[1] Aubin, J.-P., Frankowska, H.: Set-valued analysis. Modern Birkh¨auser Classics. Birkhh¨auser Boston, Inc., Boston, MA (2009)
[2] Ambrosio, L., Tortorelli, V. M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43, 999–1036 (1990)
[3] Ambrosio, L., Tortorelli, V. M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6, 105–123 (1992)
[4] Bonnivard, M., Lemenant, A., Millot, V.: On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers. Interfaces Free Bound. 20, 69–106 (2018)
[5] Braides, A.: Γ-convergence for beginners. Oxford University Press, Oxford (2002)
[6] Bronsard, L., Kohn, R. V.,: Motion by mean curvature as the singular limit of Ginzburg- Landau dynamics. J. Differential Equations 90, 211–237 (1991)
[7] Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Dif- ferential Equations 96, 116–141 (1992)
[8] de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc. 347, 1533–1589 (1995)
[9] Evans, L. C., Soner, H. M., Souganidis, P. E.: Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45, 1097–1123 (1992)
[10] Francfort, G. A., Le, N. Q., Serfaty, S.: Critical points of Ambrosio–Tortorelli converge to critical points of Mumford–Shah in the one-dimensional Dirichlet case. ESAIM Control Optim. Calc. Var., 15, 576–598 (2009)
[11] Francfort, G. A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)
[12] Fonseca, I., Liu, P.: The Weighted Ambrosio–Tortorelli Approximation Scheme. SIAM J. Math. Anal., 49(6), 4491–4520 (2017)
[13] Giacomini, A.: Ambrosio-Tortorelli approximation of quasi-static evolution of brittle frac- tures. Calc. Var. Partial Differential Equations 22, 129–172 (2005)
[14] Giga, Y.: Surface evolution equations: a level set approach. Birkh¨auser, Basel (2006)
[15] Hutchinson, J. E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn- Hilliard theory. Calc. Var. Partial Differential Equations 10, 49–84 (2000)
[16] Ito, A., Kenmochi, N., Yamazaki, N.: A phase-field model of grain boundary motion. Appl. Math. 53, 433–454 (2008)
[17] Kobayashi, R., Giga, Y.: Equations with singular diffusivity. J. Statist. Phys. 95, 1187–1220 (1999)
[18] Kobayashi, R., Warren, J. A., Carter, W. C.: Modeling grain boundaries using a phase field technique. Hokkaido University Preprint Series in Mathematics #422 (1998)
[19] Kobayashi, R., Warren, J. A., Carter, W. C.: A continuum model of grain boundaries. Physica D: Nonlinear Phenomena, 140(1–2), 141–150 (2000)
[20] Kobayashi, R., Warren, J. A., Carter, W. C.: Grain boundary model and singular diffusivity: In: Free boundary problems: theory and applications, GAKUTO Inter- nat. Ser. Math. Sci. Appl. 14, 283–294, Gakk¯otosho, Tokyo (2000)
[21] Kohn, R., Sternberg, P.: Local minimisers and singular perturbations. Proc. Roy. Soc. Edin- burgh Sect. A 111, 69–84 (1989)
[22] Lemenant, A., Santambrogio, F.: A Modica–Mortola approximation for the Steiner problem. C. R. Math. Acad. Sci. Paris 352, 451–454 (2014)
[23] Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, 123–142 (1987)
[24] Modica, L., Mortola, S.: Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5), 14, 526–529 (1977)
[25] Modica, L., Mortola, S.: Un esempio di Γ−-convergenza. Boll. Un. Mat. Ital. B (5), 14, 285–299 (1977)
[26] Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associ- ated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)
[27] Moll, S., Shirakawa, K.: Existence of solutions to the Kobayashi–Warren–Carter system. Calc. Var. Partial Differential Equations, 51(3-4):621–656 (2014)
[28] Moll, S., Shirakawa, K., Watanabe, H.: Energy dissipative solutions to the Kobayashi– Warren–Carter system. Nonlinearity, 30(7):2752–2784 (2017)
[29] Moll, S., Shirakawa, K., Watanabe, H.: Kobayashi–Warren–Carter type systems with nonho- mogeneous Dirichlet boundary data for crystalline orientation. In preparation
[30] Watanabe, H., Shirakawa, K.: Qualitative properties of a one-dimensional phase-field sys- tem associated with grain boundary. In Nonlinear analysis in interdisciplinary sciences— modellings, theory and simulations, volume 36 of GAKUTO Internat. Ser. Math. Sci. Appl., pages 301–328. Gakk¯otosho, Tokyo, 2013.
[31] Shirakawa, K., Watanabe, H.: Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete Contin. Dyn. Syst. Ser. S, 7(1):139–159 (2014)
[32] Shirakawa, K., Watanabe, H., Yamazaki, N.: Solvability of one-dimensional phase field sys- tems associated with grain boundary motion. Math. Ann. 356, 301–330 (2013)
[33] Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101, 209–260 (1988)