The Uniform Fat Segment and Uniform Cusp Properties in Shape Optimization

Introduction A large class of domains Ω in RN can be characterized by a local geometric property, the segment property, which is equivalent to the property that Ω be locally a C0-epigraph. This property is sufficient to get the density of the function space Ck(Ω) in the Sobolev space Wm,p(Ω) for any 1 ≤ m ≤ k and it plays a key role in establishing the continuity of the solution of the Neumann problem for the Laplace operator with respect to the underlying domain Ω.