We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ R, g ∈ C(R) is a superlinear and subcritical function, and Ω is a domain in R2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Oscillation and related results are given for the problem under separated end conditions, assuming 1/p, q and r ∈ L1(I). Attention focuses on the two cases (i) p > 0 with r indefinite, and (ii) p indefinite with r > 0.

We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ ℝ, g ∈ C(ℝ) is a superlinear and subcritical function, and Ω is a domain in ℝ2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

Let f : G → H be a fixed homomorphism and p′ : G * H → G and p″ : G * H → H the two projections of the free product. Then a co-action relative to f is a homomorphism s : G → G * H such that p′s = id and p″s = f. We study this notion and investigate the following questions. What restrictions does s place on the structure of the group G? What form does s take in special cases? When does s induce a co-multiplication on H? What is the relation between associativity of s and associativity of the induced co-multiplication m on H? What are the properties of the operation of Hom(H, B) on Hom(G, B) induced by s : G → G * H? In addition, we give several diverse examples of co-actions in the last section.

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxation Assume there exists an equilibrium curve A(u), such that r(u,A(u)) = 0. Under some assumptions on σ and r, we prove the existence of global (in time) solutions of bounded variation, uε, υε, for ε > 0 fixed.

As ε → 0, we prove the convergence of a subsequence of uε, υε to some u, υ that satisfy the equilibrium equations

Let L : dom L ⊂ L2(Ω) → L2(Ω) be a self-adjoint operator, Ω being open and bounded in RN. We give a description of the Fučík spectrum of L away from the essential spectrum. Let λ be a point in the discrete spectrum of L; provided that some non-degeneracy conditions are satisfied, we prove that the Fučík spectrum consists locally of a finite number of curves crossing at (λ, λ). Each of these curves can be associated to a critical point of the function H : x ↦ 〈|x|,x〉L2 restricted to the unit sphere in ker(L – λI). The corresponding critical values determine the slopes of these curves. We also give global results describing the Fučík spectrum, and existence results for semilinear equations, by performing degree computations between the Fučík curves.

We define the deformation multiplicity of a map germ f: (Cn, 0) → (Cp, 0) with respect to a Boardman symbol i of codimension less than or equal to n and establish a geometrical interpretation of this number in terms of the set of Σi points that appear in a generic deformation of f. Moreover, this number is equal to the algebraic multiplicity of f with respect to i when the corresponding associated ring is Cohen-Macaulay. Finally, we study how algebraic multiplicity behaves with weighted homogeneous map germs.

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

We prove the existence of a very singular solution to when 1 < p < (N + 2)/(N + 1).

We introduce what are called regular materials for which, by definition, the corresponding solution of the classical periodic homogenization problem remains bounded in . We give examples of two types of such materials depending on whether the coefficients representing them belong to W1,∞ or not. A complete characterization is obtained in the former case.

Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension. The stable decomposition is related to a description of the integral cohomology ring.

In this paper we consider the semilinear elliptic problem in a bounded domain Ω ⊆ Rn, where μ ≥ 0, 0 ≤ α ≤ 2, 2α* := 2(n − α)/(n − 2), f : Ω → R+ is measurable, f > 0 a.e, having a lower-order singularity than |x|-2 at the origin, and g : R → R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions Ip for which the embedding is compact. When p = 2, α = 2, f ∈ I2 and 0 ≤ μ < (½(n − 2))2, we prove that the linear problem has -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n − 2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 ≤ α < 2.

Linked equations are studied on [0,1] subject to boundary conditions of the form Results are given on existence, location, asymptotics and perturbation of the eigenvalues λj and oscillation of the eigenfunctions yi.

We consider the maximal operator defined on the real line by which is related to the Cesàro convergence of the singular integrals. We characterize the weights w for which Mα is of weak type, strong type and restricted weak type (p, p) with respect to the measure w(x) dx.

We consider how the shape of a domain affects the number of positive solutions of a nonlinear elliptic problem. In fact, we show that if a bounded domain Ω is sufficiently close to a union of disjoint bounded domains Ω1,…, Ωm, the number of positive solutions of a nonlinear elliptic problem on Ω is at least 2m −1.

In this paper we construct a compact set K of zero Hausdorff dimension that satisfies certain ‘arithmetic-type’ thickness properties. The concept of ‘arithmetic thickness’ has its origins in applications to harmonic analysis, introduced in a paper by Lebedev and Olevskiĭ. For example, there are no spectral sets whose ‘essential boundary’ can contain the above set K.

Lower semicontinuity and relaxation results in BV are obtained for multiple integralswhere the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.