The relation between quasi-convexity and k-quasi-convexity, k ≥ 2, is investigated. It is shown that every smooth strictly k-quasi-convex integrand with p-growth at infinity, p > 1, is the restriction to kth-order symmetric tensors of a quasi-convex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for kth-order variational problems are deduced as corollaries of well-known first-order theorems. This generalizes a previous work by Dal Maso et al., in which the case where k = 2 was treated.

We consider the Lp norms of sums of characteristic functions of affine subspaces of a vector space V over a finite field under certain restrictions on p, dim V and the dimensions of the subspaces involved. We investigate the conditions under which these norms are increased when the affine subspaces are replaced by their parallel translates passing through 0. Applications to extremal configurations for Kakeya maximal-type inequalities are given and open questions are raised.

This paper deals with the non-negative boundary blow-up solutions of the equation ∆u = b(x)up + c(x)uσ|∇u|q in Ω ⊂ ℝ,N, where b(x), c(x) ∈ Cγ (Ω,ℝ+) for some 0 < γ < 1 and can be vanishing or singular on the boundary, and p, σ and q are non-negative constants. The existence and asymptotic behaviour of such a solution near the boundary are investigated, and we show how the nonlinear gradient term affects the results. As a consequence of the asymptotic behaviour, we also show the uniqueness result.

We study, via an asymptotic analysis, an elliptic eigenvalue problem in a 1D–1D multidomain and in a 1D–2D multidomain filled with anisotropic material. The corresponding isotropic cases were considered in a previous work by Gaudiello and Sili.

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

We consider a Cauchy problem for a semilinear heat equation

with p > 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),

where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero of

then u is called a backward self-similar solution blowing up at t = T. We show that, for all p > 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

It is well known that the loss of the coerciveness for governing monotone nonlinearities in evolution equations/inclusions can lead to the problem having no solution for given data, and the rule for choosing appropriate data has to be prescribed. Allowing constitutive functions in the evolution relations for elasto-/visco-plastic models of monotone type to be non-coercive, we first give a new (relaxed) meaning to the solvability of the systems of equations under consideration and then we define criteria for choosing admissible data, which guarantees the solvability in the defined sense. Realizing this strategy, a slight extension of the well-developed monotone-operator method to our needs is performed. The theory is applied to some well-known models in elasto-/visco-plasticity. The relations between the standard notion of the solvability and defined one are investigated.

An initial–boundary-value problem for a class of wave equations with nonlinear damping and source terms in a bounded domain is considered. We establish the non-existence result of global solutions with the initial energy controlled above by a critical value via the method introduced in a work by Autuori et al. in 2010. This improves the 2009 result of Liu and Wang.

We study domain walls that are topological solitons in one dimension. We present an existence theory for the solutions of the basic governing equations of some extended geometrically constrained domain-wall models. When the cross-section and potential density are both even, we establish the existence of an odd domain-wall solution realizing the phase-transition process between two adjacent domain phases. When the cross-section satisfies a certain integrability condition, we prove that a domain-wall solution always exists that links two arbitrarily designated domain phases.