Existence of a solution for a nonlinearly elastic plane membrane 
“under tension”

Daniel Coutand

Displaying similar documents to “Existence of a solution for a nonlinearly elastic plane membrane “under tension””

Young-Measure approximations for elastodynamics with non-monotone stress-strain relations

Carsten Carstensen, Marc Oliver Rieger (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density . Their time-evolution leads to a nonlinear wave equation $u_{t t} = div S (D u)$ with the non-monotone stress-strain relation $S = D φ$ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in...

The nonlinear membrane model: a Young measure and varifold formulation

Med Lamine Leghmizi, Christian Licht, Gérard Michaille (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We establish two new formulations of the membrane problem by working in the space of $W_{Γ_{0}}^{1, p} (Ω, 𝐑^{3})$ -Young measures and $W_{Γ_{0}}^{1, p} (Ω, 𝐑^{3})$ -varifolds. The energy functional related to these formulations is obtained as a limit of the formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing...

Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces

Stefano Lisini (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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We study existence and approximation of non-negative solutions of partial differential equations of the type   $\partial_{t} u - div (A (\nabla (f (u)) + u \nabla V)) = 0 in (0, + \infty) \times ℝ^{n}, (0.1)$ where is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f : [0, + \infty) \to [0, + \infty)$ is a suitable non decreasing function, $V : ℝ^{n} \to ℝ$ is a convex function. Introducing the energy functional $φ (u) = \int_{ℝ^{n}} F (u (x)) d x + \int_{ℝ^{n}} V (x) u (x) d x$ , where is a convex function linked to by $f (u) = u F^{'} (u) - F (u)$ , we show that is the “gradient flow” of with respect to the 2-Wasserstein distance between probability measures on the...

Expansion for the superheating field in a semi-infinite film in the weak- limit

Pierre Del Castillo (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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Dorsey, Di Bartolo and Dolgert (Di Bartolo , 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak- limit. These authors deduced a formal expansion for the superheating field in powers of $κ^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion...

An example in the gradient theory of phase transitions

Camillo De Lellis (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove by giving an example that when ≥ 3 the asymptotic behavior of functionals $\int_{Ω} ε | \nabla^{2} {u |}^{2} {+ (1 - | \nabla u |}^{2})^{2} / ε$ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

Coupling the Stokes and Navier–Stokes equations with two scalar nonlinear parabolic equations

Macarena Gómez Mármol, Francisco Ortegón Gallego (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the components of the velocity field coupled with two scalar quantities and . The system presents nonlinear turbulent viscosity $A (θ, ϕ)$ and nonlinear source terms of the form $θ^{2} {| \nabla u |}^{2}$ and ${θ ϕ | \nabla u |}^{2}$ lying in . Some existence results are shown in this paper, including $L^{\infty}$ -estimates and positivity for both and .