Revisited for existence proof of optimal solution in Bernoulli free boundary problem using an energy-gap cost functional

TL;DR

This paper revisits the Bernoulli free boundary problem, focusing on the existence proof of optimal solutions using an energy-gap cost functional, and corrects a previous proof error related to solution boundedness.

Contribution

It provides a corrected proof for the existence of optimal solutions in Bernoulli free boundary problems using an energy-gap functional, addressing a flaw in earlier work.

Findings

Corrected the proof of existence of optimal solutions.

Established boundedness of solution sequences via Poincaré-Friedrichs inequality.

Clarified the limitations of previous estimates using Cauchy-Schwarz inequality.

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 Poincar\'e-Friedrichs inequality.