Symmetric positive systems with boundary characteristic of constant multiplicity

Author:
Jeffrey Rauch

Journal:
Trans. Amer. Math. Soc. **291** (1985), 167-187

MSC:
Primary 35L50

DOI:
https://doi.org/10.1090/S0002-9947-1985-0797053-4

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.

**[1]**R. Agemi,*The initial boundary value problem for inviscid barotropic fluid motion*, Hokkaido Math. J.**10**(1981), 186-182. 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), 377-408. MR**645322 (84j:35012)****[3]**D. Ebin,*The initial boundary value problem for subsonic fluid motion*, Comm. Pure Appl. Math.**32**(1979), 1-19. MR**508916 (80i:76026)****[4]**K. O. Friedrichs,*The identity of weak and strong extension of differential operators*, Trans. Amer. Math. Soc.**55**(1944), 132-151. MR**0009701 (5:188b)****[5]**-,*Symmetric hyperbolic linear differential equations*, Comm. Pure Appl. Math.**7**(1954), 345-392. MR**0062932 (16:44c)****[6]**-,*Symmetric positive linear differential equations*, Comm. Pure Appl. Math.**11**(1958), 333-418. MR**0100718 (20:7147)****[7]**Choa-Hao Gu,*Differentiable solutions of symmetric positive partial equations*, Chinese J. Math**5**(1964), 541-545.**[8]**L. Hormander,*Linear partial differential operators*, Springer-Verlag, 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), 427-454. MR**0118949 (22:9718)****[10]**J. L. Lions and E. Magenes,*Non-homogeneous boundary value problems and applications*, Springer-Verlag, Berlin and New York, 1972.**[11]**A. Majda and S. Osher,*Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary*, Comm. Pure Appl. Math.**28**(1975), 607-675. MR**0410107 (53:13857)****[12]**R. Moyer,*On the nonidentity of weak and strong extensions of differential operators*, Proc. Amer. Math. Soc.**19**(1968), 487-488. MR**0222720 (36:5770)****[13]**T. Nishida and J. Rauch,*Local existence for smooth inviscid compressible flows in bounded domains*(to appear).**[14]**S. Osher,*An ill-posed problem for a hyperbolic equation near a corner*, Bull, Amer. Math. Soc.**79**(1973), 1043-1044. MR**0350211 (50:2704)****[15]**G. Peyser,*On the differentiability of solutions of symmetric hyperbolic systems*, Proc. Amer. Math. Soc.**14**(1963), 963-969. MR**0156102 (27:6034)****[16]**L. Sarason,*Differentiable solutions of symmetrizable and singular symmetric first order systems*, Arch. Rational Mech. Anal.**26**(1967), 357-384. MR**0228808 (37:4387)****[17]**-,*On weak and strong solutions of boundary value problems*, Comm. Pure Appl. Math.**15**(1962), 237-288. MR**0150462 (27:460)****[18]**R. S. Phillips,*Dissipative hyperbolic systems*, Trans. Amer. Math. Soc.**86**(1957), 109-173. MR**0090748 (19:863d)****[19]**-,*Dissipative operators and hyperbolic systems of partial differential equations*, Trans. Amer. Math. Soc.**90**(1959), 249-276. MR**0104919 (21:3669)****[20]**R. S. Phillips and L. Sarason,*Singular symmetric positive first order differential operators*, J. Math. Mech.**15**(1966), 235-272. MR**0186902 (32:4357)****[21]**J. Rauch and F. Massey,*Differentiability of solutions to hyperbolic initial-boundary value problems*, Trans. Amer. Math. Soc.**189**(1974), 303-318. 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), 1113-1129. MR**0440182 (55:13061)****[24]**M. Tsuji,*Analyticity of solutions of hyperbolic mixed problems*, J. Math. Kyoto Univ.**13**(1973), 323-371. MR**0320540 (47:9077)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0797053-4

Keywords:
Symmetric positive boundary value problem,
characteristic boundary,
dissipative boundary conditions,
mixed initial boundary value problem

Article copyright:
© Copyright 1985
American Mathematical Society