We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fučik, Nečas, Souček, and Souček. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p(x)-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.This paper is dedicated with esteem and gratitude to Professor Carlo Sbordone on the occasion of his 70th birthday 1
The problem of splitting a homotopy equivalence along a submanifold is
closely related to the surgery exact sequence and to the problem of surgery of
manifold pairs. In classical surgery theory there exist two approaches to
surgery in the category of manifolds with boundaries. In the $ rel \partial$
case the surgery on a manifold pair is considered with the given fixed manifold
structure on the boundary. In the relative case the surgery on the manifold
with boundary is considered without fixing maps on the boundary. Consider a
normal map to a manifold pair $(Y, \partial Y)\subset (X, \partial X)$ with
boundary which is a simple homotopy equivalence on the boundary $\partial X$.
This map defines a mixed structure on the manifold with the boundary in the
sense of Wall. We introduce and study groups of obstructions to splitting of
such mixed structures along submanifold with boundary $(Y, \partial Y)$. We
describe relations of these groups to classical surgery and splitting
obstruction groups. We also consider several geometric examples.Comment: Abh. Math. Sem. Univ. Hamburg, to appea
Recent research in coarse geometry revealed similarities between certain
concepts of analysis, large scale geometry, and topology. Property A of G.Yu is
the coarse analog of amenability for groups and its generalization (exact
spaces) was later strengthened to be the large scale analog of paracompact
spaces using partitions of unity. In this paper we go deeper into divulging
analogies between coarse amenability and paracompactness. In particular, we
define a new coarse analog of paracompactness modelled on the defining
characteristics of expanders. That analog gives an easy proof of three
categories of spaces being coarsely non-amenable: expander sequences, graph
spaces with girth approaching infinity, and unions of powers of a finite
non-trivial group.Comment: 24 pages, version 3 as a result of comments by a great refere
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