We discuss convexity for several free boundary value problems in exterior domains that are generally formulated as $$ \Delta u = f(u) \quad \text{in } \Omega \setminus D, \qquad |\nabla u | = g \quad \text{on } \partial \Omega , \qquad u\geq 0 \quad \text{in } \Bbb R^n $$ where $u$ is assumed to be continuous in $ \Bbb R^n$, $ \Omega = \{u > 0\}$ (is unknown), $u=1$ on $\partial D$, and $D$ is a bounded domain in $\Bbb R^n$ ($n\geq 2$). Here $g= g(x)$ is a given smooth function which is either strictly positive (Bernoulli-type) or identically zero (obstacle type). Properties for $f$ will be spelled out in exact terms in the text.

We make the following general assumptions on $f$ (see [Fr-Ph] for similar type of problems) $$ \begin{cases} f(t) > 0 \qquad & \text{for} \,\,t > 0 , \\ f(t)=0 \qquad & \text{for} \,\, t\leq 0 \\ f(t)= b t^{\alpha} + o(t^{\alpha}) \qquad & \text{for}\,\, t < t_1, \,\, \text{some}\,\, t_1 >0 \,\,\text{and} \,\, -1< \alpha< 1,\\ \end{cases} $$ where $ b \geq 0$. We also assume $$ \hbox{ $f$ is either left- or right-continuous and its discontinuities are isolated}. $$

Theorem:

Let $D$ be a bounded convex domain in $\Bbb R^n$ ($n \geq 2$), and $f$ satisfy the above properties. Let $g(x)$ be a $C^\beta$-function in the entire space satisfying $\max (b, g(x)) \geq b_0$ for some $b_0$, and $g(x) $ be either identically zero or a strict positive. When $g > 0$ we assume $1/g$ to be concave function.

Then there exists a non-negative function $u$ with $\Omega := \{ u >0\}$ solving the free boundary problem $$ \begin{cases} \Delta u = f(u) \qquad & \text{in } \Omega \setminus D , \\ u = 1 \qquad& \text{on } \partial D ,\\ |\nabla u |= g \qquad& \text{on } \partial \Omega , \end{cases} $$ with convex super-level sets i.e. $\{u > l \}$ is convex, for all $0 \leq l \leq 1$.

If in addition there is a point $z \in D$, such that $$ tg(t(x-z) + z) \ \hbox{is non-decreasing in } t, \ \hbox{ for all } x \in \Bbb R^n, $$ then $u$ is the unique solution.

Abstract: We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, seems likely to adapt to more general domains, as well as other types of operators.

We consider operators of the form $$ \mathcal{L} u = (a^{ij}u_i)_j + b^i u_i + cu , $$ with the following ellipticity conditions $$ \Lambda^{-1} |\xi|^2 \leq \langle a^{ij}(x) \xi, \xi \rangle \leq \Lambda |\xi|^2, $$ for some $\Lambda >0$ and for all nonzero $\xi \in \R^n$. Furthermore, $a^{ij}(x)$ is a real $n \times n$ matrix. For the lower order terms we assume $$|c(x)|, \sum |b^i(x)| \leq \Lambda-1$$ and that $c(x) \leq 0$. We say $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R})$ if $$ \begin{aligned} \mathcal{L} u(x) &=0 \text{ in } \Omega_{L,R} \cap B_R ,\\ u(x)&=0 \text{ on } \Omega_{L,R}^c\cap B_R, \end{aligned} $$ and that $u \in \mathcal{S_{\mathcal{L}}}(\Omega_{L,R},d^{\gamma})$ if $$ \begin{aligned} |\mathcal{L} u(x)| &\leq (\text{dist}(x,\partial \Omega_{L,R} \cap B_R))^{\gamma} \ \text{ in } \Omega_{L,R} , \\ u(x)&=0 \ \text{ on } \Omega_{L,R}^c\cap B_R. \end{aligned} $$ To apply the Hölder continuity estimates for elliptic operators we will require that $\gamma>-2/n $. Since the boundary is Lipschitz, this will ensure the correct integrability assumptions for the right hand side.

Theorem:

Let $0 \in \partial \Omega_L$ with $L< M$, and fix $x^0 \in \Omega_L$. Assume further that $B_1 \cap \{x_n > 1/4\} \subseteq \Omega_L$. Assume $u,v \geq 0$ and $u,v \in \mathcal{S}(\Omega_{L},d^{\gamma})$ with $\mathcal L u, \mathcal L v \leq 0$ and $u(x^0)=v(x^0)=1$, and also assume that $2-\alpha+\gamma>0$, with $\gamma>-2/n$ and $\alpha$ be such that $$ \sup_{B_{r}(x)} u \geq c_1 u(e_n/2) r^{\alpha}. $$

Then there exists a uniform constant $C>0$ (depending only on dimension $n$, Lipschitz constant $M$, $M-L$, and dist$(x^0,\partial \Omega_L)$) such that \begin{equation} \label{e:ine3} C^{-1} v(x) \leq u(x) \leq Cv(x) \end{equation} for all $x \in B_{1/2}$.