Lp-Stability of a Class of Volterra Systems

This paper presents some new and explicit stability results for Volterra systems from two different approaches. The first approach is based on monomial domination of the Volterra system's memoryless output nonlinearity and the second on its Lipschitz-norm. The former yields more widely applicable results, but introduces nonconvexity in the signal spaces for certain parameter values.

Lp-Stability of a Class of Volterra Systems

This paper presents some new and explicit stability results for Volterra systems from two different approaches. The first approach is based on monomial domination of the Volterra system's memoryless output nonlinearity and the second on its Lipschitz-norm. The former yields more widely applicable results, but introduces nonconvexity in the signal spaces for certain parameter values.

On the regularity of the total variation minimizers

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term [Formula: see text] is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension [Formula: see text] and (in case of the global regularity) for convex domains.

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games

We prove that the Burkill-Cesari integral is a value on a subspace ofACand then discuss its continuity with respect to both theBVand the Lipschitz norm. We provide an example of value on a subspace ofACstrictly containingpNAas well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley’s one, on a subspace ofAC∞.

Boundary Regularity for Capillary Surfaces

For solutions of capillarity problems with the boundary contact angle being bounded away from 0 and π and the mean curvature being bounded from above and below, we show the Lipschitz continuity of a solution up to the boundary locally in any neighborhood in which the solution is bounded and ∂Ω is 𝐶2; the Lipschitz norm is determined completely by the upper bound of | cos θ|, together with the lower and upper bounds of 𝐻, the upper bound of the absolute value of the principal curvatures of ∂Ω and the dimension 𝑛.