Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maximal positive boundary value problems as limits of singular perturbation problems

Authors: Claude Bardos and Jeffrey Rauch
Journal: Trans. Amer. Math. Soc. 270 (1982), 377-408
MSC: Primary 35B25; Secondary 35F05, 35L40
MathSciNet review: 645322
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain $ \Omega \subset {{\mathbf{R}}^n}$. In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of $ \Omega $. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in $ \Omega $ and outside $ \Omega $. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in $ \Omega $. The boundary condition is determined in a simple way from the system and the singular terms.

References [Enhancements On Off] (What's this?)

  • [1] C. Bardos, D. Brézis, and H. Brezis, Perturbations singulières et prolongements maximaux d’opérateurs positifs, Arch. Rational Mech. Anal. 53 (1973/74), 69–100 (French). MR 0348247
  • [2] K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392. MR 0062932
  • [3] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 0100718
  • [4] -, Well-posed problems of mathematical physics, mimeographed lecture notes, New York Univ.
  • [5] L. Hörmander, Linear partial differential operators, 2nd rev. printing, Springer-Verlag, Berlin, 1964.
  • [6] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [7] Tosio Kato, Singular perturbation and semigroup theory, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Springer, Berlin, 1976, pp. 104–112. Lecture Notes in Math., Vol. 565. MR 0458244
  • [8] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427–455. MR 0118949
  • [9] J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR 0600331
  • [10] Jeffrey Rauch and Michael Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975), 27–59. MR 0377303
  • [11] Leonard Sarason, On weak and strong solutions of boundary value problems, Comm. Pure Appl. Math. 15 (1962), 237–288. MR 0150462
  • [12] Leonard Sarason, Differentiable solutions of symmetrizable and singular symmetric first order systems, Arch. Rational Mech. Anal. 26 (1967), 357–384. MR 0228808
  • [13] David S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1971/72), 1113–1129. MR 0440182
  • [14] M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
  • [15] M. I. Višik and L. A. Ljusternik, The asymptotic behaviour of solutions of linear differential equations with large or quickly changing coefficients and boundary conditions, Russian Math. Surveys 15 (1960), no. 4, 23–91. MR 0124607
  • [16] Calvin H. Wilcox, Wave operators and asymptotic solutions of wave propagation problems of classical physics, Arch. Rational Mech. Anal. 22 (1966), 37–78. MR 0199531
  • [17] J. Rauch, Boundary value problems as limits of problems in all space, Séminaire Goulaouic-Schwartz (1978/1979), École Polytech., Palaiseau, 1979, pp. Exp. No. 3, 17. MR 557514

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35B25, 35F05, 35L40

Retrieve articles in all journals with MSC: 35B25, 35F05, 35L40

Additional Information

Keywords: Singular perturbations, boundary layers, maximal positive boundary value problems
Article copyright: © Copyright 1982 American Mathematical Society