arXiv subjects

Friedemann Schuricht

Publications and source records attributed to Friedemann Schuricht.

7 recordsLinked to original sources

arXiv2026

Density measures and applications

The paper, that continuous some previous work of Schönherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive measures. We study their basic properties and investigate related integrals. Measures taking only the values 0 and 1 are considered as special case. The results are first applied to weak convergence in $\mathcal{L}^\infty(Ω)$. Then we derive integral representations by means of such measures for several notions of differentiability for integrable functions and we show a kind of mean value theorem for some class of Sobolev functions. Finally we provide a new approach to the generalized Jacobians in the sense of Clarke.

math.AP↗

arXiv2026

Density Measures

The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied to the precise representative of general integrable functions and then they are specialized to functions of bounded variation. Moreover, a new representation of the generalized gradients in the sense of Clarke is given for the finite dimensional case.

math.AP↗

arXiv2022

A theory of traces and the divergence theorem

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function. For integrable vector fields with distributional divergence being a measure, we also obtain Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed in general. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. The Gauss-Green formulas are also available for Sobolev and BV functions. Finally, for any open set the existence of a weak solution of a boundary value problem is shown as application of the trace theory.

math.AP↗

arXiv2019

A nonsmooth nonconvex descent algorithm

The paper presents a new descent algorithm for locally Lipschitz continuous functions $f:X\to\mathbb{R}$. The selection of a descent direction at some iteration point $x$ combines an approximation of the set-valued gradient of $f$ on a suitable neighborhood of $x$ (recently introduced by Mankau & Schuricht) with an Armijo type step control. The algorithm is analytically justified and it is shown that accumulation points of iteration points are critical points of $f$. Finally the algorithm is tested for numerous benchmark problems and the results are compared with simulations found in the literature.

math.NA↗

arXiv2018

Gradients on Sets

For a locally Lipschitz continuous function $f:X\to\mathbb{R}$ the generalized gradient $\partial f(x)$ of Clarke is used to develop some (set-valued) gradient on a set $A\subset X$. Existence, uniqueness and some approximation are considered for optimal descent directions on set $A$. The results serve as basis for nonsmooth numerical descent algorithms that can be found in subsequent papers.

math.OC↗

arXiv2017

Pure Measures, Density Measures and the Dual of L-infinity

Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The main results presented here are a more precise representation of the dual of the space of essentially bounded functions, leading to the notion of pure measures, and the definition and analysis of density measures which constitute a large class of such measures. It is shown that density measures have applications in the context of traces. In particular, new and meaningful examples of pure measures are given on Rn, in contrast to common examples in the literature, which are usually constructed on N.

math.MG↗

arXiv2017

A General Theorem of Gauß Using Pure Measures

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield a Gauß formula for essentially bounded vector fields having divergence measure. The second extends a result of Silhavy [19] on normal traces. In particular, it is shown that a Gauß Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field. All of these measures are shown to have their core on the boundary of the domain of integration.

math.AP↗