Journal of Functional Analysis最新文献

Mixed volumes in n-dimensional Euclidean space are functionals of n-tuples of convex bodies $K,L,C_1,\dots ,C_{n-2}$. The Alexandrov–Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov–Fenchel inequality remains a challenging open problem. Major recent progress was made by Yair Shenfeld and Ramon van Handel [13], [14], in particular they resolved the problem in the cases where $C_1,\dots ,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some dimensional restriction). In [6] we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution (and under a dimensional restriction), we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball $B^n$ and $C_1,\dots ,C_{n-2}$. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary $(n-1)$-tuple of polyoids (or smooth bodies). The result thus (partially) confirms a long-standing conjecture by Rolf Schneider in the case of polyoids, and hence in particular it covers the case of zonoids and triangle bodies.

For an étale correspondence $\Omega :G\to H$ of étale groupoids, we construct an induction functor ${\mathrm{Ind}}_{\Omega }:{\mathrm{KK}}^H\to {\mathrm{KK}}^G$ between equivariant Kasparov categories. We introduce the crossed product of an H-equivariant correspondence by Ω, and use this to build a natural transformation ${\alpha }_{\Omega }:K_{⁎}(G⋉{\mathrm{Ind}}_{\Omega }-)\Rightarrow K_{⁎}(H⋉-)$. When Ω is proper these constructions naturally sit above an induced map in K-theory $K_{⁎}\left(C^{⁎}\right(G\left)\right)\to K_{⁎}\left(C^{⁎}\right(H\left)\right)$.

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely$\{\begin{array}{cc}-\Delta u=\lambda e^u& \text{ in }\Omega \\ u=0& \text{ on }\partial \Omega ,\end{array}$ where $\Omega \subset R^2$ is a bounded smooth domain and $\lambda >0$ is a small parameter.

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of ideals and almost isometric ideals in Banach spaces. We prove that given two metric spaces $N\subseteq M$ there always exists an almost isometric local retract $S\subseteq M$ with $N\subseteq S$ and $dens\left(N\right)=dens\left(S\right)$. We also prove that metric spaces which are local retracts (respectively almost isometric local retracts) can be characterised in terms of a condition of extendability of Lipschitz functions (respectively almost isometries) between finite metric spaces. Various examples and counterexamples are exhibited.

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland, and prove that transversally Rockland operators yield a K-homology class. We construct an equivariant KK-class for transversally Rockland transverse symbols, and show a Poincaré duality type result linking the class of an operator and its symbol.

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

We prove that the image of an isometric embedding into $R^3$ of a two dimensional complete Riemannian manifold $(\Sigma ,g)$ without boundary is a convex surface, provided that, first, both the embedding and the metric g enjoy a $C^{1,\alpha }$ regularity for some $\alpha >2/3$, and second, the distributional Gaussian curvature of g is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a general phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form.

In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

A classical result in the study of Ginzburg-Landau equations is that, for Dirichlet or Neumann boundary conditions, if a sequence of functions has energy uniformly bounded on a logarithmic scale then we can find a subsequence whose Jacobians are convergent in suitable dual spaces and whose renormalized energy is at least the sum of absolute degrees of vortices. However, the corresponding question for the case of tangential or normal boundary conditions has not been considered. In addition, the question of convergence of up to the boundary is not very well understood. Here, we consider these questions for a bounded, connected, open set of $R^2$ with $C^{2,1}$ boundary.