# Approximation of a semilinear elliptic problem in an unbounded domain

Messaoud Kolli; Michelle Schatzman

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 1, page 117-132
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topKolli, Messaoud, and Schatzman, Michelle. "Approximation of a semilinear elliptic problem in an unbounded domain." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 117-132. <http://eudml.org/doc/194148>.

@article{Kolli2010,

abstract = {
Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ increases on
[0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $
\xR_\{+\}^\{2\}$ with homogeneous Dirichlet boundary conditions by the
solution of $-\Delta u_\{L\}+f(u_\{L\})=0,$ on ]0,L[2 with adequate
non-homogeneous Dirichlet conditions.
We show that the error uL - u
tends to zero exponentially fast, in the uniform norm.
},

author = {Kolli, Messaoud, Schatzman, Michelle},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Semilinear elliptic equations; full-space problems;
approximation by finite domains.; semilinear elliptic equations; approximation by finite domains},

language = {eng},

month = {3},

number = {1},

pages = {117-132},

publisher = {EDP Sciences},

title = {Approximation of a semilinear elliptic problem in an unbounded domain},

url = {http://eudml.org/doc/194148},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Kolli, Messaoud

AU - Schatzman, Michelle

TI - Approximation of a semilinear elliptic problem in an unbounded domain

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 1

SP - 117

EP - 132

AB -
Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ increases on
[0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $
\xR_{+}^{2}$ with homogeneous Dirichlet boundary conditions by the
solution of $-\Delta u_{L}+f(u_{L})=0,$ on ]0,L[2 with adequate
non-homogeneous Dirichlet conditions.
We show that the error uL - u
tends to zero exponentially fast, in the uniform norm.

LA - eng

KW - Semilinear elliptic equations; full-space problems;
approximation by finite domains.; semilinear elliptic equations; approximation by finite domains

UR - http://eudml.org/doc/194148

ER -

## References

top- 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.
- H. Brezis, Analyse fonctionnelle. Masson, Paris (1983). Théorie et applications [Theory and applications].
- Xinfu Chen, Generationand propagation of interfaces for reaction-diffusion equations. J. Differential Equations96 (1992) 116-141.
- E.A. Coddington and N. Levinson, Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York, Toronto, London (1955).
- Ha Dang, P.C. Fife and L.A. Peletier, Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys.43 (1992) 984-998.
- F.R. de Hoog and R. Weiss, An approximation theory for boundary value problems on infinite intervals. Computing24 (1980) 227-239.
- P. de Mottoni and M. Schatzman, Development of interfaces in ${\mathbb{R}}^{N}$. Proc. Roy. Soc. Edinburgh Sect. A116 (1990) 207-220.
- P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc.347 (1995) 1533-1589.
- B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comp.31 (1977) 629-651.
- 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.
- D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition.
- 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.
- 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.
- T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Differential Geom.38 (1993) 417-461.
- 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.
- A. Jepson, Asymptotic boundary conditions for ordinary differential equations. Ph.D. thesis, California Institute of Technology (1980).
- 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.
- M. Schatzman, On the stability of the saddle solution of Allen-Cahn's equation. Proc. Roy. Soc. Edinburgh Sect. A125 (1995) 1241-1275.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.