82 research outputs found
Quantitative anisotropic isoperimetric and Brunn-Minkowski inequalities for convex sets with improved defect estimates
In this paper we revisit the anisotropic isoperimetric and the
Brunn-Minkowski inequalities for convex sets. The best known constant
depending on the space dimension in both inequalities is due
to Segal [\ref{bib:Seg.}]. We improve that constant to for convex sets
and to for centrally symmetric convex sets. We also conjecture, that the
best constant in both inequalities must be of the form i.e., quadratic
in The tools are the Brenier's mapping from the theory of mass
transportation combined with new sharp geometric-arithmetic mean and some
algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.Comment: Recently Emanuel Milman from Technion pointed out a mistake, where I
missed an n^{1/2} on the right hand side of (3.18). The correction of the
mistake leads to a reproduction of the already known (Legal [35]) constant
Cn^
Gaussian curvature as an identifier of shell rigidity
In the paper we deal with shells with non-zero Gaussian curvature. We derive
sharp Korn's first (linear geometric rigidity estimate) and second inequalities
on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type
boundary conditions. We prove that if the Gaussian curvature is positive, then
the optimal constant in the first Korn inequality scales like and if the
Gaussian curvature is negative, then the Korn constant scales like
where is the thickness of the shell. These results have classical flavour
in continuum mechanics, in particular shell theory. The Korn first inequalities
are the linear version of the famous geometric rigidity estimate by Friesecke,
James and M\"uller for plates [14] (where they show that the Korn constant in
the nonlinear Korn's first inequality scales like ), extended to shells
with nonzero curvature. We also recover the uniform Korn-Poincar\'e inequality
proven for "boundary-less" shells by Lewicka and M\"uller in [37] in the
setting of our problem. The new estimates can also be applied to find the
scaling law for the critical buckling load of the shell under in-plane loads as
well as to derive energy scaling laws in the pre-buckled regime. The exponents
and in the present work appear for the first time in any sharp
geometric rigidity estimate.Comment: 25 page
The asymptotically sharp Korn interpolation and second inequalities for shells
We consider shells in three dimensional Euclidean space which have bounded
principal curvatures. We prove Korn's interpolation (or the so called first and
a half\footnote{The inequality first introduced in [6]}) and second
inequalities on that kind of shells for \Bu\in W^{1,2} vector fields,
imposing no boundary or normalization conditions on \Bu. The constants in the
estimates are optimal in terms of the asymptotics in the shell thickness
having the scalings or The Korn interpolation inequality reduces
the problem of deriving any linear Korn type estimate for shells to simply
proving a Poincar\'e type estimate with the symmetrized gradient on the right
hand side. In particular this applies to linear geometric rigidity estimates
for shells, i.e., Korn's fist inequality without boundary conditions.Comment: 5, short note. arXiv admin note: text overlap with arXiv:1709.0457
New asymptotically sharp Korn and Korn-like inequalities in thin domains
It is well known that Korn inequality plays a central role in the theory of
linear elasticity. In the present work we prove new asymptotically sharp Korn
and Korn-like inequalities in thin curved domains with a non-constant
thickness. This new results will be useful when studying the buckling of
compressed shells, in particular when calculating the critical buckling load.Comment: 16 page
Estimation of transaction costs during tax collection process
Tigran Jrbashyan; Davit Harutyunya
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