Differentiability properties for a class of non-convex functions

Closed sets K ⊂ satisfying an external sphere condition with uniform radius (called ϕ-convexity or proximal smoothness) are considered. It is shown that for -a.e. x ∊ ∂K the proximal normal cone to K at x has dimension one. Moreover if K is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to ∂ K and the unit proximal normal equals -a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : with ϕ-convex epigraph are shown, among other results, to be locally BV and twice -a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where f is not differentiable is studied. Finally we show that for -a.e. x there exists δ (x) > 0 such that f is semiconvex on B(x,δ(x)). We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used. Work partially supported by M.I.U.R., project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations.”     

Differentiability and semismoothness properties of integral functions and their applications

In this paper we study differentiability and semismoothness properties of functions defined as integrals of parameterized functions. We also discuss applications of the developed theory to the problems of shape-preserving interpolation, option pricing and semi-infinite programming.Supported by the Hong Kong Research Grant Council under grant PolyU 5296/02P.     

Certain geometric properties of normalized Bessel functions

In this paper, we give a set of sufficient conditions for the normalized form of the generalized Bessel function to be univalent in the open unit disk, and further we obtain certain inequalities containing normalized Bessel functions.     

Bessel capacities and rectangular differentiation in Besov spaces

We consider the differentiation of integrals of functions in Besov spaces with respect to the basis of arbitrarily oriented rectangular parallelepipeds in Rn. We study almost everywhere convergence with respect to Bessel capacities. These outer measures are more sensitive than n-dimensional Lebesgue measure, and therefore we improve the positive results in [H. Aimar, L. Forzani, V. Naibo, Rectangular differentiation of integrals of Besov functions, Math. Res. Lett. 9 (2-3) (2002) 173-189].     

Optimal Besov Differentiability for Entropy Solutions of the Eikonal Equation

In this paper we study the eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production, and the validity of a kinetic formulation. © 2019 Wiley Periodicals, Inc.     

Continuity of Bessel potentials

It is shown that certain capacities associated with potentials of functions in Lebesgue classes are non-increasing under orthogonal projection of sets. This inequality is then used to discuss continuity of traces of potentials on subspaces of possibly low dimension. The case of principal interest is the Bessel potential. This research was partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant No. 883-67.     

Differentiability of multivariate refinable functions and factorization

The paper develops a necessary condition for the regularity of a multivariate refinable function in terms of a factorization property of the associated subdivision mask. The extension to arbitrary isotropic dilation matrices necessitates the introduction of the concepts of restricted and renormalized convergence of a subdivision scheme as well as the notion of subconvergence, i.e., the convergence of only a subsequence of the iterations of the subdivision scheme. Since, in addition, factorization methods pass even from scalar to matrix valued refinable functions, those results have to be formulated in terms of matrix refinable functions or vector subdivision schemes, respectively, in order to be suitable for iterated application. Moreover, it is shown for a particular case that the the condition is not only a necessary but also a sufficient one. Dedicated to Charles A. Micchelli, a unique person, friend, mathematician and collaborator, on the occasion of his sixtieth birthday Mathematics subject classifications (2000) 65T60, 65D99.     

Integrals of Bessel functions

We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians.     

On higher order Bessel potentials

The solutions of the equation in , where are investigated, Bessel potentials of higher order are defined, and recurrence relations between these solutions and these Bessel potentials are obtained. It is also proved that these solutions and the solutions of , under certain conditions, are identical. Received: 6 November 2002     

Generalization of Bessel and hypergeometric functions

A set of new functions, which generalize the generalized hypergeometric functions, is presented. A preliminary analysis is given, including generating functions, differential equations, series and integral representations.     

Bounds for Bessel functions

We establish lower and upper bounds for the Bessel functionJ _v (x) and the modified Bessel functionI _v(x) of the first kind. Our chief tool is the differential equation satisfied by these functions.     

Differentiability of type-2 fuzzy number-valued functions

In this paper, we define a differentiability of the type-2 fuzzy number-valued functions. The definition is based on type-2 Hukuhara difference which is defined in the paper as well. The related theorem of the differentiability of the type-2 fuzzy number-valued functions is derived. In addition, a parametric closed form of the perfect triangular quasi type-2 fuzzy numbers is introduced, and finally, the applicability and an approach to solving type-2 fuzzy differential equations are illustrated using some examples and cases.