A Boundary Value Problem for Quasilinear Hyperbolic Systems In
Total Page：16
File Type：pdf, Size：1020Kb
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze LAMBERTO CESARI A boundary value problem for quasilinear hyperbolic systems in the Schauder canonic form Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 1, no 3-4 (1974), p. 311-358 <http://www.numdam.org/item?id=ASNSP_1974_4_1_3-4_311_0> © Scuola Normale Superiore, Pisa, 1974, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A Boundary Value Problem for Quasilinear Hyperbolic Systems in the Schautder Canonic Form (*). LAMBERTO CESARI (**) 1. - Introduction. In the present paper we take into consideration the following Schauder canonic form of quasilinear hyperbolic systems in a slab 1. = ~x ~0 c x c a] . Thus, whenever the m X m matrix [Aii] is the identity matrix, system (1.1) reduces to the Lax-Courant ca- nonic form Instead of usual Cauchy data at x = 0, we shall take into considera- tion here more general types of boundary data (I, II, III below). I. For instance, we may assume that certain functions y E .Err i = 1, ... , m, and an integer m’, 0 c m’ m, are assigned, and we may re- (*) This research was partially supported by AFOSR Research Project 71-2122 at the University of Michigan. (**) University of Michigan, Ann Arbor. Pervenuto alla Redazione il 24 Gennaio 1974 e in forma definitiva il 5 Otto- bre 1974. 312 quest that For m’= m (as well as for m’= 0) we have the usual Cauchy problem. we assume II. More generally,y may that certain numbers ~0~~ and functions i = 1, - - ., m, are assigned, and we may request that III. In a more general setting, we may assume that certain numbers ai, 0 c az c a, functions ~~ (y), YEEr, i = 1, ... , m, and an mXm matrix [bi3(y), E are det and we that il j:--:: 1, ..., m],9 y Er, assigned, (bii) :A 0, may request If is the identity matrix, then this boundary condition III reduces = = = a = to II. If furthermorel ai 0 f or i 1~ ..., m’, ai f or i m’ -~- 1, ..., m, then we have problem I. In the present paper we prove a theorem of existence, uniqueness, and continuous dependence upon the data, for Schauder hyperbolic systems (1.1) with boundary conditions III when both the matrix [Aij] and the matrix have « dominant » main diagonal. Thus, problems I and II the identity matrix) for system (1.2) ([Aii] the identity matrix) are always included. In § 2 we give a new proof with needed estimates of the existence theorem for the Cauchy problem for Schauder’s system (1.1), proof based on Banach’s fixed point theorem (see [7, 8] for a previous proof). In §3 we then prove the existence theorem for system (1.1) with boundary condi- tions III (thus, including boundary conditions I and II). The proof is also based on Banach’s fixed point theorem, and on the precise estimates obtained in § 2. We proved a slightly simpler theorem in [1, 3] for systems (1.2) with boundary conditions III (problems I and II being always included). When the « dominant main diagonal condition » is not satisfied, the conclusions of the same theorems may not hold, as simple counterexamples show [2]. Since we obtain the solution as the fixed point of transformations which are contractions in the uniform topology, the usual iterative method is uniformly convergent to the unique solution. 313 The boundary value problems under consideration, in the present gen- erality, are new. However, problem I, for very particular systems, was considered by O. Niccoletti [11 ], and aspects of these problems were discussed anew later by different authors (see e.g. [12-21]). Leaving aside Goursat problems and analogous ones, let us mention here that boundary value problems for linear symmetric systems have been studied by Friedrichs [9] and Sarason [13]. Finally, various periodicity requirements as boundary value problems for canonic forms of nonlinear hyperbolic systems in the plane, including the wave equation, have been stu- died by a number of authors, in particular by Cesari [5] and Hale [10] making use of alternative methods (see these two papers for further references). 2. - The main existence theorem for the Cauchy problem. We consider here quasilinear hyperbolic systems of the Schauder canonic form. Thus, x is a scalar, y = ( yl , ... , yr ) is an r-vector, and z(x, y ) = - (z1, ..., z~) is the m-vector of unknown functions zi(x, Y1’ ..., Y.), i=1, ..., m. We denote by jyj = Max, BYkB the norm of y in Er and by lzl = magi Izil the norm of z in Em. We consider first the Cauchy problem for the differential system in an infinite strip with initial data THEOREM I (an existence theorem for the Cauchy Problem (2.1), (2.2)). Let Ia denote an interval and, if is a given constant, let S~ also denote the interval [- Q, Q]m c Em. Let = be continuous functions on Ai j(x, y, z), 1, ..., m, Ia. ao > 0, with in for some constant ,u, and let us assume that there are constants C ~ 0 and a function E L1[0, ao], such that, for all (x, y, z), (x, y, z), (x, y, z) E Iae X Er = we and all i, j 1, ..., m, have 21 - Annali della Scuola lVorm. Sup. di Pisa 314 = Let y, z), Y7 0)7 ~ 1, ..., m, k = 1, ..., r, be functions defined in Ia.xErxQ, measurable in x for every (y, z), continuous in (y, z) for every x, and let us assume that there are nonnegative functions m(x), I(x), n(x), E all E 0 x ao,7 m, l, n , II .Ll [o, ao],7 such that, for (x, y, z), (x, y, z ) we have E I a X Er X S2, i = 1, ..., m, k = 1, ..., r, Let y E Er, i = 1, ..., m, be given functions continuous in Er, and let us assume that there are constants c~, A, 0 o c~ C S~, ~l ~ 0, such that, for all and i == 1, ..., m, we have Then, for a sufficiently small, 0 there are a constant Q > 0, a function x(x) E .Ll[o, a], and functions z(x, y) = z(x, yl, all E ..., yr) =(zl, ... zm), continuous in such that for (x, y), (x, y), (x, y) C-IaxEr9 and i = 1, ... , m, we have satisfying (2.2) everywhere in Er and satisfying (2.1) a.e. in Da . Further- more, z(x, y) is unique and depends continuously on (qJ1’ ..., qJm) in the classes which are described in the proof below. PROOF. The proof is divided into parts (a),..., (g). (a) Choice of constants p, Q, f unction X, and estimates for a. As usual we denote by otij the cofactor of Åij in the m x m matrix divided by Since relations (2.3-5) yield analogous relations for the elements (Xij. Thus, there are constants C’, and a 0, such that for all (xy,z), (x, y, z), and i, j = 1, ... , m, we have 315 Note that the functions z), y, z) are absolutely continuous in each single variable x, yh, zs with Analogously, the functions (!ik(X, y, z), y, z) are absolutely continuous in each yh and in each zs with For every we define the following constants: Let us choose constants Let us take and, for every a, We first can choose4 sufficiently small so that 316 and we denote by A the constant 2 - (I - L,(l + Q))-1, so that 1 and certainly We shall have to impose on a further limitations from above. Though this could well be done at this stage, we prefer to mention the further restrictions on the size of a as need comes in the course of the argument. (b) The classes ~o and ~1. We denote by Da and 1,, the regions Let Ko be the set of all systems of continuous functions gik in L1a satisfying the following conditions Thus, each function gik is absolutely continuous in ~ for every (x, y), and we have the set of all systems Then relations (2.21-23) become 317 Thus, fori we have that is, the functions hik are uniformly bounded in Also Finally, for the r-vector functions we also have Note that jlo is a subset of the Banach spaceI with norm We also consider the set Ki of all systems of continuous bounded functions zi in Da satisfying the following conditions E = m. for all (x, y), (x, 9), (x, y) Da, i 1, ..., Thus, each zi is absolutely continuous in z for every y, Lipschitzian in y for every x, and we have 318 we also have for all (x, y), (x, y), (~, y) E Da . Here, is a subset of the Banach space with norm (c) The transformation Tz. For every fixed ZEX1, let us consider the transformation T~ defined on Ko, say G = Txg, or - [Gi.], by taking Note that the functions G ik are obviously continuous, and that for all used here inequalities 319 By comparison of (2.34-36) with (2.21-23) we conclude that G = belongs to Ko .