Wavelet and Fourier regularization methods are effective for the nonparametric regression problem. We prove that the loss function evaluated for the regularization parameter chosen through GCV or Mallows criteria is asymptotically equivalent in probability to its minimum over the regularization parameter. © 2001 Elsevier Science B.V.

We focus on three different convexity principles for local and nonlocal variational integrals. We prove various generalizations of them, as well as their equivalences. Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given. We also prove a measure-theoretic minimum principle for nonlocal and non- linear positive eigenfunctions.

A highly nonlinear eigenvalue problem is studied in a Sobolev space with variable exponent. The Euler-Lagrange equation for the minimization of a Rayleigh quotient of two Luxemburg norms is derived. The asymptotic case with a "variable infinity" is treated. Local uniqueness is proved for the viscosity solutions.

We investigate the switching dynamics of multistable nematic liquid crystal devices. In particular, we identify a remarkably simple two-dimensional device which exploits hybrid alignment at the surfaces to yield a bistable response. We also consider a three-dimensional tristable nematic device with patterned anchoring, recently implemented in practice, and discuss how the director and disclination patterns change during switching.

In this paper, we numerically investigate the breakup dynamics of droplets in an emulsion flowing in a tapered microchannel with a narrow constriction. The mesoscale approach for multicomponent fluids with near contact interactions is shown to capture the deformation and breakup dynamics of droplets interacting within the constriction, in agreement with experimental evidence. In addition, it permits us to investigate in detail the hydrodynamic phenomena occurring during breakup stages.

Given an open set ?, we consider the problem of providing sharp lower bounds for ? (?), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong-Krahn-Szego inequality, asserting that the disjoint unions of two equal balls minimize ? among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ? are considered as well. © 2012 Springer-Verlag Berlin Heidelberg.

We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue ?, providing various equivalent characterizations for it. We also prove an upper bound for ? in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock-Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian.

Given a bounded open set in [Formula presented], [Formula presented], and a sequence [Formula presented] of compact sets converging to an [Formula presented]-dimensional manifold [Formula presented], we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on [Formula presented], with Neumann boundary conditions on [Formula presented].