On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation

Multilevel frames for sparse tensor product spaces

For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A ^(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.     

The method of mothers for non-overlapping non-matching DDM

In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering, to fix the ideas, the homogeneous Dirichlet problem for in , our variables are i) the approximations of u in each sub-domain (each on its own grid), and ii) an approximation of u on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform). The novelty is in the way to derive, from , the values of each trace of on the boundary of each . We do it by solving an auxiliary problem on each that resembles the mortar method but is more flexible. Under suitable assumptions, quasi-optimal error estimates are proved, uniformly with respect to the number and size of the subdomains. A preliminary version of the method and of its theoretical analysis has been presented in Bertoluzza et al. (15th international conference on domain decomposition methods, 2002).     

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Operator space tensor products of <Emphasis Type="Italic">C</Emphasis>*-algebras

For C*-algebras A and B, the identity map from into A _λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A⊗B of two C*-algebras A and B with centers Z(A) and Z(B) under operator space projective norm is equal to . A characterization of isometric automorphisms of and A _h B is also obtained. Dedicated to Ajit Iqbal Singh on her 65th birthday.     

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

Infinitely many solutions of some nonlinear variational equations

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

Algebraic cycles on products of elliptic curves over <Emphasis Type="Italic">p</Emphasis>-adic fields

We show that for any odd prime p there is a smooth projective threefold W defined over a p-adic field such that the Chow group CH²(W)/ and the Griffiths group Griff²(W)/ are infinite for suitable primes . We further give examples of smooth projective fourfolds over these p-adic fields for which the -torsion subgroup CH³ is infinite.     

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

Linear connections of a framed distribution of hyperplane elements in conformal space

We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions $ \mathcal{M} We study the geometry of affine and normal connections induced by a complete normalization of mutually orthogonal distributions and in conformal space C _n , where is a distribution of hyperplane elements, and is a distribution of line elements. We consider invariant fields of pencils that are parallel with respect to the normal connection along any curve belonging to the distribution . Original Russian Text ? A.M. Matveeva, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 7, pp. 79–84.