We present a comprehensive study of interpolation inequalities for periodic functions with zero mean, including the existence of and the asymptotic expansions for the extremals, best constants, various remainder terms, etc. Most attention is paid to the critical (logarithmic) Sobolev inequality in the two-dimensional case, although a number of results concerning the best constants in the algebraic case and different space dimensions are also obtained.

We completely classify non-degenerate surfaces with parallel second fundamental form in three-dimensional Riemannian and Lorentzian reducible spaces.

The oscillation of the nonlinear differential equation

where Φ is an increasing odd homeomorphism, is considered when the weight b is not summable near infinity. We extend previous results, stated for equations with the classical p-Laplacian, by obtaining necessary and sufficient conditions of integral type for the oscillation. The role of the boundedness of Im Φ [Dom Φ] is analysed in detail. Our results includes the case Φ* ◦ F linear near zero or near infinity, where Φ* is the inverse of Φ. Several examples, concerning the curvature or relativity operator, illustrate our results.

We study the homogenization of the linearized system of elasticity standing for the equilibrium equation of a highly periodic heterogeneous elastic medium submitted to small deformations and made of two different materials: a very rigid material located in a set Fε (ε being the size of the period of the medium) of vertical fibres surrounded by a soft elastic material localized in the set Mε. The ratio between the coefficients of the elasticity tensor of the two materials is assumed to be 1/ε4. We deal with the general case without any special assumption, such as isotropy, on the material.

We study the semilinear elliptic inequality –Δu ≥ φ(δK (x))f(u) in ℝN / K, where φ, f are positive and non-increasing continuous functions. Here K ⊂ ℝN (N ≥ 3) is a compact set with finitely many components, each of which is either the closure of a C2 domain or an isolated point, and δK (x) = dist(x, ∂K). We obtain optimal conditions in terms of φ and f for the existence of C2-positive solutions. Under these conditions we prove the existence of a minimal solution and we investigate its behaviour around ∂K as well as the removability of the (possible) isolated singularities.

A new construction of derived equivalences is given, which relates different endomorphism rings and, more generally, cohomological endomorphism rings, including higher extensions, of objects in triangulated categories. These objects need to be connected by certain universal maps that are cohomological approximations and that exist in very general circumstances. The construction turns out to be applicable to a wide variety of situations, covering finite-dimensional algebras as well as certain infinite-dimensional algebras, Frobenius categories and n-Calabi–Yau categories.

We study the Emden–Fowler equation whose coefficient function is even in the interval (—1, 1), negative near t = 0 and positive near t = ±1. Then we prove that a least energy solution is not even. Therefore, the equation has an even positive solution and a non-even positive solution.

We study the Cauchy problem for the degenerate and uniformly parabolic equations with gradient term. The local existence, global existence and non-existence of solutions are obtained. In the case of global solvability, we get the exact estimates of a solution. In particular, we obtain the global existence of solutions in the limiting case.