Gradient theory of phase transitions in composite media

SIAM Journal on Mathematical Analysis, 2012

We consider the van der Waals' free energy functional, with scaling parameter ε, in the plane domain R + × R + , with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries R + ×{0} and R + ×{∞}, and free boundary conditions on {∞}×R +. Finally, the datum on {0} × R + is chosen in such a way that the interface between the pure phases is pinned at some point (0, y). We show that there exists a critical scaling, y = yε, such that, as ε → 0, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.

The Quarterly Journal of Mathematics, 2017

In this paper we study, via Γ-convergence techniques, the asymptotic behaviour of a family of coupled singular perturbations of a non-convex functional of the type as a variational model to address two-phase transition problems under the volume constraints and where the additional unknown ρ interplays with ∇u in the formation of interfaces.

A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy, precisely,

2003

Asymptotic Analysis, 2009

We introduce a simplified model for a multi-material made up of two elastic bodies connected by a strong thin material layer whose stiffness grows as 1 . The model is obtained by identifying the Γ-limit of the stored strain energy functional of the physical problem when the thickness of the intermediate layer tends to zero. The intermediate layer behaves as a stiffening elastic membrane. Furthermore, in the linear anisotropic case, we establish the strong convergence of the exact solution toward the solution of the limit problem.

Nonlinear Analysis: Theory, Methods & Applications, 2012

We are concerned with the Lane-Emden-Fowler equation − u = λk(x)u q ± h(x)u p in Ω, subject to the Dirichlet boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N , k and h are variable potential functions, and 0 < q < 1 < p. Our analysis combines monotonicity methods with variational arguments.

Applying the general non-linear theory of shells undergoing phase transitions, we derive the balance equations along the singular curve modelling the phase interface in the shell. From the integral forms of balance laws of linear momentum, angular momentum, and energy as well as the entropy inequality we obtain the local static balance equation along the curvilinear phase interface. We also derive the thermodynamic condition allowing one to determine the interface position within the deformed shell midsurface. The special case of the pure mechanical theory is also considered.

2019

We study homogenization of a locally periodic two-scale dual-continuum system where each continuum interacts with the other. Equations for each continuum are written separately with interaction terms (exchange terms) added. The homogenization limit depends strongly on the scale of this continuum interaction term with respect to the microscopic scale. In J. S. R. Park and V. H. Hoang, Hierarchical multiscale finite element method for multicontinuum media, arXiv:1906.04635, we study in details the case where the interaction terms are scaled as O(1/ǫ) where ǫ is the microscale of the problem. We establish rigorously homogenization limit for this case where we show that in the homogenization limit, the dual-continuum structure disappears. In this paper, we consider the case where this term is scaled as O(1/ǫ). This case is far more interesting and difficult as the homogenized problem is a dual-continuum system which contains features that are not in the original two scale problem. In pa...

ESAIM: Mathematical Modelling and Numerical Analysis, 2001

In this paper, we study how solutions to elliptic problems with periodically oscillating coefficients behave in the neighborhood of the boundary of a domain. We extend the results known for flat boundaries to domains with curved boundaries in the case of a layered medium. This is done by generalizing the notion of boundary layer and by defining boundary correctors which lead to an approximation of order ε in the energy norm. Résumé. Onétudie ici le comportement au voisinage de la frontière du domaine de solutions de problèmes elliptiquesà coefficients oscillant périodiquement. Les résultats, connus pour des frontières plannes, sontétendus au cas de frontières courbes et pour un milieu stratifié. On généralise pour cela la notion de couche limite et on définit des correcteurs de frontière qui conduisentà une approximation d'ordre ε dans la normeénergie.