The aim of this paper is to determine in a natural manner the subspace of the space of Hilbert modular newforms of level n which correspond to eigenforms of an appropriate quaternion algebra, in the sense of having the same eigenvalues with respect to the corresponding Hecke operators. This study may be seen as a particular case of the Jacquet–Langlands correspondence.

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Aε = −div(aε (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to an operator Ahom = −div(ahom(x, ∇u)). We show that any limit point λ of a sequence of eigenvalues λε is an eigenvalue of the limit operator Ahom, where λε is an eigenvalue corresponding to the operator Aε. We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.

The properties of convex representatives of polyconvex functions W : R2×2 → R are studied. The structure of the largest convex representative ϕW : R5 → R is given, and this is used to derive a representation formula for ϕW when W is C1 and satisfies a mild growth condition. The regularity of ϕW is discussed when W is C1; in particular, when W is C1, grows super-quadratically and is strictly polyconvex, it is shown that W has a C1 convex representative.

Using a standard theory of differential operators in Lebesgue spaces, we re-prove and generalize some results of Chernyavskaya and Shuster, giving (mostly sufficient) conditions that minimal operators determined by expressions of the form −(ry′)′ + qy with domain and range in possibly different Lp spaces on intervals with at least one singular endpoint have bounded inverses.

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.

We prove that several definitions of integral multilinear mappings given in the literature are equivalent. We show that what we call S-factorizable multilinear mappings are integral, but that the converse is not true (contrary to earlier claims). Using S-factorizable polynomials we give characterizations of L∞-spaces, Asplund spaces, spaces not containing ℓ1 and spaces with the compact range property. Some of these characterizations seem to be new even for linear operators.

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.

The existence of solutions of the Dirichlet problems for the prescribed mean-curvature equation on some unbounded domains in Rn(n ≥ 2) is proved. The results are proved using a modified version of the Perron method, where a subsolution is a solution to the minimal surface equation, while a supersolution is not constructed; instead, the role played by a supersolution is replaced by the estimates on the uniform bounds on the liftings of subfunctions on compact sets.

A three spectral inverse Sturm–Liouville problem is considered arising in the theory of vibrating strings. It is shown that in some special cases the solution of this problem is unique.

Lp–Lq decay estimates of solutions to Cauchy problems of linear thermoelastic systems with second sound in three spatial variables are studied. By carefully analysis of the different effects of higher, middle and lower frequencies in phase space, the asymptotic behaviour of characteristic roots of coefficient matrices is obtained. Then, with the help of the information on characteristic roots and the interpolation theorem, a decay estimate of parabolic type for the coupled system of the potential part of displacement, temperature and heat flux is obtained. Finally, Lp–Lq decay estimates of hyperbolic type for the original thermoelastic systems with second sound are obtained.

We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.