We examine the phase dependent Helfrich energy and show an Euler-Lagrange equation on the phase seperation line. This result has already been observed by e.g. Jülicher-Lipowski and later Elliot-Stinner. Here we are able to lower the regularity assumption for this result down to $C^{1,1}$ for the seperation line. In the proof we employ a carefully choosen test function utilising the signed distance function.
We investigate equilibrium configurations for surface energies which contain the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$ coupled with the elastic energy of the boundary curve, which we studied previously in [23]. It is shown that if a critical surface for this type of functional is axially symmetric, then it satisfies a simpler second order variational problem. Many examples of solutions of this are given.
In this paper we introduce a mathematical model for small deformations induced by external forces of closed surfaces that are minimisers of Helfrich-type energies. Our model is suitable for the study of deformations of cell membranes induced by the cytoskeleton. We describe the deformation of the surface as a graph over the undeformed surface. A new Lagrangian and the associated Euler-Lagrange equations for the height function of the graph are derived. This is the natural generalisation of the well known linearisation in the Monge gauge for initially flat surfaces. We discuss energy perturbati…
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, followed by proofs of classical results, including a solution to the relevant inverse problem, Noether's theorem on symmetries and conservation laws, and several aspects of minimal hypersurfaces. In the second chapter, the equivalence problem for Poincare-Cartan forms is solved, giving…
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a class of Bäcklund transformations relating two hyperbolic Euler-Lagrange systems of distinct types.
We prove that on the condition of non-trivial solutions, the Euler-Lagrange and Noether equations are equivalent for the variational problem of nonlinear Poisson equation and a class of more general Lagrangians, including position independent and of p-Laplacian type. As applications we prove certain propositions concerning the nonlinear Poisson equation and its generalisations, the equivalence of admissible and inner variations and discuss the inverse problem of determining the Lagrangian from conservation or symmetry laws.
We make some remarks on the Euler-Lagrange equation of energy functional $I(u)=\int_Ωf(\det Du)\,dx,$ where $f\in C^1(\mathbb R).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $Ω$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $Ω.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicty, radial solutions, and some special properties and questions in the 2-D cases.
Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler-Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part $K-W$ and a forcing term. We consider two situations: $G$ satisfying $Δ_2\cap\nabla_2$ in infinity and globally. We give conditions on the growth of the potential near zero for both situations.
In this paper we study the effects of simultaneous homogenization and dimension reduction in the context of convergence of stationary points for thin nonhomogeneous rods under the assumption of the von Kármán scaling. Assuming stationarity condition for a sequence of deformations close to a rigid body motion, we prove that the corresponding sequences of scaled displacements and twist functions converge to a limit point, which is the stationary point of the homogenized von Kármán rod model. The analogous result holds true for the von Kármán plate model.
Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE expanded. The hypotheses on F do not guarrantee existence of minimising weak solutions and include the singular p-Laplacian for 1<p<2. A much deeper converse is also true, if K={0} and extra natural assumptions are satisfied. Our main advance is that we introduce systematic "flat" sup-convolution regularisations which apply to general singular nonlinear PDEs i…
Flows of one-dimensional continuum in Lagrangian coordinates are studied in the paper. Equations describing these flows are reduced to a single Euler-Lagrange equation which contains two undefined functions. Particular choices of the undefined functions correspond to isentropic flows of an ideal gas, different forms of the hyperbolic shallow water equations. Complete group classification of the equation with respect to these functions is performed. Using Noether's theorem, all conservation laws are obtained. Their analogs in Eulerian coordinates are given.
We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form \begin{equation*} c|ξA(ω,x)|^p\leq f(ω,x,ξ)\leq |ξA(ω,x)|^p+Λ(ω,x) \end{equation*} for some $p\in (1,+\infty)$ and with a stationary and ergodic diagonal matrix $A$ such that its norm and the norm of its inverse satisfy minimal integrability assumptions. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of $f$ w…
The Evans function has become a standard tool in the mathematical study of nonlinear wave stability. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). We report on an unexpected complication that frustrates this computation for viscous shock profiles in gas dynamics. Although this phenomenon---related to the choice of Eulerian or Lagrangian coordinate system used to describe the gas---is present already in the one-dimensional setting, its …
The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_ΩF(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(Ω, \mathbb R^N )\cap K$, where $K$ is a closed convex subset of the Dirichlet class $\mathrm W^{k,1}_{g}(Ω, \mathbb R^N ),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathscr F$. We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity…
A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general functional of linear growth depending on the modulus of the gradient. We emphasize that Serrin ([1]) in fact proved the removability of singularities on sets of $(n-1)$-dimensional Hausdorff measure zero in an even more general setting. Our main interest is to generalize the comparison principles as outlined, for instance, in Section 10 of [2] without having the par…
We prove that any distribution $q$ satisfying the equation $\nabla q=÷{\bf f}$ for some tensor ${\bf f}=(f^i_j), f^i_j\in h^r(U)$ ($1\leq r<\infty$) -the {\it local Hardy space}, $q$ is in $h^r$, and is locally represented by the sum of singular integrals of $f^i_j$ with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure $p$ (modulo constant) associated with incompressible elastic energy-minimizing deformation ${\bf u}$ satisfying $|\nabla {\bf u}|^2, |{\rm cof}\nabla{\bf u}|^2\in h^1$. We also derive the system of Euler-L…
We study stochastic homogenization for convex integral functionals $$u\mapsto \int_D W(ω,\tfrac{x}\varepsilon,\nabla u)\,\mathrm{d}x,\quad\mbox{where}\quad u:D\subset \mathbb{R}^d\to\mathbb{R}^m,$$ defined on Sobolev spaces. Assuming only stochastic integrability of the map $ω\mapsto W(ω,0,ξ)$, we prove homogenization results under two different sets of assumptions, namely $\bullet_1\quad$ $W$ satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate $W^*(\cdot,0,ξ)$ and a mild monotonicity condition that ensures that the functional does not increase too …
We study uniaxial solutions of the Euler-Lagrange equations for a Landau-de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. In the elastic isotropic case, we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a director field of certain symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the "escape into third dimension" director cannot correspond to a purely uniaxial solution of the Landau-de Gennes Euler-Lagrange equations and we do not use artificia…
