×

zbMATH — the first resource for mathematics

Symmetric positive systems with boundary characteristic of constant multiplicity. (English) Zbl 0549.35099
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 neighbourhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.

MSC:
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A30 Geometric theory, characteristics, transformations in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Rentaro Agemi, The initial-boundary value problem for inviscid barotropic fluid motion, Hokkaido Math. J. 10 (1981), no. 1, 156 – 182. · Zbl 0472.76065 · doi:10.14492/hokmj/1381758108 · doi.org
[2] Claude Bardos and Jeffrey Rauch, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc. 270 (1982), no. 2, 377 – 408. · Zbl 0485.35010
[3] David G. Ebin, The initial-boundary value problem for subsonic fluid motion, Comm. Pure Appl. Math. 32 (1979), no. 1, 1 – 19. · Zbl 0378.76043 · doi:10.1002/cpa.3160320102 · doi.org
[4] K. O. Friedrichs, The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc. 55 (1944), 132 – 151. · Zbl 0061.26201
[5] K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345 – 392. · Zbl 0059.08902 · doi:10.1002/cpa.3160070206 · doi.org
[6] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333 – 418. · Zbl 0083.31802 · doi:10.1002/cpa.3160110306 · doi.org
[7] Choa-Hao Gu, Differentiable solutions of symmetric positive partial equations, Chinese J. Math 5 (1964), 541-545.
[8] Lars Hörmander, Linear partial differential operators, Springer Verlag, Berlin-New York, 1976. · Zbl 0321.35001
[9] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427 – 455. · Zbl 0094.07502 · doi:10.1002/cpa.3160130307 · doi.org
[10] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, Berlin and New York, 1972. · Zbl 0223.35039
[11] Andrew Majda and Stanley Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), no. 5, 607 – 675. · Zbl 0314.35061 · doi:10.1002/cpa.3160280504 · doi.org
[12] Robert D. Moyer, On the nonidentity of weak and strong extensions of differential operators, Proc. Amer. Math. Soc. 19 (1968), 487 – 488. · Zbl 0173.13204
[13] T. Nishida and J. Rauch, Local existence for smooth inviscid compressible flows in bounded domains (to appear).
[14] Stanley Osher, An ill posed problem for a hyperbolic equation near a corner, Bull. Amer. Math. Soc. 79 (1973), 1043 – 1044. · Zbl 0268.35064
[15] Gideon Peyser, On the differentiability of solutions of symmetric hyperbolic systems, Proc. Amer. Math. Soc. 14 (1963), 963 – 969. · Zbl 0171.07203
[16] Leonard Sarason, Differentiable solutions of symmetrizable and singular symmetric first order systems, Arch. Rational Mech. Anal. 26 (1967), 357 – 384. · Zbl 0162.40901 · doi:10.1007/BF00281640 · doi.org
[17] Leonard Sarason, On weak and strong solutions of boundary value problems, Comm. Pure Appl. Math. 15 (1962), 237 – 288. · Zbl 0139.28302 · doi:10.1002/cpa.3160150301 · doi.org
[18] R. S. Phillips, Dissipative hyperbolic systems, Trans. Amer. Math. Soc. 86 (1957), 109 – 173. · Zbl 0081.31102
[19] R. S. Phillips, Dissipative operators and hyperbolic systems of partial differential equations, Trans. Amer. Math. Soc. 90 (1959), 193 – 254. · Zbl 0093.10001
[20] R. S. Phillips and Leonard Sarason, Singular symmetric positive first order differential operators, J. Math. Mech. 15 (1966), 235 – 271. · Zbl 0141.28701
[21] Jeffrey B. Rauch and Frank J. Massey III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303 – 318. · Zbl 0282.35014
[22] Steve Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), no. 1, 49 – 75. · Zbl 0612.76082
[23] David S. Tartakoff, Regularity of solutions to boundary value problems for first order systems, Indiana Univ. Math. J. 21 (1971/72), 1113 – 1129. · Zbl 0235.35019 · doi:10.1512/iumj.1972.21.21089 · doi.org
[24] Mikio Tsuji, Analyticity of solutions of hyperbolic mixed problems, J. Math. Kyoto Univ. 13 (1973), 323 – 371. · Zbl 0264.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.