We prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potential

and study the stability and oscillation properties of these solutions. We also show that if the equation on ℝN has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝN a family of the exterior Dirichlet problems in ℝN \ K admits a continuum of stable slow-decay infinite-energy solutions.

The Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

We answer the following question posed by Lechuga: given a simply connected space X with both H* (X; ℚ) and π*(X) ⊗ ℚ being finite dimensional, what is the computational complexity of an algorithm computing the cup length and the rational Lusternik—Schnirelmann category of X?

Basically, by a reduction from the decision problem of whether a given graph is k-colourable for k ≥ 3, we show that even stricter versions of the problems above are NP-hard.

In the present paper we classify del Pezzo surfaces with log terminal singularities admitting an action of a finite simple group.

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite-dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-) Riemannian geodesics and constant mean curvature hypersurfaces.

The goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kähler.

We introduce a ‘double’ version of Γ-convergence, which we have named ‘double Γ-convergence’, and apply it to obtain the Γ-limit of double-perturbed energy functionals as p → 1 and p → +∞, respectively. The limit of (p, q)-type capacity as p → 1 and p → +∞, respectively, is also obtained in this manner.

Consider the equation

where ƒ ∈ Lp(ℝ), p ∈ (1, ∞) and

By a solution of (*), we mean any function y absolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the space Lp(ℝ), i.e.

(1) for any function , there exists a unique solution y ∈ Lp(ℝ) of (*);

(2) there exists an absolute constant c1(p) > 0 such that the solution y ∈ Lp(ℝ) of (*) satisfies the inequality

We study the following problem on the strengthening estimate (**). Let a non-negative function be given. We have to find minimal additional restrictions for θ under which the following inequality holds:

Here, y is a solution of (*) from the class Lp(ℝ), and c2 (p) is an absolute positive constant.

We classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic 0. We consider the commuting variety of the nilradical of the Lie algebra of a Borel subgroup B of G. In case B acts on with only a finite number of orbits, we verify that is equidimensional and that the irreducible components are in correspondence with the distinguishedB-orbits in . We observe that in general is not equidimensional, and determine the irreducible components of in the minimal cases where there are infinitely many B-orbits in .

For the third-order differential equation y′″ = ƒ(t, y, y′, y″), where , questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

We prove continuity in generalized parabolic Morrey spaces of sublinear operators generated by the parabolic Calderón—Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. As a consequence, we obtain a global -regularity result for the Cauchy—Dirichlet problem for linear uniformly parabolic equations with vanishing mean oscillation (VMO) coefficients.

A long-standing conjecture of Faith in ring theory states that a left self-injective semi-primary ring A is necessarily a quasi-Frobenius ring. We propose a new method for approaching this conjecture, and prove it under some mild conditions; we show that if the simple A-modules are at most countably generated over a subring of the centre of A, then the conjecture holds. Also, the conjecture holds for algebras A over sufficiently large fields, i.e. if the cardinality of is larger than the dimension of the simple A-modules (or of A/Jac(A)). This effectively proves the conjecture in many situations, and we obtain several previously known results on this problem as a consequence.

We show results on Lp-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lamé system. Here, we rely on resolvent estimates recently established by Mitrea and Monniaux.