Uniform Cusp Property, Boundary Integral, and Compactness for Shape Optimization

In this paper we consider the family of sets verifying the uniform cusp property introduced in [2] and extended in [4] to cusp functions only continuous a t the origin. In the latter case we show that to any extended cusp function, we can associate a continuous, non-negative, and monotone strictly increasing cusp function of the type introduced in [2]. We construct an example of a bounded set in R~ with a CUSP function of the form cJOIa, 0 < CY < 1, for which its boundary integral is infinite and the Hausdorff dimension of its boundary is exactly N a. We then give compactness theorems for the family of subsets of a bounded open holdall verifying a uniform cusp property with a uniform bound on either the De Georgi [6] or the y-density perimeter of Bucur and Zol&o [I]. We also give their uniform local Co-graph versions following [4]. *This research has been supported by National Sciences and Engineering Research Council of Canada discovery grant A-8730 and by a FQRNT grant from the Ministere de 1'Education du QuBbec. SYSTEM MODELING AND OPTIMIZATION This class forms a much larger family than the one of subsets verifying a uniform cone property.