In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an $L^p$ best approximation problem under divergence constraints to the shape tensor introduced in Laurain and Sturm: ESAIM Math. Model. Numer. Anal. 50 (2016). More precisely, the main result of this paper states that the $L^p$ distance of the above approximation problem is equal to the dual norm of the shape derivative considered as a functional on $W^{1,p^\ast}$ (where $1/p + 1/p^\ast = 1$). This implies that for any given shape, one can evaluate its distance from being a stationary one with respect to the shape derivative by simply solving the associated $L^p$-type least mean approximation problem. Moreover, the Lagrange multiplier for the divergence constraint turns out to be the shape deformation of steepest descent. This provides a way, as an alternative to the approach by Deckelnick, Herbert and Hinze: ESAIM Control Optim. Calc. Var. 28 (2022), for computing shape gradients in $W^{1,p^\ast}$ for $p^\ast \in ( 2 , \infty )$. The discretization of the least mean approximation problem is done with (lowest-order) matrix-valued Raviart-Thomas finite element spaces leading to piecewise constant approximations of the shape deformation acting as Lagrange multiplier. Admissible deformations in $W^{1,p^\ast}$ to be used in a shape gradient iteration are reconstructed locally. Our computational results confirm that the $L^p$ distance of the best approximation does indeed measure the distance of the considered shape to optimality. Also confirmed by our computational tests are the observations that choosing $p^\ast$ (much) larger than 2 (which means that $p$ must be close to 1 in our best approximation problem) decreases the chance of encountering mesh degeneracy during the shape gradient iteration.

翻译：本文中，受PDE约束的形状优化问题被重新表述为在散度约束下对Laurain与Sturm所引入形状张量的$L^p$最佳逼近问题（Laurain, Sturm: ESAIM Math. Model. Numer. Anal. 50 (2016)）。更精确地说，本文主要结果表明，上述逼近问题的$L^p$距离等于形状导数作为$W^{1,p^\ast}$上泛函的对偶范数（其中$1/p + 1/p^\ast = 1$）。这意味着，对任意给定形状，仅需求解相应的$L^p$型最小平均逼近问题，即可评估该形状偏离形状导数平稳点的程度。此外，散度约束的拉格朗日乘子恰好为最速下降形状变形。这提供了除Deckelnick, Herbert与Hinze方法（Deckelnick, Herbert, Hinze: ESAIM Control Optim. Calc. Var. 28 (2022)）之外，另一种在$W^{1,p^\ast}$（$p^\ast \in (2, \infty)$）中计算形状梯度的方法。最小平均逼近问题的离散化采用（最低阶）矩阵值Raviart-Thomas有限元空间，从而得到作为拉格朗日乘子的形状变形的分片常数逼近。形状梯度迭代中允许的$W^{1,p^\ast}$可容许变形通过局部重构获得。我们的计算结果证实，最佳逼近的$L^p$距离确实度量了所考虑形状与最优性的偏离程度。计算测试还验证了如下观察：选择$p^\ast$（远）大于2（这意味着在我们的最佳逼近问题中$p$需接近1）可降低形状梯度迭代过程中网格退化的概率。