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Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

For two-dimensional, immersed closed surfaces f : Σ → ℝ n , we study the curvature functionals ℰ p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of ℰ p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional condition....

Two-dimensional models of fabrics

Denis Caillerie, Hervé Tollenaere (1995)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Two-Layer Flow with One Viscous Layer in Inclined Channels

O. K. Matar, G. M. Sisoev, C. J. Lawrence (2008)

Mathematical Modelling of Natural Phenomena

We study pressure-driven, two-layer flow in inclined channels with high density and viscosity contrasts. We use a combination of asymptotic reduction, boundary-layer theory and the Karman-Polhausen approximation to derive evolution equations that describe the interfacial dynamics. Two distinguished limits are considered: where the viscosity ratio is small with density ratios of order unity, and where both density and viscosity ratios are small. The evolution equations account for the presence of...

Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation

Ahmed Abbas Mizeal, Mudhir A. Abdul Hussain (2012)

Archivum Mathematicum

In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.

Two-scale div-curl lemma

Augusto Visintin (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied...

Two-scale homogenization for a model in strain gradient plasticity

Alessandro Giacomini, Alessandro Musesti (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

Two-scale homogenization for a model in strain gradient plasticity

Alessandro Giacomini, Alessandro Musesti (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

Sergey Korotov (2007)

Applications of Mathematics

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered...

Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators

Andrey Andreev, Milena Racheva (2014)

Applications of Mathematics

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which...

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