A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman

• Published in: SIAM journal on mathematical analysis 2009-01, Vol.41 (1), p.318-359
• Main Authors: Martínez, Sandra, Wolanski, Noemi
• Format: Article
• Language: English
• Subjects: Constraining; Free boundaries; Mathematical analysis; Operators; Partial differential equations; Singular perturbation; Viscosity
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/070703740

In this paper we study the following problem. For $\varepsilon>0$, take $u^{\varepsilon}$ as a solution of $\mathcal{L}u^{\varepsilon}:=\mathrm{div}\,(\frac{g(|\nabla u^{\varepsilon}|)}{|\nabla u^{\varepsilon}|}\nabla u^{\varepsilon})=\beta_{\varepsilon}(u^{\varepsilon})$, $u^{\varepsilon}\geq0$. A solution to $(P_{\varepsilon})$ is a function $u^{\varepsilon}\in W^{1,G}(\Omega)\cap L^{\infty}(\Omega)$ such that $\int_{\Omega}g(|\nabla u^{\varepsilon}|)\frac{\nabla u^{\varepsilon}}{|\nabla u^{\varepsilon}|}\nabla\varphi\,dx =-\int_{\Omega}\varphi\,\beta_{\varepsilon}(u^{\varepsilon})\,dx$ for every $\varphi\in C_0^{\infty}(\Omega)$. Here $\beta_{\varepsilon}(s)=\frac{1}{\varepsilon}\beta\left(\frac{s}{\varepsilon}\right)$, with $\beta\in\mathrm{Lip}(\mathbb{R})$, $\beta>0$ in $(0,1)$ and $\beta=0$ otherwise. We are interested in the limiting problem, when $\varepsilon\to 0$. As in previous work with $\mathcal{L}=\Delta$ or $\mathcal{L}=\Delta_p$ we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a $C^{1,\alpha}$ surface. This result is new even for $\Delta_p$. Throughout the paper, we assume that $g$ satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].