A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

We prove partial regularity for local minimizers of quasiconvex integrals of the form I(v) = ∫ΩF(Dv(x))dx, where the integrand f(ξ) has sub-quadratic growth, i.e. |F(ξ)| ≤ L(1 + |ξ|p), with 1 < p < 2. A function u ∈ W1,p(Ω;RN) is a W1,q(Ω;RN) local minimizer of I(v) if there exists a δ > 0 such that I(u) ≤ I(v) whenever v and ‖Dv − Du‖q ≤ δ.

We show that, if f : M2×2 → R is rank-one convex on the hyperboloid is the set of 2 × 2 real symmetric matrices, then f can be approximated by quasi-convex functions on M2×2 uniformly on compact subsets of . Equivalently, every gradient Young measure supported on a compact subset of is a laminate.

For certain Lie subalgebras L of associative algebras Q, we characterize the multi-additive skew-symmetric maps f : Ln → Q, which are covariant under the action of L.

In this work we consider the non-autonomous problem Δu = a(x)um in the unit ball B ⊂ RN, with the boundary condition u|∂B = +∞, and m > 0. Assuming that a is a continuous radial function with a(x) ˜ C0 dist(x, ∂B)−γ as dist(x, ∂B) → 0, for some C0 > 0, γ > 0, we completely determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of m and γ. The case 0 < m ≤ 1, as well as estimates for solutions to the linear problem m = 1, are a significant part of our results.

In a planar domain, to each function w ∈ W2,2 satisfying a suitable condition we associate a non-divergence elliptic differential operator 𝔏 such that 𝔏w = 0. For a given converging sequence {wk}, we study G-convergence of the corresponding sequence {𝔏k} of operators.

We study the unique solvability of density-dependent incompressible Navier-Stokes equations in the whole space RN (N ≥ 2). The celebrated results by Fujita and Kato devoted to the constant density case are generalized to the case when the initial density is close to a constant: we find local well posedness for large initial velocity, and global well posedness for initial velocity small with respect to the viscosity. Our functional setting is very close to the one used by Fujita and Kato.

Let D be a domain in C and let f be a transcendental meromorphic function on C such that C* \f(C) = ∅, {∞} or {α, β}, where α and β are two distinct values in C* = C ∪ {∞}. Then the family of functions g that are analytic on D and such that f ∘ g has no fixpoints on D is normal.

A line congruence is a two-parameter family of lines in R3. In this paper we study singularities of line congruences. We show that generic singularities of general line congruences are the same as those of stable mappings between three-dimensional manifolds. Moreover, we also study singularities of normal congruences and equiaffine normal congruences from the viewpoint of the theory of Lagrangian singularities.

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form If(u) = ∫Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.

By applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

For positive rational integers λ, we study the Hecke L-series attached to elliptic curves y2 = x3 − 2433Dλ over the quadratic field Q(√−3) and obtain various bounds of p(= 2, 3)-adic valuations of their values at s = 1 according to the cases of D and λ. In particular, for the case of even λ, we obtain a criterion of reaching the bounds of 3-adic valuations. From this, combining with the work of Coates and Wiles and Rubin, we obtain some results about the conjecture of Birch and Swinnerton-Dyer of these curves.

By combining a technique inspired to the theory of sublinear elliptic equations with the Emden-Fowler inversion technique of Atkinson and Peletier, we obtain uniqueness of positive solutions of the following equation where B ⊂ Rn is the ball of radius one, λ > 0 and 1 < ϑ ≤ 2.