We consider shape optimization problems for elasticity systems in architecture. A typical objective in this context is to identify a structure of maximal stability that is close to an initially proposed one. For structures without external forces on varying parts, classical methods allow proving the existence of optimal shapes within well-known classes of bounded uniformly Lipschitz domains. We discuss this for maximally stable roof structures. We then introduce a more general framework that includes external forces on varying parts (for instance, caused by loads of snow on roofs) and prove the existence of optimal shapes, now in a subclass of bounded uniformly Lipschitz domains, endowed with generalized surface measures on their boundaries. These optimal shapes realize the infimum of the corresponding energy of the system. Generalizing further to yet another, very new framework, now involving classes of bounded uniform domains with fractal measures on their boundaries, we finally prove the existence of optimal architectural shapes that actually realize the minimum of the energy. As a by-product we establish the well-posedness of the elasticity system on such domains. In an auxiliary result we show the convergence of energy functionals along a sequence of suitably converging domains. This result is helpful for an efficient approximation of an optimal shape by shapes that can be constructed in practice.

我们考虑建筑弹性系统的形状优化问题。在这种情况下，典型的目标是确定一种最大稳定性的结构，该结构接近最初提出的结构。对于在不同部分上没有外力的结构，经典方法可以证明在有界的​​均匀Lipschitz域的众所周知的类中存在最佳形状。我们将讨论最大稳定屋顶结构。然后，我们引入一个更通用的框架，该框架包括不同部分上的外力（例如，由屋顶上的积雪引起的外力），并证明了最佳形状的存在（现在属于有界均匀Lipschitz域的一个子类），并赋予了他们的界限。这些最佳形状实现了系统相应能量的最小化。进一步推广到另一个非常新的框架，该框架现在包含有界的均匀域类，并且在它们的边界上进行分形测量，我们最终证明存在最佳的建筑形状，这些形状实际上实现了能量的最小值。作为副产品，我们在这些区域上建立弹性系统的适定性。在一个辅助结果中，我们显示了沿一系列适当收敛的域的能量泛函的收敛。该结果有助于通过在实践中构造的形状有效地近似最佳形状。作为副产品，我们在这些区域上建立弹性系统的适定性。在一个辅助结果中，我们显示了沿一系列适当收敛的域的能量泛函的收敛。该结果有助于通过在实践中构造的形状有效地近似最佳形状。作为副产品，我们在这些区域上建立弹性系统的适定性。在一个辅助结果中，我们显示了沿一系列适当收敛的域的能量泛函的收敛。该结果有助于通过在实践中构造的形状有效地近似最佳形状。