UNIVERSITY OF KANSAS DEPARTMENT OF MATHEMATICS
STUDIES IN EIGENVALUE PROBLEMS
Technical Report 21 CHARACTERISTICS AND ...
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UNIVERSITY OF KANSAS DEPARTMENT OF MATHEMATICS
STUDIES IN EIGENVALUE PROBLEMS
Technical Report 21 CHARACTERISTICS AND CAUCHY PROBLEM FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER by
Jacek Szarski
Paper written under Contract Nonr 58304 with Office of Naval Research Lawrence, Kansas 1959
TABLE OF CONTENTS Introduction ..................................................
1
Chapter I. 1.
Partial differential inequalities of first order............
2.
Uniqueness and continuous dependence on initial
values of the solution of the Cauchy problem .............
2
9
Chapter II. 3.
Characteristic strips of first order .....................
4.
Local existence of the solution of the Cauchy problem for a non-linear equation. . ...... 6 .... 6 ........... .....
5.
13
20
General properties of characteristics and Plis's theorems ......................................
37
Chapter III. 6.
Characteristic strips of second order ...................
54
7.
Existence theorem and estimation of the existence domain
64
8.
Generalized Cauchy problem for a system of partial
differential equations of first order .....................
70
Chapter IV.
9.
Systems of equations with total differentials .............
78
10.
Involutory systems of equations ........................
88
Bibliography .................................................
99
CHARACTERISTICS AND CAUCHY PROBLEM FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER. 1. by
J. Szarski
Introduction.
The present paper deals with the theory of Cauchy
characteristics for the equation of the form zx = f(x, y1, ..., Yn, z, zyl.... , zyn)
(1. 0)
and with the Cauchy) problem for equation (1. 0) and for systems of equa-
tions of the form (2.0)
zi = fi(x, yI, .... Yn, zl! .... zm,izl ,..., zl
(3.0)
zi
(4. 0)
V
yn yl = piv(xI, .... xp, z1, ..., zm ) 0=1120...'M; V=1,2# .... P)
zx = flv (xI, ..., xP, yl, ...)i/n, zl
v-
zm zl ,..., zl yl
yn
)
(i = 1$ 2..., m; v= 1,..., P)
The paper is divided into four chapters. Chapter I deals with partial differential inequalities of first order and with the uniqueness of the solution of the Cauchy problem for equation (1. 0) and for systems (2. 0). The classi-
cal theory of characteristic strips of first order for equation (1. 0) together
with recent results obtained by A. Plis is treated in Chapter II. Characteristic strips of second order for equation (1. 0) and existence theorems for equation (1. 0) and for systems (2. 0) are dealt with in Chapter III. Chapter IV is 1.
Paper written under Contract Nonr 58304 with the Office of Naval Research
concerned with uniqueness and existence theorems for systems (3. 0) and (4. 0).
Chapter I contains published results obtained by T. Wad ewski and
by the author. The first part of Chapter II consists of classical material
and results published by A. Plis constitute the bulk of this chapter. Chapter III contains published results obtained by A. Plie. The first part of Chapter IV is classical the geometrical interpretation of compatibility conditions for systems (3. 0), given by T. Wazewski and the author, is to appear soon in Annales Polonici Mathematici{ the existence theorem for systems (4. 0) has not yet been published.
Numbers in square brackets
refer to bibliography.
CHAPTER I F31.
Partial differential inequalities of first order. We consider a system of partial differential inequalities of first order
is < ft(x, Y, Z, 4Y) where Y =
Yn),
Z=
zm),
(i = 1, 2,..., m)
zY = (zi
,...,
Yl
zl ) yn
inequality contains only derivatives of the i-th function
and the i-th z(x, Y)
We introduce the following Assumptions A.
fi(x, Y, Z, Q), where Q = (ql, ..., qn) are defined for (x, Y)
belonging to the set (2.1)
0<x fi°(x*. Y*. V(x*, Y*), vY (x*, Y*))
1.
The last hypothesis of our theorem is obviously satisfied if we assume that for all indices i and for all points (x*, Y*), where x* > 0, belonging to i° (2.1) u and v ° have first derivatives and possess StolzIs differential in case of (x*, Y*) being a point on the side surface of the pyramid (2.1), and satisfy inequalities (6.1). 1.
-3-
Under these assumptions the inequalities ui(x, Y) < v (x, Y)
(7.1)
(i = 1, 2...., m)
hold in (2.1), Proof.
By (5.1) and by the continuity of ul and v the set of x,
0 < x < a, such that (7.1) holds true in (2. 1) for 0 < x < x is not empty.
Let us denote by x* the least upper bound of such
We have to prove
that x* = a. Suppose it is not true and hence (8.1)
Then there exists an index io'and a point Y* such that (x*, Y*) belongs to (2.1) and (i = 1, 2,..., m), for 0 < x < x*
ui(x, Y) < v (x, Y),
(9.1)
u°(x*, 1`) = v°(x*, Y*)
(10.1)
Now, there are two cases to be distinguished. Case I.
Suppose (x*, Y*) Is an interior point of (2.1). i
i
Then by (10.1) and
by the hypothesis of the theorem u° and v° have first derivatives at (x*, Y*).
Consider the function u °(x*, Y) - v °(x*, Y) depending on the Y.
By (9.1) and (10.1) it attains maximum at Y* and hence, Y* being an in-
terior point, we have (11.1)
uYjo (x*, Y*) = v Yj°(x*, Y*)
(j = 1,..., n)
Similarly the function u°(x, Y*) - v°(x, Y*) depending on x attains its maximum in the interval 0 < x < x* at the point x*. Therefore (12.1)
to
to
ux (x*, Y*) - vX (x*, Y*) > 0
On the other hand, by (4.1), (6.1), (9.1), (10.1), and (11.1) we have successively -4-
fi°(x*, Y# U(x Y*). uY (x*, Y*))
uX (x*, Y*) - vX O(X*, Y#)
oo
Proof. By (8.6) and (10.6) it follows from theorem 4.4 that there
exists a common neighborhood G* in which z(x, Y) and zv(x, Y) are defined.
These solutions are constructed by means of characteristics
in the following way (see theorem 2.4). Denote by -57-
Y*(x, H), zl`(x, H), Q*(x, H) ,
(13.6)
where H = (rill..., nZ), the solution of characteristic equations (5.4) satisfying initial conditions Y*(x*, H) _
z"(x*, H) =
Q"(x", H) = w* (H)
(H),
Functions (13.6) are of class Cl in a neighborhood V*of the point x = x H = Y". Denote similarly by Y*"(x,
(13.6")
Q*'(x, H)
z*v(x, H),
H),
the solution of equations (5.4) with f
replaced by f', satisfying the
initial conditions Y*"(x*,
H) = H,
z" (x", H) = WV (H),
Q*"(x*, H) = tvY(1i)
Functions (13.6') are of class CZ in V*. By (9.6) and (11.6) we have (14.6)
lim D.z*" = D z*, lim D Q*" = DQ*
lim D.Y*V = D
V - co
V -s- m
7
j
"+co j
J
(j = 0, 1)
uniformly in V*. Since 8 y"`
Det (
J
88 rJ i
)
x = x*
we can assume, by (14.6), V" to be so small that *
(15.6)
Det (anj ),
Denote now by (16.6)
respectively by (16.6")
k"
Deta (te) > i
(" = 1, 2.... ) in V*.
the inverse transformations of Y = Y*(x, H) and Y = Y*v(x, H) respectively. H*(x, Y) and H*v(x, Y) are of class Cl and C2 respec-
tively in G* and we may assume G* to be so small that the image of G* by (16.6) and by (16.6v) be contained in V*. By (14.6) and (15.6) we have litn DJH*v. = DJH*
(17.6)
(j = 0,1) uniformly in G*.
V->oo Moreover, the relations
(18.6)
z(x, Y) = z*(x, H*(x, Y)),
zv(x, Y) = z"V(x, H*v(x, H))
(19.6)
zy(x, Y) = Q*(x,H*(x, Y)),
zY(x, Y) = Q*V(x,H*V(x, H))
hold true in G*. From (19.6) it follows that z(x, Y) and zv(x, Y) are of class C2 and C3 respectively in G*. By (14.6), (17.6), (18.6) and (19.6) we get (12.6).
Definition 1. 6. A sequence of functions Y(x), z(x), Q(x), T(x) _ (til(x),..., tnn(x))
satisfying equations (5.4) and (compare (3.6)) n
n
dtij
:' m=1fgkgmtki tmj + k=1 (fgkyi + fgk. qi) tkj
+
n
(20.6)
+
(fgkyj + fgkz gj) tki + fz tij + fyiyj k=1
+ fyiz qj + fyjzgi + fzZ gigj
+
(i, j = 1, 2,,.., n)
is called characteristic strip of second order of equation (1.4). Remark 1. 6.
Equations (5.4) are obviously independent of equations (20.6)
-59-
and can be integrated separately. If we have a solution Y(x), z(x), Q(x)
of (5.4), then, substituting in (20.6) Y = Y(x), z = z(x), Q = Q(x) we get for the unknown functions tij(x) a system of equations whose right
hand members are polynomials of second degree in tij with continuous coefficients depending only on x. Therefore, the solutions of the system
thus obtained are uniquely determined by the initial data.
Theorem 2.6. [7]. Suppose the assumptions B (§4) to be satisfied. Let z(x, Y) be a solution of (1.4) and (2.4), which is of class C2 in a do-
main D containing the characteristic (29.4). Under these assumptions functions (21.6)
Y(x),
ii
tij(x) = zy y (x, Y(x))
z(x), Q(x),
satisfy equations (5.4) and (20.6) in the interval (27.4) and the initial conditions (28.4) and (22.6)
ti,)(x)
= wYiy j(Y) .
Proof. It is sufficient to show that for any x*e (ai, a2) functions (21.6) satisfy system (20.6). To this effect, set w*(Y) = z(x*, Y) .
Y* = Y(x*),
The function w*(Y) is then of class C2 in a neighborhood D* of Y*. f(x,Y,z,Q) is of class C2 in a neighborhood a* of (x*,Y*,w*(Y*),wY(Y*))
and we can assume that for some constant M > 0 we have (23.6)
Dkw* I
<M
(k = 0, 1, 2)
in
D*
(24.6)
IDkfi < M
(k = 0, 1, 2)
in
Sl*
On the other hand, there exist sequences wv(Y) and fv(x, Y, z, Q) of class
-60-
C3 in D* and 11 * respectively such that (9.6) and (11.6) hold true. By (9.6), (11.6), (23.6), and (24.6) we can assume that (8.6) and (10.6)
are satisfied. Thus, all assumptions of lemma 1. 6 are fulfilled and hence, there exists a neighborhood G* in which z(x, Y) and the solutions
zv(x, Y) of (1.4v) and (2.6v) are defined.
zv(x, Y) are of class
C3 in G* and we have (12.6). Now,
z(x, Y) being of class C2 we have by theorem 1. 5 and by
the uniqueness of solutions of (5.4) z(x) = z(x, Y(x)),
(f being of class C2)
Q(x) = zy(x, Y(x))
,
and hence, the functions Y(x), z(x), Q(x) satisfy the conditions (25.6)
Y(x')
Ay(x*,Y*) =w* (Y*)
*, ?(x*) = z(x*,Y*) = w*(Y*), Q(x*)
Denote by Yv(x), zv(x), QV(x) the solution of system (5.4) with f re-
placed by fv, satisfying the conditions Yv(x*) =
(26.6)
Y#0
zv(x*) = wv(y*),
Qv(x*) = wYV (Y*)
These solutions exist in some common neighborhood of x* (independent
on y). Now, zv(x, Y) being of class C3 we have by theorem 1. 6 that the functions (27.6)
YV(xl, zv(x), Qv(x), til(x) = 7V (x, Y''(x)) yiyj
satisfy (20.6v) where (20.6v) is obtained from (20.6) by replacing f by fv. In virtue of (9.6), (11.6), (25.6) and (26.6) we have (28.6)
lim Yv(x) = Y(x),
lim zv(x) = z(x), v -- co
-61-
lim Qv(x) = Q(x)
v-
Co
(29.6)
lim tij(x) = lim zy v
ijy (x, Yv(x)) = zyi jy(x,Y(x)) = tij()
Now, by (11.6), (28.6), and (29.6) the right-hand members of (20.6v) with
the substitution Y = Y'(x), z = zv(x), Q = QV(x), tij = tij(x) tend uniformly to the right-hand members of (20.6) with the substitution Y = Y(x), z = z(x), Q = Q(x), tij = tij(x). Hence it follows that functions (21.6) satisfy (20.6) in a neighborhood of x*. Theorem 3. 6. [7]. Let the assumptions B (94) be satisfied and suppose
that the solution tij(x) of the system (20.6) with the substitution Y = Y(x), z = z(x), Q = Q(x), satisfying initial conditions (22.6) exists in the inter-
val (27.4). Under these assumptions there exists a solution z(x, Y) of
(1. 4) and (2. 4), which is of class C2 in a domain D containing the charac-
teristic (29.4). Proof. (30.6)
In view of theorem 3. 4 it is sufficient to prove that Dot (uij(x) + 0
in the interval al < x < a2 (see the notations introduced in theorem 3.4). Now, since Det (uij(x)) = 1,
inequality (30.6) holds true in some neighborhood of z. Let cl < x < c2
be the largest subinterval of (a,, a2)
n which (30.6) is satisfied. To com-
plete the proof, we have to show that cI = a1, c2 = a2. Suppose for instance that (31.6)
c2 < a2
.
Since all the assumptions of theorem 3.4. are satisfied in the interval
cI < x < c2, there exists a solution z(x, Y) of (1.4) and (2.4) which is of -6z-
class C2 in a domain containing the characteristic
cl <x 0,
of class C1 for arbitrary p and H from some neighborhood of 0. Since for r = 0 we have by (13.9) (15.9)
hi((p, 0) ° 0
,
and the solution h1((p, 0) . is defined for arbitrary
cp,
it follows that for
sufficiently small r > 0 the functions h1(cp, r) are defined and of class C1 in the interval 0
cp
!
Zr.
Let us denote by 3Ci(cp,H,r) the expression in brackets in the righthand side of (12.9).
Then by (13.9) we have
i
h (Z7r, r)
r 2 ?r
Zr
= 1
r7r
S 3C1(cp,H(r,cp), r) dy 0
Since by (15.9) 3C1(cp,H(p,0), 0) = Q1 cos cp - P1 sin cp and consequently 27r
3C1((p'H(9,0), 0) dy 0
we can write -83-
=0
h(2r,r) r
la =
1
_ [C1(cp,H(p,r), r)
- XCi(W,H(p,0). o)] dcp
0
Now, JCi(cp,H(p,r), r) being of class Cl we have
rlim0
(16.9)
r
[3i(p,H(wr),r)
lim hl( r --r 0 r
- 3i((p,H(p,o),o)] = d 3C i(p,H((,,r),r)
r= 0
rZrr dcp
= a J0
r
.
r= 0
But, by the definition of Xi we get m (17.9)
3C1(9,H(p,r),r)
= cos p[di coe x (P
r=0
y
sin
J=1
Oi h)r (cp,0)]
z
m^^
- sing[ PXCoecp+Pyeincp+ L
hr (q),0)]
J-1
On the other hand, by (12.9) dhr(cp,0) =
'ti Cos cp - °Pi sin cp ,
dcp
and hence, since by (13.9), (18.9)
hr(0, 0) - 0,
h=(rp,0) = Qi sin cp + Pi(cos cp - 1)
From (16.9), (17.9) and (18.9) follows (14.9).
We will prove now the following existence theorem. Theorem 3.9. Suppose that Piv(X,Z) are of class Cl and that the compatibility conditions (9.9) hold true in the domain -84-
X(xv
- xv)2 < r
,
vx(-i
- zi)2 < R.
i
V
Assume, furthermore, that (Piv)2 < M
(19.9) i, v
Under these assumptions there exists a unique solution zi(X) (i=1,2,...,m) of (1.9) and (2.9), which is of class C2 in the domain (xv - xv)2 < 6 =min (r,
(20.9)
)
V
Proof. Uniqueness follows from theorem 1.9. To prove the existence, denote by zl = zl(x, A)
the solution of system (5.9) satisfying the initial conditions (21.9)
zi(O, A) = zi
We have obviously (22.9)
zi(x, O) - zi
The right-hand members of (5.9) are of class C1 for (23.9)
0 < x < aA ,
yiw-
i
and (24.9)
L( v)2
1.
()Iv)2
V
We will prove that for A = ()L1...., Ap) satisfying inequality (24.9) the functions 21(x, A) are defined in the interval
0<x 1, Suppose the contrary, then
PA < 1