Shape optimization for the Neumann problem of the Laplace equation is important for application and also from the numerical point of view. Mathematical analysis of such problem in the half space is not available. In this paper we prove the shape differentiability of solutions in appropriate weighted Sobolev spaces which describe the behavior of solutions at infinity. We will consider two differ...

Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...

Abstract In this paper, we study capital allocation for dynamic risk measures, with an axiomatic approach but also by exploiting the relation between measures and BSDEs. Although there is a wide literature on rules in static setting only few recent papers work and, moreover, those mainly focus gradient approach. To fill gap, then discuss new perspectives to problem going beyond already existing...

We show that there is an operator space notion of Lipschitz embeddability between spaces which strictly weaker than its linear counterpart but still strong enough to impose restrictions on structures. This shows a nontrivial theory nonlinear geometry for and it answers question in Braga et al. (Proc Am Math Soc 149(3):1139–1149, 2021). For that, we introduce the version Lipschitz-free Banach pr...

We propose a notion of smoothness of nonexpected utility functions, which extends the variational analysis of nonexpected utility functions to more general settings. In particular, our theory applies to state dependent utilities, as well as the multiple prior expected utility model, both of which are not possible in previous literatures. Other nonexpected utility models are shown to satisfy smo...

In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of differentiability of convex functions (cf. the book...

This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization find superconductor minimizes a certain cost functional under given target on electric field over specific domain interest. For governing PDE-model, we consider elliptic curl-curl variational inequality (VI) second kind with L1-type nonlin...