Remarks on the convexity of free boundaries (Scalar and system cases)

Abstract:

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

The interest is in the particular case where $D$ is star-shaped or convex. The focus is on the case where $f(u)$ lacks monotonicity, so that the recently developed tool of quasiconvex rearrangement is not applicable directly. Nevertheless, such quasiconvexity is used in a slightly different manner, and in combination with scaling and asymptotic expansion of solutions at regular points. The latter heavily relies on the regularity theory of free boundaries.

Also, convexity for several systems of equations in a general framework is discussed, and some ideas along with several open problems are presented.