Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Monneau, Régis; Roquejoffre, Jean-Michel; Roussier-Michon, Violaine

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 2, pp. 217-248.

Nous construisons des ondes progressives sous la forme de graphes $z=-ct+\phi \left(x\right)$, $\phi :x\in {ℝ}^{N-1}\mapsto \phi \left(x\right)\in ℝ$, $N\ge 2$, solutions du mouvement par courbure moyenne forcée $V_n=-c_0+\kappa$ ($c\ge c_0$) en dimension $N$ d’espace et avec un comportement asymptotique prescrit. Pour toute solution de viscosité ${\phi }_{\infty }$, $1$-homogène en espace, de l’équation eikonale $|D{\phi }_{\infty }|=\sqrt{{(c/c_0)}^2-1}$, nous mettons en évidence une solution régulière et concave du mouvement par courbure moyenne forcée dont le comportement asymptotique est donné par ${\phi }_{\infty }$. Nous décrivons aussi ${\phi }_{\infty }$ en terme d’une mesure de probabilité sur la sphère ${𝕊}^{N-2}$.

We construct travelling wave graphs of the form $z=-ct+\phi \left(x\right)$, $\phi :x\in {ℝ}^{N-1}\mapsto \phi \left(x\right)\in ℝ$, $N\ge 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\ge c_0$) with prescribed asymptotics. For any $1$-homogeneous function ${\phi }_{\infty }$, viscosity solution to the eikonal equation $|D{\phi }_{\infty }|=\sqrt{{(c/c_0)}^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by ${\phi }_{\infty }$. We also describe ${\phi }_{\infty }$ in terms of a probability measure on ${𝕊}^{N-2}$.

[1] G. Barles, H. M. Soner & P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), 439-469. | MR

[2] L. A. Caffarelli & W. Littman, Representation formulas for solutions to $\Delta u-u=0$ in ${𝐑}^n$, in Studies in partial differential equations, MAA Stud. Math. 23, Math. Assoc. America, 1982, 249-263. | MR

[3] M. G. Crandall, H. Ishii & P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 1-67. | MR

[4] L. C. Evans, Partial differential equations, Graduate Studies in Math. 19, Amer. Math. Soc., 1998. | MR

[5] P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics 53, Society for Industrial and Applied Mathematics (SIAM), 1988. | MR

[6] D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, Grundl. Math. Wiss. 224, Springer, 1977. | MR

[7] F. Hamel, R. Monneau & J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst. 13 (2005), 1069-1096. | MR

[8] F. Hamel, R. Monneau & J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst. 14 (2006), 75-92. | MR

[9] C. Imbert, Convexity of solutions and $C^{1,1}$ estimates for fully nonlinear elliptic equations, J. Math. Pures Appl. 85 (2006), 791-807. | MR

[10] J. I. Kanelʼ, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (101) (1962), 245-288. | MR

[11] P. De Mottoni & M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. 347 (1995), 1533-1589. | MR

[12] H. Ninomiya & M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, in Free boundary problems: theory and applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl. 13, Gakkōtosho, 2000, 206-221. | MR

[13] H. Ninomiya & M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations 213 (2005), 204-233. | MR

[14] M. Taniguchi, Multi-dimensional traveling fronts in bistable reaction-diffusion equations, Discrete Contin. Dyn. Syst. 32 (2012), 1011-1046. | MR