We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution $u:B_1\subset {\mathbb R}^n \to {\mathbb R}^m$ to the elliptic system \begin{equation*} \hbox{div} ((A + (B-A)\chi_D )\nabla u) = 0 \quad \text{in }B_1, \end{equation*} where $A$ and $B$ are Dini continuous, uniformly elliptic matrices, we prove that if $\nabla u \in L^{\infty} (D)$ then $u$ is Lipschitz in $B_{1/2}$. A similar result is also derived for the parabolic counterpart of this problem.

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

In this paper we prove symmetry for solutions to the semi-linear elliptic equation $$ \Delta u = f(u) \quad \hbox{ in } B_1, \qquad 0 \leq u < M, \quad \hbox{ in } B_1, \qquad u = M, \quad \hbox{ on } \partial B_1, $$ where $M>0$ is a constant, and $B_1$ is the unit ball. Under certain assumptions on the r.h.s. $f (u)$, the $C^1$-regularity of the free boundary $\partial \{u>0\}$ and a second order asymptotic expansion for $u$ at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin's celebrated boundary point lemma, which is not available in our case due to lack of $C^2$-regularity of solutions.

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.