The thermodynamic limit of quantum Coulomb systems Part I. General theory

This article is the first in a series dealing with the thermodynamic properties of quantum Coulomb systems.In this first part, we consider a general real-valued function E defined on all bounded open sets of R³. Our aim is to give sufficient conditions such that E has a thermodynamic limit. This means that the limit E(Ωn)|Ωn|⁻¹ exists for all ‘regular enough’ sequence Ωn with growing volume, |Ωn|→∞, and is independent of the considered sequence.The sufficient conditions presented in our work all have a clear physical interpretation. In the next paper, we show that the free energies of many different quantum Coulomb systems satisfy these assumptions, hence have a thermodynamic limit.     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

Spectral controllability for 2D and 3D linear Schrödinger equations

We consider a quantum particle in an infinite square potential well of Rn, n=2,3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: (Kal) if Ω is the bottom of the well, then for every eigenvalue λ of , the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in Rn. In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time Tmin(Ω)>0 for spectral controllability, i.e., if T>Tmin(Ω), one has spectral controllability in time T if condition (Kal) holds true for (Ω) and, if T<Tmin(Ω) and the dipolar moment is one-directional, then one does not have spectral controllability in time T. We next characterize a necessary and sufficient condition on the dipolar moment insuring that spectral controllability in time T>Tmin(Ω) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory.     

Suppleness of the sheaf of algebras of generalized functions on manifolds

We show that the sheaves of algebras of generalized functions Ω→G(Ω) and Ω→G^∞(Ω), Ω are open sets in a manifold X, are supple, contrary to the non-suppleness of the sheaf of distributions.     

Separate continuity of the Lempert function of the spectral ball

We find all matrices A from the spectral unit ball Ωn such that the Lempert function lΩn(A,⋅) is continuous.     

Asymptotic Optimality of Isoperimetric Constants

In this paper, we investigate the existence of L ²(π)-spectral gaps for π-irreducible, positive recurrent Markov chains with a general state space Ω. We obtain necessary and sufficient conditions for the existence of L ²(π)-spectral gaps in terms of a sequence of isoperimetric constants. For reversible Markov chains, it turns out that the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. These results are used to recover classical results concerning uniform ergodicity and the spectral gap property as well as other new results. As an application of our result, we present a rather short proof for the fact that geometric ergodicity implies the spectral gap property. Moreover, the main result of this paper suggests that sharp upper bounds for the spectral gap should be expected when evaluating the isoperimetric flow for certain sets. We provide several examples where the obtained upper bounds are exact.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

The Regularity of the Inverses of Sobolev Homeomorphisms with Finite Distortion

Let Ω and Ω′ be bounded open sets in ? ⁿ , n≥2, and let Hom(Ω;Ω′) be the class of homeomorphisms f:Ω→Ω′. If f∈Hom(Ω;Ω′)∩W ^1,n?1(Ω;Ω′) is a homeomorphism with finite inner distortion, we deduce regularity properties of the inverse f ^?1 from the regularity of the distortion function of f.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

A curious behavior of capillary surfaces

It is known that for a given base domainΩ and «contact angle»γ, there will be a critical γ₀, 0 ?γ₀, ?π/2., such that a solution overΩ of the scalar capillarity equation in the absence of external (e.g. gravity) force field will exist if γ₀, 0 ?γ, ?π/2, and will fail to exist if 0 ?γ ?γ₀. In the particular case for whichΩ is a circular disk, a solution exists for everyγ, and is unique up to an additive constant. For a piecewise smoothΩ that is not circular, there will be a boundary pointP of maximal inward directed curvatureK _M (at a protruding corner we defineK _M=∞). We suppose that a solution exists for such a domain, and ask whether a solution continues to exist if the domain is made closer to circular by smoothing with an inscribed interior arc of constant curvatureK<K _M. That is so in many cases, for example it is true ifΩ is a rectangle, and in fact in that case smoothing decreases γ₀. In the present note, we show that the answer can also be negative. To that effect, we give an example of a convex domain Ω at which the valueK _M is achieved only at a single isolated point, and for which smoothing at that point changes existence to non-existence.     

On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

We study variational problems of the form $$\inf\{\lambda_k(\Omega): \Omega\ \mbox{open in}\ \mathbb{R}^m,\ T(\Omega ) \le1 \},$$ where λ _k (Ω) is the k-th eigenvalue of the Dirichlet Laplacian acting in L ²(Ω), and where T is a non-negative set function defined on the open sets in ? ^m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T(B)=1 is a minimizer for k=1. Upper bounds are obtained for the number of components of any bounded minimizer if T satisfies a scaling relation. For example, we show that if T is Lebesgue measure and if k≤m+1 then any bounded minimizer has at most 7 components. We also consider variational problems over open sets Ω in ? ^m involving the (m?1)-dimensional Hausdorff measure of ?Ω.     

On the best Sobolev trace constant and extremals in domains with holes

We study the dependence on the subset A⊂Ω of the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measure. In the case that Ω is a ball we prove that there exists an optimal hole that is spherically symmetric. In the case p=2 we prove that every optimal hole is spherically symmetric. Then, we study the behavior of the best constant when the hole is allowed to have zero Lebesgue measure. We show that this constant depends continuously on the subset and we discuss when it is equal to the Sobolev trace constant without the vanishing restriction.     

On a characterization of Arakelian sets

Let K be a compact set in the complex plane, such that its complement in the Riemann sphere, (? ∪ {∞}) / K, is connected. Also, let U ? ? be an open set which contains K. Then there exists a simply connected open set V ? ? such that K ? V ? U. We show that if K is replaced by a closed set F ? ?, then the preceding result is equivalent to the fact that F is an Arakelian set in ?. This holds in more general case when ? is replaced by any simply connected open set Ω ? ?. In the case of an arbitrary open set Ω ? ?, the above extends to the one point compactification of Ω. If we do not require (? ∪ {∞}) /K to be connected, we can demand that each component of (? ∪ {∞}) / V intersects a prescribed set A containing one point in each component of (? ∪ {∞}) / K. Using the previous result, we prove that again if we replace K by a closed set F, the latter is equivalent to the fact that F is a set of uniform meromorphic approximation with poles lying entirely in A.     

On a capacity for modular spaces

The purpose of this article is to define a capacity on certain topological measure spaces X with respect to certain function spaces V consisting of measurable functions. In this general theory we will not fix the space V but we emphasize that V can be the classical Sobolev space W1,p(Ω), the classical Orlicz-Sobolev space W1,Φ(Ω), the Haj?asz-Sobolev space M1,p(Ω), the Musielak-Orlicz-Sobolev space (or generalized Orlicz-Sobolev space) and many other spaces. Of particular interest is the space given as the closure of in W1,p(Ω). In this case every function u∈V (a priori defined only on Ω) has a trace on the boundary ∂Ω which is unique up to a Capp,Ω-polar set.     

Few distinct distances implies no heavy lines or circles

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set \(\mathcal{P}\) of n points determines o(n) distinct distances, then no line contains Ω(n ^7/8) points of \(\mathcal{P}\) and no circle contains Ω(n ^5/6) points of \(\mathcal{P}\).We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.     

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω⊂Rn with boundary conditions on ∂Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L²(Ω)⊕(L²m(∂Ω)), which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.     

Über den Freudenthalschen Spektralsatz

For a vector lattice E with the principal projection property, the following generalization of H.Freudenthal's spectral theorem is proved: There exists a measure space (Ω,R,π) such that integration with respect to π establishes a vector lattice isomorphism from L¹(π) to E. Here π:?→E is a σ -additive vector measure on some δ-ring R which, for [σ-] Dedekind complete E, may be chosen to be the δ-ring of relatively compact [Baire-] Borel sets in a locally compact space. Among others Kakutani's representation of abstract L-spaces as concrete L¹ -spaces is an immediate consequence.     

A New Class of Harmonic Measure Distribution Functions

Let Ω be a planar domain containing 0. Let h _Ω (r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h _Ω is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X _n such that \(h_{X_{n}}\) converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h _Ω , where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X _n . Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h _Ω for a unique symmetric bounded simply connected domain Ω.     

On a problem of magnetohydrodynamics in a multi-connected domain

We consider the following problem in the MHD approximation: the vessel Ω₁⊂Ω is filled with an incompressible, electrically conducting fluid, and is surrounded by a dielectric or by vacuum, occupying the bounded domain Ω₂=Ω?Ω₁. In Ω we have a magnetic and electric field and the external surface S=∂Ω is an ideal conductor. The emphasis in the paper is on when Ω is not simply connected, in which case the MHD system is degenerate. We use Hodge-type decomposition theorems to obtain strong solutions locally in time or global for small enough initial data, and a linearization principle for the stability of a stationary solution.     

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Let Ω^Ω be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of Ω^Ω, then we write U≈V if there exists a finite subset F of Ω^Ω such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in Ω^Ω is the least cardinality of a subset A of Ω^Ω such that the union of U and A generates Ω^Ω. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω.The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in Ω^Ω where d is the minimum cardinality of a dominating family for N^N. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω^Ω are 0, 1, 2, and d.We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2^ℵ₀.