Search results for "Bounded"

Notions of Dirichlet problem for functions of least gradient in metric measure spaces

2019

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazón-Rossi–De León, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain. Peer reviewed

Weyl's Theorems and Extensions of Bounded Linear Operators

2012

A bounded operator $T\in L(X)$, $X$ a Banach space, is said to satisfy Weyl's theorem if the set of all spectral points that do not belong to the Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues and having finite multiplicity. In this article we give sufficient conditions for which Weyl's theorem for an extension $\overline T$ of $T$ (respectively, for $T$) entails that Weyl's theorem holds for $T$ (respectively, for $\overline T$).

Fixpunktmengen von halbeinfachen Automorphismen in halbeinfachen Lie-Algebren

1976

Let g be a semisimple Lie algebra over an algebraically closed field of characteristic 0. The set of fixed points of a semisimple inner automorphism of g is a regular reductive subalgebra of maximal rank [1], so it is defined by a subsystem of the root system Φ of g relative to a suitable Cartan subalgebra. The main theorem of the article characterizes the corresponding subsystems of Φ. The second part of the article shows how to compute the fixed point algebras of semisimple outer automorphisms of g. A complete list of all fixed point algebras is then easily obtainable. The results are applied to bounded symmetric domains. References

Multiplizit�ten ?unendlich-ferner? Spitzen

1979

LetX be the quotient of a bounded symmetric domainD by an arithmetically defined subgroup Γ of all analytic automorphisms ofD and letX * be theSatake-compactification ofX. In the present note, the multiplicities of the local rings of the zero-dimensional boundary components ofX * will be computed in a completely elementary manner using reduction-theory in selfadjoint homogeneous cones.

Techniques in the Theory of Local Bifurcations: Cyclicity and Desingularization

1993

A fundamental open question of the bifurcation theory of vector fields in dimension 2 is whether the number of locally bifurcating limit cycles in an analytic unfolding is bounded, or more precisely, whether any limit periodic set has finite cyclicity. In these notes we introduce several techniques for attacking this question: asymptotic expansion of return maps, ideal of coefficients, desingularization of parametrized families. Moreover, because of their practical interest, we present some partial results obtained by these techniques.

Generalized prequojections and bounded maps

1989

Geometric rigidity of conformal matrices

2009

We provide a geometric rigidity estimate a la Friesecke-James-Muller for conformal matrices. Namely, we replace SO(n) by a arbitrary compact subset of conformal matrices, bounded away from 0 and invariant under SO(n), and rigid motions by Mobius transformations.

Sharp estimate on the inner distance in planar domains

2020

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlev\'e length bound $\kappa(E) \le\pi \mathcal{H}^1(E)$ is sharp.

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

2010

Abstract We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is us…

La topologie à l'infini des variétés à géométrie bornée et croissance linéaire

1997

Abstract We study the topology at infinity of a non compact riemannian manifold with bounded geometry and linear growth-type.