An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data

Abstract

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

Appendix: the theorem of Correa and Seeger

We first introduce some notations. Consider a functional $G:[0,{\varepsilon }]\times X \times Y \rightarrow \mathbb {R}$, for some ${\varepsilon } > 0$ and the topological spaces X and Y. For each $t \in [0, {\varepsilon }]$, define $\displaystyle g(t) = \inf \nolimits _{x\in X} \sup \nolimits _{y\in Y} G(t,x,y)$ and $\displaystyle h(t) = \sup \nolimits _{y\in Y} \inf \nolimits _{x\in X} G(t,x,y)$, and the associated sets $\displaystyle X(t) = \big \{ \hat{x} \in X : \sup \nolimits _{y\in Y} G(t,\hat{x}, y) = g(t)\big \}$ and $\displaystyle Y(t) = \big \{ \hat{y} \in Y : \inf \nolimits _{x\in X} G(t,x,\hat{y}) = h(t)\big \}$.

Given the above definitions, we introduce the set of saddle points$S(t)=\{(\hat{x},\hat{y})\in X\times Y : g(t) = G(t, \hat{x}, \hat{y}) = h(t)\}$, which may be empty. In general, the inequality $h(t) \leqslant g(t)$ holds and when $h(t) = g(t)$, we exactly have $S(t)=X(t) \times Y(t)$. Here, we are particularly interested on the situation when G admits saddle points for all $t \in [0,{\varepsilon }]$.

Now, we quote an improved version [21, Theorem 5.1, pp. 556–559] of the theorem of Correa and Seeger. This result provides realistic conditions under which the existence of the limit $dg(0) = \lim _{t\searrow 0} (g(t) - g(0))/t$ is guaranteed.

Theorem 1

([25]) Let X, Y, G and ${\varepsilon }$ be given as previously. Assume that the following assumptions hold:

• (H1) for all $t \in [0,\varepsilon ]$, the set S(t) is non-empty;

• (H2) the partial derivative $\partial _tG(t,x,y)$ exists for all $(t, x,y) \in [0,\varepsilon ] \times X \times Y$;

• (H3) for any sequence $\{t_n\}_{n\in \mathbb {N}}$, with $t_n \rightarrow 0$, there exists a subsequence $\{t_{n_k}\}_{k\in \mathbb {N}}$ and $x^0 \in X(0)$, $x_{n_k} \in X(t_{n_k})$ such that for all $y \in Y(0)$, $\liminf _{\begin{array}{c} t\searrow 0\\ k \rightarrow \infty \end{array}} \partial _tG(t, x_{n_k} , y) \geqslant \partial _tG(0, x^0, y)$; and

• (H4) for any sequence $\{t_n\}_{n\in \mathbb {N}}$, with $t_n \rightarrow 0$, there exists a subsequence $\{t_{n_k}\}_{k\in \mathbb {N}}$ and $y^0 \in Y(0)$, $y_{n_k} \in Y(t_{n_k})$ such that for all $x \in X(0)$, $\limsup _{\begin{array}{c} t\searrow 0\\ k \rightarrow \infty \end{array}} \partial _tG(t, x, y_{n_k}) \leqslant \partial _tG(0, x, y^0)$.

Then, there exists $(x^0, y^0) \in X(0) \times Y(0)$ such that $\displaystyle \frac{{\mathrm{d}}g}{{\mathrm{d}}t}(0) = \partial _tG(0, x^0, y^0)$.