Approximation of a semilinear elliptic problem in an unbounded domain

Kolli, Messaoud; Schatzman, Michelle

ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 1, pp. 117-132.

Let $f$ be an odd function of a class ${\mathrm{C}}^2$ such that $f\left(1\right)=0,f^{\text{'}}\left(0\right)<0,f^{\text{'}}\left(1\right)>0$ and $x\mapsto f\left(x\right)/x$ increases on $[0,1]$. We approximate the positive solution of $-\Delta u+f\left(u\right)=0,$ on ${ℝ}_{+}^2$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_L+f\left(u_L\right)=0,$ on ${]0,L[}^2$ with adequate non-homogeneous Dirichlet conditions. We show that the error $u_L-u$ tends to zero exponentially fast, in the uniform norm.

Classification : 35J60, 35P15
Mots clés : semilinear elliptic equations, full-space problems, approximation by finite domains

[1] S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1084-1095.

[2] H. Brezis, Analyse fonctionnelle. Masson, Paris (1983). Théorie et applications [Theory and applications]. | MR | Zbl

[3] Xinfu Chen, Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations 96 (1992) 116-141. | Zbl

[4] E.A. Coddington and N. Levinson, Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York, Toronto, London (1955). | MR | Zbl

[5] Ha Dang, P.C. Fife and L.A. Peletier, Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 43 (1992) 984-998. | Zbl

[6] F.R. De Hoog and R. Weiss, An approximation theory for boundary value problems on infinite intervals. Computing 24 (1980) 227-239. | Zbl

[7] P. De Mottoni and M. Schatzman, Development of interfaces in ${ℝ}^N$. Proc. Roy. Soc. Edinburgh Sect. A 116 (1990) 207-220. | Zbl

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

[9] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31 (1977) 629-651. | Zbl

[10] L.C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992) 1097-1123. | Zbl

[11] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[12] T.M. Hagstrom and H.B. Keller, Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains. Math. Comp. 48 (1987) 449-470. | Zbl

[13] T. Hagstrom and H.B. Keller, Exact boundary conditions at an artificial boundary for partial differential equations in cylinders. SIAM J. Math. Anal. 17 (1986) 322-341. | Zbl

[14] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Differential Geom. 38 (1993) 417-461. | Zbl

[15] A.D. Jepson and H.B. Keller, Steady state and periodic solution paths: their bifurcations and computations, in Numerical methods for bifurcation problems, Dortmund (1983). Birkhäuser, Basel (1984) 219-246. | Zbl

[16] A. Jepson, Asymptotic boundary conditions for ordinary differential equations. Ph.D. thesis, California Institute of Technology (1980).

[17] P.A. Markowich, A theory for the approximation of solutions of boundary value problems on infinite intervals. SIAM J. Math. Anal. 13 (1982) 484-513. | Zbl

[18] M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1241-1275. | Zbl