Phi : \Omega \times \mathbb {R}_{+} \times \mathbb {R}^N \rightarrow \mathbb {R}^N$ is a vector field with an appropriate p-structure, $\lambda _0$ is a non-negative and bounded function and $0\le q<p-1$. Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved $C^{\gamma }$ regularity estimates along free boundary points, namely $\mathfrak {F}_0(u, \Omega ) = \partial \{u>0\} \cap \Omega$, where the regularity exponent is given explicitly by $\gamma = \frac{p}{p-1-q} \gg 1$. Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $(N-1)$-Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator $-\Delta _p u + \lambda _0 u^q\chi _{\{u>0\}} = 0$ for any $\lambda _0>0$.