Global convergence of $W^{1,\infty}$-steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space

1. Problem Statement

The paper addresses PDE-constrained shape optimization problems where the state equation is a semilinear elliptic partial differential equation. The goal is to find an optimal domain $\Omega$ within a set of admissible domains $\mathcal{S}$ (contained in a fixed hold-all domain $D \subset \mathbb{R}^d$) that minimizes a shape functional:
$$\min_{\Omega \in \mathcal{S}} J(\Omega) := \int_{\Omega} j(x, u(x), \nabla u(x)) \, dx$$
The state variable $u$ is the unique weak solution to the semilinear elliptic boundary value problem:
$$-\Delta u + g(u) = f \quad \text{in } \Omega, \quad u = 0 \quad \text{on } \partial\Omega$$
where $g$ is a monotone nonlinearity ($g' \geq 0$).

The authors investigate the convergence of a steepest descent algorithm realized directly in the function space $W^{1,\infty}_0(D, \mathbb{R}^d)$, rather than relying on finite element discretizations. The domains are updated using the method of mappings, where $\Omega_k = \Phi_k(\hat{\Omega})$ for a fixed reference domain $\hat{\Omega}$ and bi-Lipschitz mappings $\Phi_k$.

2. Methodology

2.1 The Descent Algorithm

The authors employ a steepest descent method with an Armijo line search (Algorithm 2.1):

1. Descent Direction: At each iteration $k$, a direction $V^k$ is chosen to minimize the shape derivative $J'(\Omega_k)[V]$ subject to the constraint $|DV| \leq 1$ almost everywhere in $D$. This defines the direction in the $W^{1,\infty}$ topology.
2. Step Size: A step size $t_k$ is selected from a geometric sequence ($1/2, 1/4, \dots$) satisfying the Armijo condition:
   $$J(\Omega_{k+1}) - J(\Omega_k) \leq \gamma t_k J'(\Omega_k)[V^k]$$
   where $\Omega_{k+1} = (\text{id} + t_k V^k)(\Omega_k)$.
3. Mapping Update: The transformation is updated via composition: $\Phi_{k+1} = (\text{id} + t_k V^k) \circ \Phi_k$.

2.2 Mathematical Framework

• State and Adjoint Equations: The existence and uniform boundedness of the state $u$ and adjoint $p$ in $H^1_0(\Omega) \cap L^\infty(\Omega)$ are established under specific growth conditions on the objective function integrand $j$ and the nonlinearity $g$.
• Shape Derivative: The shape derivative $J'(\Omega)[V]$ is derived using the material derivative approach, resulting in an integral expression involving $u$, $p$, and the deformation field $V$.
• Topology: The analysis is conducted in the infinite-dimensional space $W^{1,\infty}$. The convergence of the shapes is analyzed using the Hausdorff complementary metric and $\gamma$-convergence (Mosco convergence of Sobolev spaces).

3. Key Contributions and Results

3.1 Existence of Armijo Step Size (Lemma 3.1)

The authors prove that for any non-stationary shape, there exists a valid step size $t > 0$ satisfying the Armijo condition. This relies on the specific growth conditions imposed on the second derivatives of $j$ (equations 2.2–2.6) to ensure the Taylor expansion of the objective functional behaves correctly under domain perturbations. This extends previous results from linear Poisson problems to the semilinear case.

3.2 Global Convergence in Function Space (Theorem 3.2)

The primary theoretical result is the global convergence of the algorithm in the sense that the norm of the shape derivative vanishes:
$$\|J'(\Omega_k)\| \to 0 \quad \text{as } k \to \infty$$
This is proven by contradiction. If the gradient norm did not vanish, the Armijo condition would guarantee a uniform decrease in the objective function at every step, leading to a contradiction with the lower boundedness of $J(\Omega)$.

3.3 Convergence to a Stationary Shape in 2D (Theorem 3.6)

For the specific case of two space dimensions ($d=2$), the authors prove that if the sequence of mappings $(\Phi_k)$ is bounded in $W^{1,\infty}(D, \mathbb{R}^2)$, then:

• The sequence of domains $(\Omega_k)$ possesses a subsequence converging to a limit shape $\Omega$ in the Hausdorff complementary metric.
• The limit shape $\Omega$ is a stationary point of the functional ($J'(\Omega)[V] = 0$).

Crucial Technical Tools for 2D:

• Sverak's Theorem: Used to establish $\gamma$-convergence of the domains based on the uniform boundedness of the number of connected components of the complement.
• Chambolle & Doveri's Theorem: Used to handle the convergence of Neumann problems and indicator functions, ensuring the limit shape satisfies the necessary continuity properties for the state and adjoint equations.
• Relaxation of Conditions: Unlike previous works requiring bi-Lipschitz bounds on both sides (upper and lower), this result only requires the upper bound (Lipschitz continuity) on the mappings, which is naturally satisfied by the algorithm's construction.

4. Numerical Experiments

The authors present numerical experiments using the DUNE framework to validate the theoretical findings:

• Setup: A semilinear problem with $g(u) = \frac{1}{2}e^u$ and a specific source term $f$ designed to mimic a linear case with known minimizers.
• Observations:
  • Starting from a small domain, the algorithm converges to the empty set (global minimizer in this specific setup).
  • Starting from a larger domain, the algorithm converges to a ball-shaped local minimizer.
• Boundedness Check: The experiments verify that the $W^{1,\infty}$ norm of the Jacobian of the mappings ($\|D\Phi_k\|_{L^\infty}$) remains bounded as the iteration count increases, even when using a cascadic mesh refinement strategy. This supports the assumption required for the 2D convergence theorem.

5. Significance

• Infinite-Dimensional Analysis: This work moves beyond finite-dimensional discretizations to provide a rigorous convergence analysis in the continuous function space setting for semilinear PDEs.
• Robustness of Steepest Descent: It confirms that the $W^{1,\infty}$ steepest descent method with Armijo line search is globally convergent for a broad class of semilinear elliptic problems.
• Geometric Convergence: By leveraging deep results from geometric measure theory (Sverak, Chambolle & Doveri), the paper establishes that the sequence of shapes actually converges to a stationary geometric object, not just that the gradient vanishes.
• Extension of Literature: It generalizes previous results (specifically from the authors' prior work [DHH25] on linear Poisson equations) to the more complex semilinear case, demonstrating that the core convergence mechanisms remain valid despite the added nonlinearity.

In summary, the paper provides a rigorous mathematical foundation for using steepest descent methods in shape optimization for semilinear PDEs, proving that the algorithm generates a sequence of shapes that converges to a stationary solution under appropriate topological constraints.