Sharp interface limit of a multi-phase transitions model under nonisothermal conditions

Calc. Var. (2021) 60:142 https://doi.org/10.1007/s00526-021-02008-3 Calculus of Variations Sharp interface limit of a multi-phase transitions model under nonisothermal conditions Riccardo Cristoferi1 · Giovanni Gravina2 Received: 24 February 2020 / Accepted: 31 March 2021 / Published online: 30 June 2021 © The Author(s) 2021 Abstract A vectorial Modica–Mortola functional is considered and the convergence to a sharp interface model is studied. The novelty of the paper is that the wells of the potential are not constant, but depend on the spatial position in the domain . The mass constrained minimization problem and the case of Dirichlet boundary conditions are also treated. The proofs rely on the precise understanding of minimizing geodesics for the degenerate metric induced by the potential. Mathematics Subject Classification 49J45 · 34D15 · 26B30 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Previous works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Statement of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Sketch of the strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 A discussion on the assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 -convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Existence of minimizing geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Liminf inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Limsup inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Proofs of the variants of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Communicated by F.H. Lin. B Riccardo Cristoferi Giovanni Gravina 1 Department of Mathematics, Radboud University, Nijmegen, The Netherlands 2 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic 123 142 Page 2 of 62 R. Cristoferi, G. Gravina 5.1 Proof of Theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Proof of Theorem 1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1 Introduction Phase transitions phenomena are ubiquitous in nature. Examples are the spinodal decomposition in metallic alloys, the change in the crystallographic structure in metals, the order-disorder transitions, and the alterations of the molecular structures. In view of the wide range of physical and industrial applications where phase transitions are observed, it is of primary interest to understand the different mechanisms that govern these complex processes. Many physical models have been proposed over the years to capture the behavior of these phenomena and an enormous amount of insight has been gained by performing analytical studies. For this reason, the theoretical investigation of phase transitions is still currently an active field of research in the mathematical community. In the particular case of liquid-liquid phase transitions, the preferred model was proposed by van der Waals (see [61]) and was later independently rediscovered by Cahn and Hilliard (see [18]). This theory revolves around the study of the so called Modica–Mortola energy functional (often referred to as the Ginzburg–Landau free energy in the physics literature), which is the foundation of the model we consider in this paper. The primary focus of this work is the study of the -convergence of the family of functionals    1 W (x, u(x)) + ε|∇u(x)|2 d x, Fε (u):=  ε where u ∈ H 1 (; R M ), with M ≥ 1, and W :  × R M → [0, ∞) is a locally Lipschitz potential such that, for all x ∈ , W (x, p) = 0 if and only if p ∈ {z 1 (x), . . . , z k (x)}. Here  denotes an open bounded subset of R N with Lipschitz continuous boundary and, for i ∈ {1, . . . , k}, the z i :  → R M are given Lipschitz functions. Our main contribution is the treatment of the case M ≥ 2 for x-dependent wells, thus providing a first vectorial counterpart to some of the results in [14,58], where moving wells were considered in the scalar case. For the precise statement of our results we refer the reader to Sect. 1.2. 1.1 Previous works Denote by  ⊂ R N the container of the material, and assume that the system is described by a scalar valued phase (or order) parameter u :  → R, which for instance, in the case of a mixture of two or more fluids, represents the density. Stable equilibrium configurations are local minimizers of the Gibbs free energy. Under isothermal conditions, consider  W0 (u(x)) d x, (1)  where the free energy density W0 : R → [0, ∞) is taken to be non-convex in order to support a phase transitions. If the material has two stable phases, the typical form of W0 is depicted in Fig. 1. In many situations, the physical interpretation of the phase parameter naturally 123 Sharp interface limit of a multi-phase transitions model… Page 3 of 62 142 W0 W u α u α β β Fig. 1 On the left: the typical profile of the potential W0 together with its convex envelope; the region where the two do not coincide is highlighted in red. On the right: the potential W , obtained by subtracting a linear term with slope W0 (α) from W0 (color figure online) imposes a constraint on the class of admissible functions for the minimization problem for (1). If u represents a density, this often takes the form of a volume constraint, i.e.,  u(x) d x = m, (2)  for some m ∈ R. For W0 as in Fig. 1, let (α, β) be the interval where W0 does not coincide with its convex envelope. To be precise, α and β are chosen to satisfy W0 (β) − W0 (α) = W0 (α)(β − α), W0 (α) = W0 (β). The numbers α, β, μ, wher (...truncated)