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Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian

Sven Jarohs, Tadeusz Kulczycki, Paolo Salani

TL;DR

This work analyzes the exterior Bernoulli problem for the half-Laplacian, focusing on the free boundary geometry as the Bernoulli gradient parameter $\lambda$ varies. Employing the moving-plane method, fractional Hopf lemmas, and barrier arguments, it proves monotonicity of the solution domains $\Omega_{\lambda}$, derives quantitative distance bounds between $\partial \Omega_{\lambda}$ and the fixed boundary $\partial K$, and establishes asymptotic behavior: $\mathrm{dist}(\partial\Omega_{\lambda},\partial K)$ behaves like $1/\lambda^2$ for large $\lambda$ and is controlled by a decreasing function $g_{d,r_K}$ that diverges as $\lambda\to0^+$. A key geometric consequence is that inward normal rays to $\partial\Omega_{\lambda}$ meet the convex hull of $\overline{K}$, and in the planar case they meet $K$ itself. These results extend classical exterior Bernoulli theory to the nonlocal, half-Laplacian setting and yield detailed insights into the limiting shapes of $\Omega_{\lambda}$ (approaching a ball centered at $K$ as $\lambda\to0^+$).

Abstract

In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to $0^+$ and to $+\infty$. Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meet the convex envelope of the fixed boundary.

Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian

TL;DR

This work analyzes the exterior Bernoulli problem for the half-Laplacian, focusing on the free boundary geometry as the Bernoulli gradient parameter varies. Employing the moving-plane method, fractional Hopf lemmas, and barrier arguments, it proves monotonicity of the solution domains , derives quantitative distance bounds between and the fixed boundary , and establishes asymptotic behavior: behaves like for large and is controlled by a decreasing function that diverges as . A key geometric consequence is that inward normal rays to meet the convex hull of , and in the planar case they meet itself. These results extend classical exterior Bernoulli theory to the nonlocal, half-Laplacian setting and yield detailed insights into the limiting shapes of (approaching a ball centered at as ).

Abstract

In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to and to . Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meet the convex envelope of the fixed boundary.
Paper Structure (3 sections, 12 theorems, 81 equations, 5 figures)

This paper contains 3 sections, 12 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main results

Key Result

Theorem 1.2

Assume that the bounded domain $K \subset \mathbb{R}^d$ has a $C^2$ boundary and it is starshaped with respect to a ball $B_r(x_0)$ for some $r > 0$ and $x_0 \in K$. Then for any $\lambda > 0$ there exists a unique solution $u_{\lambda}$, $\Omega_{\lambda}$ of Problem bernoulli_problem. Moreover, $\

Figures (5)

  • Figure 1: $\Omega_{\lambda_i}$ for $i=1,2,3,4$ with $\lambda_1<\lambda_2<\lambda_3<\lambda_4$ and the corresponding normal rays intersecting $K$.
  • Figure 2: Exemplification of Lemma \ref{['cornerpoint']} with $U=W\cap H$ and respectively of Lemma \ref{['normal_derivative']} with $U=\Omega_+$ (but without $K$), and $e_1=(1,0,\ldots,0)$ and $e_2=(0,1,0,\ldots,0)$.
  • Figure 3: Definition of $B_{\rho}(y_0)$ with $x_0=0$.
  • Figure 4: Picture for Lemma \ref{['normal_derivative']}.
  • Figure 5: What happens when the inward normal ray to $\partial \Omega$ at $x_0$ does not meet $\overline{K}$.

Theorems & Definitions (27)

  • Theorem 1.2: Theorem 1.6 and Proposition 2.11 in JKS2022
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.4
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2: Fractional Hopf lemma
  • proof
  • ...and 17 more