An existence result for a class of shape optimization problems

Shape Optimization under Uncertainty from a Stochastic Programming Point of View

“…From [104, p. 52] we know that this polytope is the convex hull of its vertices, and from linear programming theory [17,30,78] we thus know that there is always a vertex in the set of optimal solutions to the maximization problem in (1.25). This is an important observation which allows us to rewrite (1.22) as 26) where…”

“…a perimeter constraint). Details on the question of existence can be found in [2,8,20,19,24,25,26,27,89,96].…”

“…The descent direction we use with respect to t ist ∈ R S defined bȳ 26) which makes sense, as we can see immediately from (4.25) that B t (t, φ; γ) ,t ≤ 0. The shape derivative, i.e.…”

Optimal Shapes and Masses, and Optimal Transportation Problems

“…admits a solution (see [55]). iii) The problem is of a very special type, involving only the first two eigenvalues of the Laplace operator, where neither geometrical constraints nor monotonicity of the cost are required (see [38]).…”

Regularity and long-time behavior of nonlocal heat flows

“…Shape optimization problems arise naturally in the field of structural design, when one seeks to optimize the design of a system governed by partial differential equations. Problem (P) has been studied by many authors; see for example [2,3,4,29] and the references in [2]. Problem (P) also belongs to a broader class of partition problems that have attracted interest due to connections with spectral theory, see for example [21,22,23].…”