Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Singular perturbations for systems of linear partial differential equations


Authors: A. Livne and Z. Schuss
Journal: Trans. Amer. Math. Soc. 190 (1974), 335-343
MSC: Primary 35B25
DOI: https://doi.org/10.1090/S0002-9947-1974-0340780-6
MathSciNet review: 0340780
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the system of linear partial differential equations $ \varepsilon {A^{ij}}u_{ij}^\varepsilon + {B^i}u_i^\varepsilon + C{u^\varepsilon } = f$ where $ {A^{ij}},{B^i}$ are symmetric $ m \times m$ matrices and -- C is a sufficiently large positive definite matrix. We prove that under suitable conditions $ {\left\Vert {{u^\varepsilon } - u} \right\Vert _{{L^2}}} \leq c\surd \varepsilon {\left\Vert f \right\Vert _{{H^1}}}$ where u is the solution of a suitable boundary value problem for the system $ {B^i}{u_i} + Cu = f$.


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

  • [1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] C. Bardos, D. Brezis and H. Brezis, Perturbations singulières et prolongements maximaux d'opérateurs positifs, Arch. Rational Mech. Anal. (to appear).
  • [3] A. Friedman, Singular perturbations for partial differential equations, Arch. Rational Mech. Anal. 29 (1968), 289-303. MR 37 #1754. MR 0226164 (37:1754)
  • [4] K. O. Friedrichs and P. D. Lax, Boundary value problems for first order operators, Comm. Pure Appl. Math. 18 (1965), 355-388. MR 30 #5186. MR 0174999 (30:5186)
  • [5] W. M. Greenlee, Rate of convergence in singular perturbations, Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 2, 135-191. MR 39 #3133. MR 0241795 (39:3133)
  • [6] S. Kamenomostskaya, The first boundary problem for equations of elliptic type with a small parameter with the highest derivatives, Izv. Akad. Nauk SSSR Ser. Mat. 19 (1955), 345-360. (Russian) MR 17, 627. MR 0074667 (17:627a)
  • [7] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443-492. MR 31 #6041. MR 0181815 (31:6041)
  • [8] -, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797-872. MR 38 #2437. MR 0234118 (38:2437)
  • [9] O. A. Ladyženskaja, On equations with small parameter in the higher derivatives in linear partial differential equations, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 12 (1957), no. 7, 104-120. (Russian) MR 19, 656. MR 0089341 (19:656b)
  • [10] 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 22 #9718. MR 0118949 (22:9718)
  • [11] N. Levinson, The first boundary value problem for $ \varepsilon \Delta u + A(x,y){u_x} + B(x,y){u_y} + C(x,y)u = D(x,y)$ for small $ \varepsilon $, Ann. of Math. (2) 51 (1950), 428-445. MR 11, 439. MR 0033433 (11:439k)
  • [12] A. Livne and Z. Schuss, Singular perturbations for degenerate elliptic equations of second order, Arch. Rational Mech. Anal. (to appear). MR 0336021 (49:797)
  • [13] O. A. Oleinik, On equations of elliptic type with a small parameter in the highest derivatives, Mat. Sb. 31 (73) (1952), 104-117. (Russian) MR 14, 560. MR 0052012 (14:560b)
  • [14] S. V. Sivašinskiĭ, The introduction of 'viscosity' into first order linear symmetric systems, Vestnik Leningrad. Univ. 25 (1970), no. 19, 54-57. (Russian) MR 44 #593. MR 0283361 (44:593)
  • [15] M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk 12 (1957), no. 5 (77), 3-122; English transl., Amer. Math. Soc. Transl. (2) 20 (1962), 239-364. MR 20 #2539; 25 #322.

Similar Articles

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

Retrieve articles in all journals with MSC: 35B25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0340780-6
Keywords: First order system, singular perturbations, rate of convergence
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society