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

We prove global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations. Steepest descent is realized in the Lipschitz topology. In addition, we prove a conditional convergence result for the resulting shapes in two space dimensions.