| Abstract: |
| Given a smooth bounded domain $K$ in $\mathbb{R}^N$ and a parameter $\lambda>0$, the exterior Bernoulli problem (EBP) for the half Laplacian is to find a function $u:\mathbb{R}^N\to\R$ and a smooth open subset $\Omega$ of $\mathbb{R}^N$ such that $\overline{K}\subset \Omega$ and $u$ is a solution to the problem
$$
(-\Delta)^{1/2}u=0\quad\text{in $\Omega\setminus \overline{K}$,}\quad u=0\quad\text{in $\mathbb{R}^N\setminus \Omega$,}\quad u=1\quad\text{in $\overline{K}$,}
$$
with
$$
D_{\Omega}^{1/2}u(\theta)=\lim_{t\to 0^+}\frac{u(\theta+t\nu(\theta)}{t^{1/2}}=\lambda \quad\text{for all $\theta\in \partial\Omega$,}
$$
where $\nu(\theta)$ denotes the interior unit normal at $\theta\in \partial\Omega$.\
In this talk, the existence of a solution to the EBP with its geometric properties and resulting regularity is discussed. Furthermore qualitative properties related to the asymptotic behavior of the free boundary of solutions when the \textit{Bernoulli`s gradient parameter} $\lambda$ tends to $0^+$ or to $+\infty$ are presented.
The talk is based on two joint works with Tadeusz Kulczycki and Paolo Salani. |
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