Search results for "PSC"

Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods

2018

We consider inequalities of the Poincare–Steklov type for subspaces of \(H^1\)-functions defined in a bounded domain \(\varOmega \in \mathbb {R}^d\) with Lipschitz boundary \(\partial \varOmega \). For scalar valued functions, the subspaces are defined by zero mean condition on \(\partial \varOmega \) or on a part of \(\partial \varOmega \) having positive \(d-1\) measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of \(\partial \varOmega \) (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincare type inequalities for domains typically used in finite …

On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups.

2021

AbstractThis note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connec…

Characterisation of upper gradients on the weighted Euclidean space and applications

2020

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

C 1,ω (·) -regularity and Lipschitz-like properties of subdifferential

2012

Tangent lines and Lipschitz differentiability spaces

2015

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, whe…

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

2003

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.

Weighted Hardy inequalities beyond Lipschitz domains

2014

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent \lambda=n-1. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.

Abstract and concrete tangent modules on Lipschitz differentiability spaces

2020

We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm lip} f =|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli'…

Hypersurfaces of prescribed mean curvature over obstacles

1973

Let ~2 be a bounded domain in the euclidean space IR", n-> 2, with Lipschitz boundary ~ . We shall consider surfaces which are graphs of functions u defined on f2 having prescribed mean curvature H=H(x, u) with the side condition that they should be bounded from below by an obstacle ~b. The case H = 0 (minimal surfaces) has been discussed in detail by several authors, compare [6, 7, 12, 13, 17, 18, 20, 21, 24] of the references. Tomi [-31] has also investigated parametric surfaces with variable H. More general variational problems with obstructions have been discussed in [-9] and [-10]. During the session on "Variationsrechnung", held from June 18th to June 24th, 1972 in Oberwolfach, Mirand…

Lipschitz classes and the Hardy-Littlewood property

1993

We study the geometry of plane domains and the uniform Holder continuity properties of analytic functions.