A discussion about the homogenization of moving interfaces. (English) Zbl 1180.35070

Cardaliaguet, P.; Lions, P.-L.; Souganidis, P. E.

J. Math. Pures Appl. (9) 91, No. 4, 339-363 (2009).

The authors present a number of new results concerning the behavior of moving interfaces in periodic environments and analyze in detail their averaging behavior. They concentrate on fronts moving by either first-order oscillatory velocity or mean curvature coupled with an oscillating forcing. Under sharp assumptions, they show that the fronts either homogenize or get trapped or oscillate. Several concrete examples are also discussed.
There has been considerable interest lately in the homogenization theory for first- and second-order partial differential equations in periodic/almost-periodic and stationary, ergodic, random environments. Of special interest is the study of the averaged behavior of moving interfaces in such environments. In this note the authors expand considerably this investigation in the context of periodic media. Although it is important both for applications and interesting mathematically to consider random media, very little is known for moving interfaces in this setting except for some first-order motions.
The general framework concerns the behavior as \(\varepsilon\to 0\) of the solution \(u^\varepsilon\) of the initial value problem
\[ \begin{cases} u^\varepsilon_t+F(\varepsilon D^2u^\varepsilon,Du^\varepsilon,\frac{x} {\varepsilon})=0&\text{in }\mathbb{R}^N\times(0,\infty),\\ u^\varepsilon=u_0 &\text{on }\mathbb{R}^N \times\{0\},\end{cases} \]
where \(u_0\in UC(\mathbb{R}^N)\), the space of uniformly continuous functions on \(\mathbb{R}^N\), and \(F\) is (degenerate) elliptic, geometric and periodic. In the random setting, the homogenization of this problem is still an open problem, except for completely degenerate (first-order) and (degenerate) semilinear equations with either convex or concave gradient dependence.

Reviewer: Vasile Iftode (Bucureşti)

MSC:

35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35F21	Hamilton-Jacobi equations

Keywords:

viscosity solution; periodic environments; averaging behavior

References:

[1]	Alvarez, O., Homogenization of hamilton – jacobi equations in perforated sets, J. differential equations, 159, 2, 543-577, (1999) · Zbl 0945.35010
[2]	Barles, G., Solutions de viscosité des équations de hamilton – jacobi, Mathématiques & applications (Berlin), vol. 17, (1994), Springer-Verlag Paris · Zbl 0819.35002
[3]	Barles, G.; Souganidis, P.E., A new approach to front propagation problems: theory and applications, Arch. rational mech. anal., 141, 3, 237-296, (1998) · Zbl 0904.35034
[4]	Bhattacharya, K.; Craciun, B., Homogenization of a hamilton – jacobi equation associated with the geometric motion of an interface, Proc. roy. soc. Edinburgh sect. A, 133, 4, 773-805, (2003) · Zbl 1043.35028
[5]	Caffarelli, L.A.; Souganidis, P.E.; Wang, L., Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. pure appl. math., 58, 3, 319-361, (2005) · Zbl 1063.35025
[6]	Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. amer. math. soc. (N.S.), 27, 1, 1-67, (1992) · Zbl 0755.35015
[7]	Dirr, N.; Yip, A., Pinning an de-pinning phenomena in front propagation in heterogeneous medium, Interfaces and free boundaries, 8, 79-109, (2006) · Zbl 1101.35074
[8]	Evans, L.C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. roy. soc. Edinburgh sect. A, 120, 3-4, 245-265, (1992) · Zbl 0796.35011
[9]	Evans, L.C., The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. roy. soc. Edinburgh sect. A, 111, 3-4, 359-375, (1989) · Zbl 0679.35001
[10]	Horie, K.; Ishii, H., Homogenization of hamilton – jacobi equations on domains with small scale periodic structure, Indiana univ. math. J., 47, 3, 1011-1058, (1998) · Zbl 0924.49020
[11]	Ishii, H., Almost periodic homogenization of hamilton – jacobi equations, (), 600-605 · Zbl 0969.35018
[12]	Kosygina, E.; Rezakhanlou, F.; Varadhan, S.R.S., Stochastic homogenization of hamilton – jacobi – bellman equations, Comm. pure appl. math., 59, 10, 1489-1521, (2006) · Zbl 1111.60055
[13]	Lions, P.-L.; Souganidis, P.E., Correctors for the homogenization of hamilton – jacobi equations in the stationary ergodic setting, Comm. pure appl. math., 56, 10, 1501-1524, (2003) · Zbl 1050.35012
[14]	Lions, P.-L.; Souganidis, P.E., Homogenization of “viscous” hamilton – jacobi equations in stationary ergodic media, Comm. partial diff. equations, 30, 1-3, 335-375, (2005) · Zbl 1065.35047
[15]	Lions, P.-L.; Souganidis, P.E., Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. inst. H. Poincaré anal. non linéaire, 22, 5, 667-677, (2005) · Zbl 1135.35092
[16]	P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. Preprint
[17]	Rezakhanlou, F.; Tarver, J.E., Homogenization for stochastic hamilton – jacobi equations, Arch. rational mech. anal., 151, 4, 277-309, (2000) · Zbl 0954.35022
[18]	Soravia, P., Optimality principles and representation formulas for viscosity solutions of hamilton – jacobi equations. I. equations of unbounded and degenerate control problems without uniqueness, Adv. differential equations, 4, 2, 275-296, (1999) · Zbl 0955.49016
[19]	Souganidis, P.E., Stochastic homogenization of hamilton – jacobi equations and some applications, Asymptot. anal., 20, 1, 1-11, (1999) · Zbl 0935.35008
[20]	Souganidis, P.E., Front propagation: theory and applications, (), 186-242 · Zbl 0882.35016