56 research outputs found
A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space
For functions on generalised connected surfaces (of any dimensions) with
boundary and mean curvature, we establish an oscillation estimate in which the
mean curvature enters in a novel way. As application we prove an a priori
estimate of the geodesic diameter of compact connected smooth immersions in
terms of their boundary data and mean curvature. These results are developed in
the framework of varifolds. For this purpose, we establish that the notion of
indecomposability is the appropriate substitute for connectedness and that it
has a strong regularising effect; we thus obtain a new natural class of
varifolds to study. Finally, our development leads to a variety of questions
that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure
An isoperimetric inequality for diffused surfaces
For general varifolds in Euclidean space, we prove an isoperimetric
inequality, adapt the basic theory of generalised weakly differentiable
functions, and obtain several Sobolev type inequalities. We thereby intend to
facilitate the use of varifold theory in the study of diffused surfaces.Comment: Awaiting publication in Kodai Math. J. The final printed version will
be different. 14 pages, no figure
Properties of surfaces with spontaneous curvature
A model describing cell membranes as optimal shapes with regard to the
-deficit of their mean curvature to a given constant called spontaneous
curvature is considered. It is shown that the corresponding energy functional
is lower semi-continuous with respect to oriented varifold convergence on a
space of surfaces whose second fundamental form is uniformly bounded in .
Elementary examples are presented showing that the latter condition is
necessary. As a consequence, smoothly embedded minimisers among surfaces of
higher genus are obtained. Moreover, it is shown that the diameter of a
connected surface is controlled by the -deficit of its mean curvature to
the spontaneous curvature leading to an improved condition for the existence of
minimisers. Finally, the diameter bound can be applied to obtain an
isoperimetric inequality.Comment: 37 pages, no figure
The effects of financing rules in pay-as-you-go pension systems on the life and the business cycle
Embedded Delaunay tori and their Willmore energy
A family of embedded rotationally symmetric tori in the Euclidean 3-space
consisting of two opposite signed constant mean curvature surfaces that
converge as varifolds to a double round sphere is constructed. Using complete
elliptic integrals, it is shown that their Willmore energy lies strictly below
. Combining such a strict inequality with previous works by
Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that for every
isoperimetric ratio there exists a smoothly embedded torus minimising the
Willmore functional under isoperimetric constraint, thus completing the
solution of the isoperimetric-constrained Willmore problem for tori. Similarly,
we deduce the existence of smoothly embedded tori minimising the Helfrich
functional with small spontaneous curvature. Moreover, it is shown that the
tori degenerate in the moduli space which gives an application also to the
conformally-constrained Willmore problem. Finally, because of their symmetry,
the Delaunay tori can be used to construct spheres of high isoperimetric ratio,
leading to an alternative proof of the known result for the genus zero case.Comment: 28 pages. Final version to appear in Nonlinear Analysis (TMA
A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint
Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict
inequality between the Willmore energies of two surfaces and their connected
sum in the context of isoperimetric constraints. Building on previous work by
Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of
minimisers for the isoperimetric constrained Willmore problem in every genus,
provided the minimal energy lies strictly below . Besides the geometric
interest, such a minimisation problem has been studied in the literature as a
simplified model in the theory of lipid bilayer cell membranes.Comment: 16 pages. Final version to appear in Advances in Calculus of
Variation
A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds
For varifolds whose first variation is representable by integration, we
introduce the notion of indecomposability with respect to locally Lipschitzian
real valued functions. Unlike indecomposability, this weaker connectedness
property is inherited by varifolds associated with solutions to geometric
variational problems phrased in terms of sets, chains, and immersions; yet
it is strong enough for the subsequent deduction of substantial geometric
consequences therefrom. Our present study is based on several further concepts
for varifolds put forward in this paper: real valued functions of generalised
bounded variation thereon, partitions thereof in general, partition thereof
along a real valued generalised weakly differentiable function in particular,
and local finiteness of decompositions.Comment: This is the second paper of a series of three papers which shall
supersede arXiv:1709.05504 by the same authors. 58 pages (v1: 43 pages), no
figures, added treatment of varifolds with free boundary which includes an
isoperimetric lower density ratio bound in 9.22 and results in Theorem E and
Corollary
The fiscal and intergenerational burdens of brakes and subsidies for energy prices
We study the effects of different financing rules for untargeted energy price brakes and subsidies on intergenerational welfare in a large-scale overlapping generations model. The results indicate that, in comparison with a laissez-faire solution without any government interventions, debt-financed implementations of such measures are very detrimental for young and future generations. However, the taxation of windfall profits can significantly contribute to reduce the economic burdens of these generations; whereas, the positive effects on older generations are much less pronounced
Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities
Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds
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