In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife models. Existence and uniqueness of the solution to the related initial boundary value problem are shown. Further regularity of the solution is deduced by exploiting the so-called regularizing effect. Then, the large time behavior of such a solution is studied and several convergence properties of the trajectory as time tends to infinity are discussed.
We prove a weak-strong convergence result for functionals of the form $\int_{\mathbb{R}^N} j(x, u, Du)\,dx$ on $W^{1,p}$, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya-Szegö inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.
We consider a family of optimal control problems where the control variable is given by a boundary condition of Neumann type. This family is governed by parabolic variational inequalities of the second kind. We prove the strong convergence of the optimal controls and state systems associated to this family to a similar optimal control problem. This work solves the open problem left by the authors in IFIP TC7 CSMO2011.
We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Ration. Mech. Anal. 157 (2001)] that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof is based on a relative entropy estimation for the time-discrete approximants in an environment of Lp-theory bounds, and provides an error estimate for the approximation before the formation of shocks…
We study the behaviour of solutions of linear non-autonomous parabolic equations subject to Dirichlet or Neumann boundary conditions under perturbation of the domain. We prove that Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence of function spaces for the corresponding elliptic problems. As a consequence, we obtain convergence of solutions of non-autonomous parabolic equations under domain perturbation by variational methods using the same characterisation of domains as in elliptic case. A similar technique can be applied to obtain …
We present a stability result for a wide class doubly nonlinear equations, featuring general maximal monotone operators, and (possibly) nonconvex and nonsmooth energy functionals. The limit analysis resides on the reformulation of the differential evolution as a scalar energy-conservation equation with the aid of the so-called Fitzpatrick theory for the representation of monotone operators. In particular, our result applies to the vanishing viscosity approximation of rate-independent systems.
First, let $u_{g}$ be the unique solution of an elliptic variational inequality with source term $g$. We establish, in the general case, the error estimate between $u_{3}(μ)=μu_{g_{1}}+ (1-μ)u_{g_{2}}$ %(the convex combination of two solutions) and $u_{4}(μ)=u_{μg_{1}+ (1-μ) g_{2}}$ %(the solution corresponding to the convex combination of two data) for $μ\in [0, 1]$. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy $g$ for each positive heat transfer coefficient $h$ given on a part of the boundary of …
We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations …
In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials $(Ψ_n)_n$ with superlinear growth at infinity and a smooth energy functional $\mathcal{E}$, we enucleate sufficient conditions on them ensuring that the associated gradient systems $(Ψ_n,\mathcal{E})$ Evolutionary Gamma-converge to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosi…
Let $u_{g}$ the unique solution of a parabolic variational inequality of second kind, with a given $g$. Using a regularization method, we prove, for all $g_{1}$ and $g_{2}$, a monotony property between $μu_{g_{1}} + (1-μ)u_{g_{2}}$ and $u_{μg_{1} + (1-μ)g_{2}}$ for $μ\in [0, 1]$. This allowed us to prove the existence and uniqueness results to a family of optimal control problems over $g$ for each heat transfer coefficient $h>0$, associated to the Newton law, and of another optimal control problem associated to a Dirichlet boundary condition. We prove also, when $h\to +\infty$, the strong conv…
By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.
We present a variational principle governing the quasistatic evolution of a linearized elastoplastic material. In case of linear hardening, the novel characterization allows to recover and partly extend some known results and proves itself to be especially well-suited for discussing general approximation and convergence issues. In particular, the variational principle is exploited in order to prove in a novel setting the convergence of time and space-time discretizations as well as to provide some possible a posteriori error control.
We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using $Γ$-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.
For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal dissipation, all accounted for by a d'Alembert-Lagrange-type principle.
A mathematical continuum limit of the interaction energy of a random particle chain is shown to yield new insight into the effect of microscopic heterogeneities on macroscopic fracture laws in brittle materials. We derive a formula which yields that either elastic behaviour or a crack is energetically preferred. The formula explicitly shows the dependence on the boundary condition and the microstructure of the chain. The mathematical analysis is based on a variational convergence $Γ$-convergence of convex-concave potentials together with ergodic theorems which are common tools in stochastic ho…
This work aims to provide a comprehensive and unified numerical analysis for non linear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish an existence of the exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and non conforming numerical schemes.
We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strength…
We present some results of geometric convergence of level sets for solutions of total variation denoising as the regularization parameter tends to zero. The common feature among them is that they make use of explicit constructions of variational mean curvatures for general sets of finite perimeter. Consequently, no additional regularity of the level sets of the ideal data is assumed, and in particular the subgradient of the total variation at it could be empty. In exchange, other restrictions on the data or on the noise are required. We consider two cases: characteristic functions with a param…
We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this pa…
Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions
Freie Universität Berlin（柏林自由大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；Technical University of Darmstadt（达姆施塔特工业大学）；Clausthal University of Technology（克劳塔尔工业大学）
We derive a continuum model for incompatible elasticity as a variational limit of a family of discrete nearest-neighbor elastic models. The discrete models are based on discretizations of a smooth Riemannian manifold $(M,\mathfrak{g})$, endowed with a flat, symmetric connection $\nabla$. The metric $\mathfrak{g}$ determines local equilibrium distances between neighboring points; the connection $\nabla$ induces a lattice structure shared by all the discrete models. The limit model satisfies a fundamental rigidity property: there are no stress-free configurations, unless $\mathfrak{g}$ is flat, …
Let $Ω$ be an open subset of a Stein manifold $Σ$ and let $M$ be its boundary. It is well known that $M$ inherits a natural contact structure. In this paper we consider a family of variational functionals $F_\varepsilon$ defined by the sum of two terms: a Dirichlet-type energy associated with a sub-Riemannian structure in $Ω$ and a potential term on the boundary $M$. We prove that the functionals $F_\varepsilon$ $Γ$-converge to the intrinsic perimeter in $M$ associated with its contact structure. Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We str…
This paper is devoted to the study of geodesic distances defined on a subdomain of a given Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot-Carathéodory distance. We show that the uniform convergence (on compact sets) of these distances can be equivalently characterized in terms of $Γ$-convergence of several kinds of variational problems. Moreover, we investigate the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle.
We study a time-reversed hyperbolic heat conduction problem based upon the Maxwell--Cattaneo model of non-Fourier heat law. This heat and mass diffusion problem is a hyperbolic type equation for thermodynamics systems with thermal memory or with finite time-delayed heat flux, where the Fourier or Fick law is proven to be unsuccessful with experimental data. In this work, we show that our recent variational quasi-reversibility method for the classical time-reversed heat conduction problem, which obeys the Fourier or Fick law, can be adapted to cope with this hyperbolic scenario. We establish a …
The article studies the reiterated homogenization of linear elliptic variational inequalities arising in problems with unilateral constrains. We assume that the coefficients of the equations satisfy and abstract hypothesis covering on each scale a large set of concrete deterministic behavior such as the periodic, the almost periodic, the convergence at infinity. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is of the same type as the problem with rapidly oscillating coefficients.
The rigorous derivation of linear elasticity from finite elasticity by means of Gamma-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we comp…
Since its introduction over 50 years ago, the concept of Mosco convergence has permeated through diverse areas of mathematics and applied sciences. These include applied analysis, the theory of partial differential equations, numerical analysis, and infinite dimensional constrained optimization, among others. In this paper we explore some of the consequences of Mosco convergence on applied problems that involve moving sets, with some historical accounts, and modern trends and features. In particular, we focus on connections with density of convex intersections, finite element approximations, q…
