Wiener's criterion and Γ-convergence

Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form –u + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ⁿ . The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the energy of local solutions are obtained.     

Differential forms in Carnot groups: a Γ-convergence approach

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E ₀ ^* , d _c ) of “intrinsic” differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials d _c appear naturally as limits of usual “Riemannian” differentials d _ε , ε >?0. More precisely, we show that L ²-energies associated with ε ^?κ d _ε on 1 forms Γ-converge, as ε → 0, to the energy associated with d _c .     

On the Γ-convergence of matrix fields related to the adjugate Jacobian

Adjugate Jacobians of mappings $\text{f}{}_{\mathrm{j}}\text{:Ω?}\text{R}{}^2\text{→}\text{R}{}^2$ can be represented in terms of Jacobian matrices: $\text{adj}\text{Df}{}_{\mathrm{j}}\text{=}\text{A}{}_{\mathrm{j}}\text{(x)Df}{}^{\mathrm{t}}{}_{\mathrm{j}}$, for $\text{j=1,2,…}$, by mean of symmetric matrix fields $\text{A}{}_{\mathrm{j}}\text{(x)}$ with $\text{det}\text{A}{}_{\mathrm{j}}\text{(x)=1}$ a.e. Under suitable conditions, we prove that $\text{Df}{}_{\mathrm{j}}\text{?Df}$ weakly in $\text{L}{}^1{}_{\mathrm{<mtext>loc</mtext>}}\text{(Ω;}\text{R}{}^2\text{)}$ if and only if $\text{A}{}_{\mathrm{j}}\text{(x)}$Γ-converges to a matrix $\text{A}\text{(x)}$ with $\text{det}\text{A}\text{(x)=1}$ satisfying $\text{adj}\text{Df=}\text{A}\text{(x)Df}{}^{\mathrm{t}}$. To cite this article: C. Sbordone, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A Γ-convergence result for Néel walls in micromagnetics

Three reduced models are considered for Néel walls which are dominant transition layers in thin-film micromagnetics. Each model comes as a nonlocal and nonconvex variational principle for one-dimensional magnetizations and it depends on a small parameter ε > 0. Our aim is to study the Γ-convergence of these models as eˉ 0{\varepsilon \downarrow 0} . We prove that the limiting magnetization patterns are piecewise constant functions that correspond to a finite number of walls of the same angle. The Γ-limit energy is proportional to the number of walls of these configurations and the energetic cost of each wall is quartic for small wall angles.     

Hydrodynamic limit for the Ginzburg-Landau ∇φ interface model with boundary conditions

Hydrodynamic limit for the Ginzburg-Landau interface model was established in [6] under periodic boundary conditions. This paper studies the same problem on a bounded domain imposing Dirichlet boundary conditions. A nonlinear partial differential equation of second order with boundary conditions is derived as a macroscopic equation. The main tools are H ^–1-method used in [6] and the higher integrability of gradients in [2].The author was supported by DFG project De 663/2-2 and 2/3 from April, 2001 to March, 2002, and by JSPS reserch fellowship for young scientists from April, 2002. Mathematics Subject Classification (2000): 60K35, 82C24, 35K55     

Global Cauchy problem for a Leray-α model

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we consider the Cauchy problem for the 3D Leray-α model, introduced by Cheskidov et al.[11]. We obtain the global solution...     

About an analytical approach to a quasicontinuum method via Γ-convergence

We report on an analytical study of a quasicontinuum method in the context of fracture mechanics in a one-dimensional setting. To this end, we compare the asymptotic behaviour of a discrete model with pairwise interactions of Lennard-Jones type and its quasicontinuum approximation via Γ-convergence. In an elastic regime the limiting behavior of the orginal model and its quasicontinuum approximation coincide. In the case of fracture it turns out that it is necessary to coarse grain the quasicontinuum approximation in the continuum region and at the atomistic/continuum interface in order to capture the same behavior as the atomistic model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

About an analytical approach to a quasicontinuum method via Γ-convergence

We report on an analytical study of a quasicontinuum method in the context of fracture mechanics in a one-dimensional setting. To this end, we compare the asymptotic behaviour of a discrete model with nearest and next-to-nearest neighbour interactions of Lennard-Jones type and its quasicontinuum approximation via Γ-convergence. In case of fracture it turns out that one has to coarse grain in the continuum region and at the atomistic/continuum interface in order to capture the same behavior as the atomistic model, while this is not needed if the boundary conditions are such that the system behaves elastically. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Γ-convergence for one-dimensional Ginzburg-Landau functional with generalized Lipschitz penalizing term

We use the approach developed in the paper G. Alberti, S. Muller: A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 , 761-825 (2001) to obtain Γ-convergence for a class of Ginzburg-Landau functionals I^ε(v), where v = v(s) is appropriate Sobolev function. We generalize results from the paper A. Raguž: Relaxation of Ginzburg-Landau functional with 1-Lipschitz penalizing term in one dimension by Young measures on micropatterns, Asymptotic Anal. 41 ( 3,4 ), 331-361 (2005), where original functional was penalized by 1-Lipschitz function g = g(s). In this note we prove Γ-convergence when g = g(s, v(s), v′(s)) under suitable growth conditions imposed on g. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of doubly periodic vortices in a generalized Chern–Simons model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern–Simons model. We show that there exists a critical value of the coupling parameter such that there exist self-dual doubly periodic vortex solutions for the generalized self-dual Chern–Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern–Simons equation we can also construct doubly periodic vortex solutions to a related generalized self-dual Abelian Higgs equation.     

Two-parameter bifurcation in a two-dimensional simplified Hodgkin–Huxley model

The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.     

Existence of charged vortices in a Maxwell–Chern–Simons model

In this paper, we prove the existence of charged vortex solitons in a Maxwell–Chern–Simons model. We establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field-theoretical Lagrangian. We also show that the solutions obtained are smooth.     

Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a $2\times 2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern–Simons vortices. We show that the system admits at least two solutions when the Chern–Simons coupling parameter $\kappa >0$ is sufficiently small; while no solution exists for $\kappa >0$ sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as $\kappa \rightarrow 0$ . Then we obtain a second solution by a min-max procedure of “mountain pass” type.     

Global existence for a 1D parabolic–elliptic model for chemical aggression in permeable materials

We prove the global existence and uniqueness of smooth solutions to a nonlinear system of parabolic–elliptic equations, which describes the chemical aggression of a permeable material, like calcium carbonate rocks, in the presence of acid atmosphere. This model applies when convective flows are not negligible, due to the high permeability of the material. The global (in time) result is proven by using a weak continuation principle for the local solutions.     

Global stability for a nonlocal reaction–diffusion population model

In this paper, we study a nonlocal reaction–diffusion population model. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the model. We then analyze the global stability for the model.     

The Γ-limit of a finite-strain Cosserat model for asymptotically thin domains and a consequence for the Cosserat couple modulus μc

We study the behaviour of a geometrically exact 3D Cosserat continuum model for an asymptotically flat domain. Despite the inherent nonlinearity, the Γ-limit of a corresponding canonically rescaled problem on a domain with constant thickness can be explicitly calculated. This “membrane” limit exhibits no bending contributions scaling with h ³ (similar to classical approaches) but features a transverse shear resistance scaling with h for strictly positive Cosserat couple modulus μ_c > 0. This result is physically inacceptable for a zero-thickness “membrane” limit model. Therefore it is suggested that the physically consistent value of the Cosserat couple modulus μ_c is zero. In this case, however, the Γ-limit looses coercivity for the midsurface deformation in H ^1,2(ω , ℝ³). For numerical purposes then, a transverse shear resistance can be reintroduced, establishing coercivity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation $\bar{F}$ is a difference quotient, that is where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances. Mathematics Subject Classification (2000) 47J20, 58B20, 49J45     

Global stability and pattern formation in a nonlocal diffusive Lotka–Volterra competition model

A diffusive Lotka–Volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed. The nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species. It is shown that when there is no nonlocal intraspecific competition, the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive Lotka–Volterra competition model regardless of the strength of nonlocal interspecific competition. Global stability of nonnegative constant equilibria are proved using Lyapunov or upper–lower solution methods. On the other hand, strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity. For the weak competition case, the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients.