Revisited for existence proof of optimal solution in Bernoulli free boundary problem using an energy-gap cost functional
Shiouhe Wang, Fang Shen, Yi Yang, Xueshang Feng
TL;DR
This work revisits the existence proof of an optimal solution for the Bernoulli free boundary problem within a shape-optimization framework that employs an energy-gap cost functional $\mathcal{J}(\Omega)=\int_{\Omega} |\nabla(u_N-u_R)|^2 dx$. The two auxiliary state problems define $u_N$ and $u_R$ by solving $-\Delta u=0$ in $\Omega$ with boundary data on $\Gamma$ and $\Sigma$, and $u_R$ satisfying $\partial_n u_R + \beta u_R = \lambda$ on $\Sigma$, respectively. A corrigendum addresses a flawed $Cauchy$-$Schwarz$-based bound in Eq.$(48)$, replacing it with a bound derived via the $Poincaré$-$Friedrichs$ inequality to obtain a uniform $H^1(\Omega_k)$ bound for $u_{R_k}$. This correction ensures the coercivity of the bilinear form $a+a_{\Sigma}$, enabling the compactness of the admissible state set $\mathcal{F}$ and, together with lower semi-continuity of $\mathcal{J}$, the existence of optimal domains. The result strengthens the theoretical foundation for energy-gap based shape optimization in Bernoulli free boundary problems and clarifies the necessary steps for rigorous existence proofs.
Abstract
Bernoulli free boundary problem is numerically solved via shape optimization that minimizes a cost functional subject to state problems constraints. In \cite{1}, an energy-gap cost functional was formulated based on two auxiliary state problems, with existence of optimal solution attempted through continuity of state problems with respect to the domain. Nevertheless, there exists a corrigendum in Eq.(48) in \cite{1}, where the boundedness of solution sequences for state problems with respect to the domain cannot be directly estimated via the Cauchy-Schwarz inequality as \textbf{Claimed}. In this comment, we rectify this proof by Poincaré-Friedrichs inequality.