Comparison estimates in anisotropic variational problems

A pointwise inequality between the radially decreasing symmetrals of minimizers of (possibly) anisotropic variational problems and the minimizers of suitably symmetrized problems is established. As a consequence, a priori sharp estimates for norms of the relevant minimizers are derived.     

Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.     

Maximizing the L∞ norm of the gradient of solutions to the Poisson equation

We find the maximum of ¦Du _f ¦ _L∞ when u_f is the solution, which vanishes at infinity, of the Poisson equation Δu =f on ? ⁿ in terms of the decreasing rearrangement off. Hence, we derive sharp estimates for ¦Du _f ¦ _L∞ in terms of suitable Lorentz orL ^p norms off. We also solve the problem of maximizing ¦Du _f ^B (0)¦ whenu _f ^B is the solution, vanishing on?B, to the Poisson equation in a ballB centered at 0 and the decreasing rearrangement off is assigned.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Boundary trace inequalities and rearrangements

A new approach to boundary trace inequalities for Sobolev functions is presented, which reduces any trace inequality involving general rearrangement-invariant norms to an equivalent, considerably simpler, one-dimensional inequality for a Hardy-type operator. In particular, improvements of classical boundary trace embeddings and new optimal trace embeddings are derived. This research was partially supported by the Italian research project “Geometric properties of solutions to variational problems” of GNAMPA (INdAM) 2006, by the research project MSM 0021620839 of the Czech Ministry of Education, by grants 201/03/0935, 201/05/2033 and 201/07/0388 of the Grant Agency of the Czech Republic and by the Nečas Center for Mathematical Modeling project no. LC06052 financed by the Czech Ministry of Education.     

Moser-Trudinger trace inequalities

Sharp constants in exponential inequalities involving a general class of measures in domains Ω⊂Rn are exhibited in the limiting case of the Sobolev embedding theorem. A comprehensive approach is presented yielding, as special instances, trace inequalities on ∂Ω, on smooth submanifolds of Ω of arbitrary dimension, and also on fractal subsets of Ω, and recovering, in particular, the classical Moser-Trudinger inequality.     

Measurement of the Anisotropy of the Lamb–Mössbauer Factor in EuBa2Cu3O7

Hyperfine Interactions - The lattice dynamics of high-Tc EuBCO superconductor was studied by means of Mössbauer spectroscopy of the 151Eu ion in samples with oriented grains. Three samples...     

Relaxation phenomena in Cr3+ doped Al2O3−Fe 2 57 O3 systems

Mössbauer absorption spectra have been measured for Cr³⁺ doped Al₂O₃?Fe ₂ ⁵⁷ O₃ systems for different values of the Cr³⁺ concentration at room temperature. The cross relaxation between Fe³⁺ and Cr³⁺ ions, which destroys the paramagnetic hyperfine structure of Fe⁵⁷ observed in undoped Al₂O₃?Fe ₂ ⁵⁷ O₃, is thoroughly studied. The experimental results suggest a new kind of cross-relaxation process involving three spins, i.e. two Fe³⁺.ions and one Cr³⁺. The process, though it is a higher-order one, is highly effective because it is energy-conserving.