Symmetric positive systems with boundary characteristic of constant multiplicity
Author:
Jeffrey Rauch
Journal:
Trans. Amer. Math. Soc. 291 (1985), 167187
MSC:
Primary 35L50
MathSciNet review:
797053
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Abstract: The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.
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 [1]
 R. Agemi, The initial boundary value problem for inviscid barotropic fluid motion, Hokkaido Math. J. 10 (1981), 186182. MR 616950 (82j:35124)
 [2]
 C. Bardos and J. Rauch, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc. 270 (1982), 377408. MR 645322 (84j:35012)
 [3]
 D. Ebin, The initial boundary value problem for subsonic fluid motion, Comm. Pure Appl. Math. 32 (1979), 119. MR 508916 (80i:76026)
 [4]
 K. O. Friedrichs, The identity of weak and strong extension of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132151. MR 0009701 (5:188b)
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 ChoaHao Gu, Differentiable solutions of symmetric positive partial equations, Chinese J. Math 5 (1964), 541545.
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 L. Hormander, Linear partial differential operators, SpringerVerlag, Berlin, 1963. MR 0404822 (53:8622)
 [9]
 P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427454. MR 0118949 (22:9718)
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 J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, SpringerVerlag, Berlin and New York, 1972.
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 R. Moyer, On the nonidentity of weak and strong extensions of differential operators, Proc. Amer. Math. Soc. 19 (1968), 487488. MR 0222720 (36:5770)
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 T. Nishida and J. Rauch, Local existence for smooth inviscid compressible flows in bounded domains (to appear).
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 S. Osher, An illposed problem for a hyperbolic equation near a corner, Bull, Amer. Math. Soc. 79 (1973), 10431044. MR 0350211 (50:2704)
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 G. Peyser, On the differentiability of solutions of symmetric hyperbolic systems, Proc. Amer. Math. Soc. 14 (1963), 963969. MR 0156102 (27:6034)
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 L. Sarason, Differentiable solutions of symmetrizable and singular symmetric first order systems, Arch. Rational Mech. Anal. 26 (1967), 357384. MR 0228808 (37:4387)
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 , Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 249276. MR 0104919 (21:3669)
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 R. S. Phillips and L. Sarason, Singular symmetric positive first order differential operators, J. Math. Mech. 15 (1966), 235272. MR 0186902 (32:4357)
 [21]
 J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initialboundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303318. MR 0340832 (49:5582)
 [22]
 S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, preprint. MR 834481 (87f:76085)
 [23]
 D. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1972), 11131129. MR 0440182 (55:13061)
 [24]
 M. Tsuji, Analyticity of solutions of hyperbolic mixed problems, J. Math. Kyoto Univ. 13 (1973), 323371. MR 0320540 (47:9077)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507970534
PII:
S 00029947(1985)07970534
Keywords:
Symmetric positive boundary value problem,
characteristic boundary,
dissipative boundary conditions,
mixed initial boundary value problem
Article copyright:
© Copyright 1985 American Mathematical Society
