Some isoperimetric inequalities with respect to monomial weights

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ $ with respect to monomial weights. Let $\alpha $ and $\beta $ be real numbers such that $0\le \alpha <\beta+1$, $\beta\le 2 \alpha$. We show that, among all smooth sets $\Omega$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{\Omega } y^{\beta} dxdy$, the weighted perimeter $\int_{\partial \Omega } y^\alpha \, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a bound for eigenvalues of some nonlinear problems.

An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

Combinatorial higher dimensional isoperimetry and divergence

In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called “round” and “unfolded”, provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer–Fleming inequality for finitely generated groups.

Triangulation Based Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model

This paper presents the triangulation based isogeometric analysis of the Cahn–Hilliard phase-field model. We validate our method by convergence analysis, show detailed system evolution from a randomly perturbed initial condition and then discuss related isoperimetric problems. Lastly an example highlighting its efficacy in complex geometry is provided. Triangulation based isogeometric analysis shows time step stability and complex geometry adaptability in our experiments.