Related papers: A new isoperimetric inequality for the elasticae

We study the deformations of elastic filaments confined within slowly-shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of…

In this paper we consider a class of prescribing curvature type equations on half Euclidean balls. Under suitable assumptions on the scalar curvature function and boundary mean curvature function we prove a min-max type inequality and the…

We consider the finite difference discretization of isotropic elastic wave equations on nonuniform grids. The intended applications are seismic studies, where heterogeneity of the earth media can lead to severe oversampling for simulations…

Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an…

Understanding crystal growth over arbitrary curved surfaces with arbitrary boundaries is a formidable challenge, stemming from the complexity of formulating non-linear elasticity using geometric invariant quantities. Solutions are generally…

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic…

Let $\Omega\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary $\partial\Omega$. In this paper, we establish the following sharp form of the trace Adams' inequality in $W^{2,2}(\Omega)$ with zero mean value and zero Neumann…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic…

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $\sigma$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum,…

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved…

We propose a simple mathematical model to describe the mechanical relaxation of cells within a curved epithelial tissue layer represented by an arbitrary curve in two-dimensional space. This model generalises previous one-dimensional models…

We establish a geometric inequality relating the Dirichlet energy $E_1(f)$ and the bienergy $E_2(f)$ of smooth maps \[ f : (M,g) \to (\overline{M},\overline{g}) \] between Riemannian manifolds. Assume that $(M,g)$ is a compact, connected…

We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the…

We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the…

We derive, via simultaneous homogenization and dimension reduction, the Gamma-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the…

Let $S(G^{\sigma})$ be the skew-adjacency matrix of the oriented graph $G^{\sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $\sigma$ to each of its edges. The skew energy of an oriented graph…

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between…

In this paper we found the renormalized free energy of a interacting scalar field on a compact hyperbolic manifold explicitly. We have shown a complete expression of the free energy and entropy as a function of the curvature and the…

Let $\Omega$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus \Omega$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that…

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane…