We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.

We study the weak stability index of an immersion ϕ: M → Sn+1 (1) ⊂ Rn+2 of an n-dimensional compact Riemannian manifold. We prove that the weak stability index of a compact hypersurface M with constant scalar curvature in Sn+1 (1), which is not totally umbilical, is greater than or equal to n + 2 if the mean curvature H1 and H3 are constant, and that the equality holds if and only if M is . As an application, we show that the weak stability index of an n-dimensional compact hypersurface with constant scalar curvature in Sn+1 (1), which is neither totally umbilical nor a Clifford hypersurface, is greater than or equal to 2n + 4 if the mean curvature H1 and H3 are constant.

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.

In a recent article Aldaz proved that the weak L1 bounds for the centred maximal operator associated to finite radial measures cannot be taken independently with respect to the dimension. We show that the same result holds for the Lp bounds of such measures with decreasing densities, at least for small p near to one. We also give some concrete examples, including the Gaussian measure, where better estimates with respect to the general case are obtained.

The Mullins–Sekerka sharp-interface model for phase transitions interpolates between attachment-limited and diffusion-limited kinetics if kinetic drag is included in the Gibbs–Thomson interface condition. Heuristics suggest that the typical length-scale of patterns may exhibit a crossover in coarsening rate from l(t) ˜ t1/2 at short times to l(t) ˜ t1/3 at long times. We establish rigorous, universal one-sided bounds on energy decay that partially justify this understanding in the monopole approximation and in the associated Lifshitz–Slyozov–Wagner mean-field model. Numerical simulations for the Lifshitz–Slyozov–Wagner model illustrate the crossover behaviour. The proofs are based on a method for estimating coarsening rates introduced by Kohn and Otto, and make use of a gradient-flow structure that the monopole approximation inherits from the Mullins–Sekerka model by restricting particle geometry to spheres.

We study the problem of the homogenization of Dirichlet eigenvalue problems for the p-Laplace operator in a sequence of perforated domains with fine-grained boundary. Using the asymptotic expansion method, we derive the homogenized problem for the new equation with an additional term of capacity type. Moreover, we show that a sequence of eigenvalues for the problem in perforated domains converges to the corresponding critical levels of the homogenized problem.

We establish some eigenvalue criteria for the existence of non-trivial T-periodic solutions of a class of first-order functional differential equations with a nonlinearity f(x). The nonlinear term f(x) can take negative values and may be unbounded from below. Conditions are determined by the relationship between the behaviour of the quotient f(x)/x for x near 0 and ±∞ and the smallest positive characteristic value of an associated linear integral operator. This linear operator plays a key role in the proofs of the results and its construction is non-trivial. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:

where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , , 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on , that change sign on Ω.

We introduce the concept of quasi-completely continuous multilinear operators and use this concept to characterize, for a wide class of Banach spaces X1, …, Xk, the multilinear operators T : X1 × … × Xk → X with an X-valued Aron–Berner extension.

The Grunsky map is known to be holomorphic on the universal Teichmüller space. In this paper it is proved that the Grunsky map induces a holomorphic map on the asymptotic Teichmüller space. The Caratháodory and Kobayashi metrics on the asymptotic Teichmüller space are studied as applications.