On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains

Results on the convergence of minimizers and minimum values of integral and more general functionals J_s: W^1,p(Ω_s) → ? on the sets U_s(h_s) = {v ∈ W^1,p(Ω_s): h_s(v) ≤ 0 a.e. in Ω_s}, where p > 1, {Ω_s} is a sequence of domains contained in a bounded domain Ω of ?ⁿ (n > 2), and {h_s} is a sequence of functions on ?, are announced.     

Estimates for the Torsion Function and Sobolev Constants

Let Ω be an open set in Euclidean space, and let u : Ω → ?^?+? be the expected lifetime of Brownian motion in Ω. It is shown that if u?∈?L ^p (Ω) for some p?∈?[1, ?∞?) then (i) u?∈?L ^q (Ω) for all q?∈?[p,?∞?], and (ii) \({trace}\left(e^{t\Delta_{\Omega}}\right)<\infty\) for all t?>?0, where ??Δ_Ω is the Dirichlet Laplacian acting in L ²(Ω). Pointwise bounds are obtained for u in terms of the first Dirichlet eigenfunction for Ω, assuming that the spectrum of ??Δ_Ω is discrete. It is shown that if Ω is open, bounded and connected in the plane and \(\partial\Omega\) has an interior wedge with opening angle α at vertex v then the first Dirichlet eigenfunction and u are comparable near v if and only if α?≥?π/2. Two sided estimates are obtained for the Sobolev constant

$ C_p(\Omega):= \inf\left\{\Vert \nabla u \Vert_2^2: u \in C_0^{\infty}(\Omega),\ \Vert u\Vert_p = 1\right\}, $

where 0?p?Ω satisfies a strong Hardy inequality, and the distance to the boundary function δ?∈?L ^2p/(2???p)(Ω).

     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

Kazhdan-Lusztig cells in some weighted Coxeter groups

Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.     

Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion

We study the problem of minimizing the functional \(I(\phi ) = \int\limits_\Omega {W(x,D\phi )dx}\) on a new class of mappings. We relax summability conditions for admissible deformations to φ ∈ W _n ¹ (Ω) and growth conditions on the integrand W(x, F). To compensate for that, we require the condition \(\frac{{\left| {D\phi (x)} \right|^n }} {{J(x,\phi )}} \leqslant M(x) \in L_s (\Omega )\), s > n ? 1, on the characteristic of distortion. On assuming that the integrand W(x, F) is polyconvex and coercive, we obtain an existence theorem for the problem of minimizing the functional I(φ) on a new family of admissible deformations A.     

On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field

In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F _c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б _K and a constructive Hilbert pairing {·, ·} _c : K ₂(K) × F _c (M) → <ξ> _c , where <ξ> _c is the module of roots of [p] _c (pth degree isogeny of F _c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K ₂(L) → K ₂(K) is considered. Its images are called norms in K ₂(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} _c : = 0 ? x is a norm in K ₂(K([p] _c ^-1 (β))), where [p] _c ^-1 (β) are the roots of the equation [p] _c = β, is checked constructively.     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

On vanishing coefficients of algebraic power series over fields of positive characteristic

Let K be a field of characteristic p>0 and let f(t ₁,…,t _d ) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t ₁,…,t _d ). We prove a generalization of both Derksen’s recent analogue of the Skolem–Mahler–Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices (n ₁,…,n _d )∈? ^d for which the coefficient of \(t_{1}^{n_{1}}\cdots t_{d}^{n_{d}}\) in f(t ₁,…,t _d ) is zero is a p-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to S-unit equations and more generally to the Mordell–Lang Theorem over fields of positive characteristic.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λ_j,φ_j} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u ³, u ², (u - p), u ²(u ² - p), are completed and some conjectures are presented.     

Ergodic theorems with arithmetical weights

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L ^p (ν), p > 1, the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$

exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J _s (n) = Σ _d|n d ^s μ(n/d), s > 0.

     

K?hler Manifolds with Almost Non-negative Ricci Curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> ?∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

Almost Empty Monochromatic Quadrilaterals in Planar Point Sets

For positive integers c, s ≥ 1, r ≥ 3, let W _r (c, s) be the least integer such that if a set of at least W _r (c, s) points in the plane, no three of which are collinear, is colored with c colors, then this set contains a monochromatic r-gon with at most s interior points. As is known, if r = 3, then W _r (c, s)=W _r (c, s). In this paper, we extend these results to the case r = 4. We prove that W4(2, 1) = 11, W4(3, 2) ≤ 120, and the least integer μ4(c) such that W4(c, μ4(c)) < ∞ is bounded by \(\left\lfloor {\frac{{c - 1}}{2}} \right\rfloor \cdot 2 \leqslant \mu 4\left( c \right) \leqslant 2c - 3\),where c ≥ 2.     

Weak convergence of the past and future of Brownian motion given the present

In this paper, we show that for t > 0, the joint distribution of the past {W _t?s : 0 ≤ s ≤ t} and the future {W _{t + s} :s ≥ 0} of a d-dimensional standard Brownian motion (W _s ), conditioned on {W _t ∈ U}, where U is a bounded open set in ? ^d , converges weakly in C[0,∞)×C[0,∞) as t→∞. The limiting distribution is that of a pair of coupled processes Y + B ¹,Y + B ² where Y,B ¹,B ² are independent, Y is uniformly distributed on U and B ¹,B ² are standard d-dimensional Brownian motions. Let σ _t ,d _t be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {W _t ∈U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B ¹ and Y + B ² respectively.