On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

On the power of standard information for <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> approximation in the randomized setting

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L _∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n ⁻¹ when n function values are used.     

On the structural properties of the weight space L p(x),ω for 0 < p(x) < 1

The main purpose of this paper is to study the weight space L _p(x),ω for 0 < p(x) < 1 as well as the topology of this space. Embeddings between different Lebesgue spaces with variable exponent of summability are established. In particular, it is proved that the set of all linear continuous functionals over L _p(x),ω for 0 < p(x) < 1 consists only of the zero functional.     

On the Cohen-Macaulay property for monomial curves in ℙ K 3

The arithmetical Cohen-Macaulay property for monomial curves in _K ³ with generic zero was shown in [2] to be true forn ₃>(n ₂–1)(n ₂–n ₁),n ₃>n ₂, (n ₁,n ₂,n ₃)=1. Here we establish necessary and sufficient arithmetic conditions in terms ofn ₁,n ₂,n ₃ in order for the indicated curves not to be arithmetically Cohen-Macaulay.     

On the Jordan–Hölder property for geometric derived categories

We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X   do not satisfy the Jordan–Hölder property. More precisely, there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. Assuming the Noetherian property for semiorthogonal decompositions, one can define, following Kuznetsov, the Clemens–Griffiths component CG(D)$\mathrm{CG}(\mathfrak{D})$ for each fixed maximal decomposition D$\mathfrak{D}$. We then show that D^b(X)${\mathrm{D}}^b(X)$ has two different maximal decompositions for which the Clemens–Griffiths components differ. Moreover, we produce examples of rational fourfolds whose derived categories also violate the Jordan–Hölder property.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).     

On Asymptotically Isometric Copies of l~β(0 < β < 1)

We get the characterizations of the family of all nonnegative,subadditive,β-absolutely homogeneous and continuous functionals defined on X,when the β-normed space X contains an asymptotically isometric copy of lβ.Moreover,it is proved that if a closed bounded β-convex subset K of a β-normed space contains an asymptotically isometric lβ-basis,then K contains a closed β-convex subset C which fails the fixed point property.     

On estimates for the Fourier-Bessel integral transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Subscript>+</Subscript>)

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L ₂(?₊) as applied to some classes of functions characterized by a generalized modulus of continuity.     

On the boundedness of first derivatives for solutions to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of the regularity of solutions to the Neumann problem on convex polyhedra are given. Bibliography: 27 titles. Dedicated to Nina Uraltseva with affection on the occasion of her birthday Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 105–112.     

Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609–626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431–457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115–142, 2004) in the L ² region. We prove the convergence of the law of the environment seen by the particle.     

On n-term approximations with respect to frames bounded in L p (0, 1), 2 < p < ∞

In this paper, best canonical n-term approximations in the norm of the spaces L ²(0, 1) of the family I of characteristic functions of intervals are studied.     

On the definition of the Monge-Ampère operator in ℂ<Superscript>2</Superscript>

We show that if u is a plurisubharmonic function defined on an open subset of ² then the Monge-Ampère measure (dd^cu)² can be well defined if and only if u belongs to the Sobolev space W^1,2_loc().Partially supported by KBN Grant #2 P03A 028 19     

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

On the asymptotic boundedness of the energy of solutions of the wave equation u tt ? a(t)Δu?=?0

Applying a class of quadratic forms introduced by Manfrin (Port. Math. 65(4):447?C484, 2008), we study the asymptotic behavior, as t ?? +??, of the energy of the solutions to the Cauchy problem for wave equations with time dependent propagation speed.     

On the asymptotic behavior of radial entire solutions for the equation (−Δ)3u = up in Rn

Our main task in this note is to prove the existence and to classify the exact growth at infinity of radial positive $C^6$-solutions of ${(?\mathrm{\Delta })}^{3}u=u^p$ in ${\mathbf{R}}^n$, where $n?15$ and p is bounded from below by the sixth-order Joseph–Lundgren exponent. Following the main work of Winkler, we introduce the sub- and super-solution method and comparison principle to conclude the asymptotic behavior of solutions.