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-Lagrange equations for incompressible local minimizers ${\bf u}$ that are in the space $K^{1,3}_{\rm loc}$; partially resolving a long standing problem. For Hölder continuous pressure $p$, we obtain partial regularity of area-preserving minimizers.
In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the exist…
In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.
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.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which als…
We investigate the convergence as $p\searrow1$ of the minimizers of the $W^{s,p}$-energy for $s\in(0,1)$ and $p\in(1,\infty)$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $Γ$-convergence. We also address the convergence of the corresponding Euler-Lagrange equations, and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy.
We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the $H^1$ projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct $L^r$ solutions of the Navier-Stokes equation for a short time interval.
A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.
On a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities: Energy equality and Lyapunov stability
Universität Regensburg（雷根斯堡大学）；Fudan University（复旦大学）
We study the minimizer u of a convex functional in the plane which is not Gâteaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.
A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plate…
We prove the local Hölder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}λ|\nabla u|^{2}+h(\det \nabla u) \,dx\] subject to a condition of `positive twist'. The latter turns out to be equivalent to requiring that $u$ maps circles to suitably star-shaped sets. The convex function $h(s)$ grows logarithmically as $s\to 0+$, linearly as $s \to +\infty$, and satisfies $h(s)=+\infty$ if $s \leq 0$. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to p…
We study the behaviour of global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of $\bbS^1$-equivariant configurations. It is known from our previous paper \cite{DMP2} that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of \emph{torus} or of \emph{split} type. Here, starting from a nematic…
In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $Ω\subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(Ω;{\mathbb R}^{N\times n})$, the space of functions of Bounded Mean Oscillation of John & Nirenberg. A version of Taylor's theorem is first shown to be valid provided the integrand $W$ has polynomial growth. This result is then used to demonstrate that, for the Dirichlet, Neumann, and mixed problems, every Lipschitz-continuous solution of the corresponding Euler-Lagrange equations at which the se…
We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/ε]$, for initial data of the form $εU_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/ε]$, provided that $ε$ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies …
As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhib…
In this paper we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider the displacement problem in which the outer boundary of the body is subjected to a Dirichlet type boundary condition. For a spherically symmetric body occupying the unit ball $\mathcal{B}\in\mathbb{R}^3$, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler--Lagrange equations, and sho…
In this paper we address nonlocal vector variational principles obtained by substitution of the classical gradient by the Riesz fractional gradient. We show the existence of minimizers in Bessel fractional spaces under the main assumption of polyconvexity of the energy density, and, as a consequence, the existence of solutions to the associated Euler--Lagrange system of nonlinear fractional PDE. The main ingredient is the fractional Piola identity, which establishes that the fractional divergence of the cofactor matrix of the fractional gradient vanishes. This identity implies the weak converg…
In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim χ_{\{u > 0 \}}$. The Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate and nondegeneracy of minima. We also address weak and strong regularity properties of free boundary. We show the set $\{u > 0 \}$ has locally finite perimeter…
This work deals with the entropic regularization of the Brenier problem for perfect incompressible fluids introduced by Arnaudon, Cruzeiro, Léonard and Zambrini. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms.
We prove that if $(u,Γ)$ is a minimizer of the functional $$ J(u,Γ)=\int_{B_1(0)\setminus Γ}|\nabla u|^2dx +\H^1(Γ) $$ and $Γ$ connects $\partial B_1(0)$ to a point in the interior, then $Γ$ satisfies a point-wise $C^{2,α}$-estimate at the crack-tip. This means that the Mumford-Shah functional satisfies an additional, and previously unknown, Euler-Lagrange condition. ******* The previous version of the paper contained some mistakes, which has been fixed. More explanations/details has been added in Section 6.
In this paper, we study a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three dimensional case, we prove the existence/uniqueness of local strong solutions for arbitrary initial data as well as global strong solutions under the large viscosity assumption. We also establish some regularity criteria in terms of the velocity for local smooth solutions. Finally, we investigate the stability of the system near local minimizers of the elast…
