##
**Shape optimisation problems governed by nonlinear state equations.**
*(English)*
Zbl 0918.49030

The aim of the paper is the study of the closure, in the Hausdorff complementary topology, of certain classes of domains defined by means of weak constraints given in terms of capacity, in the ambit of shape optimization problems governed by nonlinear state equations.

The main theorem proved is the following.

Theorem. Let \(B\) be a fixed ball in \(\mathbb{R}^n\), and \(1<p<+\infty\). Then the class \(W(B)\) of subdomains of \(B\) is introduced so that, if \(\{\Omega_h\}\subseteq W(B)\), and \(\{\Omega_n\}\) converges in the Hausdorff complementary topology to \(\Omega\), then \(\Omega\in W(B)\), and for every \(f\in H^{-1,p'}(B)\) the zero extensions of the solutions of the Dirichlet problems \[ (\text{DP})\begin{cases} -\Delta_p u_h= f\quad\text{in }\Omega_h\\ u_h\in H^{1,p}_0(\Omega_h)\end{cases} \] converge strongly in \(H^{1,p}_0(B)\) to the solution of the corresponding problem in \(\Omega\).

It is to be emphasized that, in general, the class of all the open subsets of \(B\) is not closed with respect to the above type of convergence, and the ‘limit’ problem can be of ‘relaxed type’, and can contain terms depending on a Borel measure. The approach to the problem is different from the one of the case with linear state equations and comes from the theory of generalized harmonic functions and monotone operators. As a consequence, a result of Šverák is extended.

If \(n-1<p\leq n\), \(k\in\mathbb{N}\), and \(\{\Omega_h\}\) is a sequence of open subsets of \(B\) such that for every \(h\) the number of the connected components of \(\mathbb{R}^n\setminus\Omega_h\) is less than or equal to \(k\), and that converges in the Hausdorff complementary topology to \(\Omega\), then the number of the connected components of \(\mathbb{R}^n\setminus \Omega\) is less than or equal to \(k\) and for every \(f\in H^{-1,p'}(B)\) the zero extensions of the solutions of (DP) converge strongly in \(H^{1,p}_0(B)\) to the solution of the corresponding problem in \(\Omega\).

Finally, the results for the \(p\)-Laplacian are extended, but only for what concerns the convergence of the solutions of (DP), to more general classes of monotone operators.

The main theorem proved is the following.

Theorem. Let \(B\) be a fixed ball in \(\mathbb{R}^n\), and \(1<p<+\infty\). Then the class \(W(B)\) of subdomains of \(B\) is introduced so that, if \(\{\Omega_h\}\subseteq W(B)\), and \(\{\Omega_n\}\) converges in the Hausdorff complementary topology to \(\Omega\), then \(\Omega\in W(B)\), and for every \(f\in H^{-1,p'}(B)\) the zero extensions of the solutions of the Dirichlet problems \[ (\text{DP})\begin{cases} -\Delta_p u_h= f\quad\text{in }\Omega_h\\ u_h\in H^{1,p}_0(\Omega_h)\end{cases} \] converge strongly in \(H^{1,p}_0(B)\) to the solution of the corresponding problem in \(\Omega\).

It is to be emphasized that, in general, the class of all the open subsets of \(B\) is not closed with respect to the above type of convergence, and the ‘limit’ problem can be of ‘relaxed type’, and can contain terms depending on a Borel measure. The approach to the problem is different from the one of the case with linear state equations and comes from the theory of generalized harmonic functions and monotone operators. As a consequence, a result of Šverák is extended.

If \(n-1<p\leq n\), \(k\in\mathbb{N}\), and \(\{\Omega_h\}\) is a sequence of open subsets of \(B\) such that for every \(h\) the number of the connected components of \(\mathbb{R}^n\setminus\Omega_h\) is less than or equal to \(k\), and that converges in the Hausdorff complementary topology to \(\Omega\), then the number of the connected components of \(\mathbb{R}^n\setminus \Omega\) is less than or equal to \(k\) and for every \(f\in H^{-1,p'}(B)\) the zero extensions of the solutions of (DP) converge strongly in \(H^{1,p}_0(B)\) to the solution of the corresponding problem in \(\Omega\).

Finally, the results for the \(p\)-Laplacian are extended, but only for what concerns the convergence of the solutions of (DP), to more general classes of monotone operators.

Reviewer: R.De Arcangelis (Napoli)

### MSC:

49Q10 | Optimization of shapes other than minimal surfaces |

49J20 | Existence theories for optimal control problems involving partial differential equations |

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\textit{D. Bucur} and \textit{P. Trebeschi}, Proc. R. Soc. Edinb., Sect. A, Math. 128, No. 5, 945--963 (1998; Zbl 0918.49030)

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