Uniform Fat Segment and Cusp Properties for Compactness in Shape Optimization

Abstract

A general compactness theorem for shape/geometric analysis and optimization is given for a family of subsets verifying the uniform fat segment property in a bounded open holdall with or without constraints on the De Giorgi [11] or the γ-density perimeter of Bucur and Zolesio [3]. The uniform fat segment property is shown to be equivalent to the uniform cusp property introduced in [9] with a continuous non-negative cusp function. This equivalence remains true for cusp functions that are only continuous at the origin. The equivalence of sets verifying a segment property with their C0-graph representation is further sharpened for sets with a compact boundary. Our C0-graph characterization is shown to be equivalent to both the uniform cusp property and the uniform segment property. It is used to formulate sufficient conditions on the local graphs of a family of subsets of a bounded open holdall to get compactness. A first condition assumes that the local graphs are bounded above by a cusp function; a second condition which requires that the local graphs be equicontinuous turns out to be equivalent to the first one. The respective solutions of the Laplacian with homogeneous Dirichlet or Neumann boundary condition are shown to be continuous with respect to domains in that family. In the Dirichlet case for 1 < p < ∞, we prove the (1,p)-stability of compact sets in the sense of Herdberg [14] under the weaker almost everywhere assumption rather than quasi everywhere. It is also shown that for the family of measurable crack free sets \(\Omega\) in a bounded open holdall \(D\colon \ v\in \{w\in W^{1,p}_0(D)\colon \ w= 0 \mbox{ almost everywhere on } D\backslash \Omega\}\) implies \(v|_{{\rm int} {\Omega}}\in W^{1,p}_0({\rm int} {\Omega})\).