Actes, Congrès intern. Math., 1970, Tome 2, p. 771 à 777.
ON LINEAR SECOND ORDER EQUATIONS
WITH NON-NEGATIVE CHARACTERISTIC FORM
by
O.A.
OLEINIK
The equation
(1)
L(u) = ak/(x)uXkXi
+ bh(x)uXh + c(x)u = f(x)
is called a second order equation with non-negative characteristic form at the set
G, if at every point x belonging to G, ahf (x) £k fy > 0 for all f = (£ 1 ,. .., fm).
(Here and everywhere in the paper summation with respect to repeated indices
is carried out from 1 to m).
Interest in equations, which do not preserve their type in a considered domain,
has existed for a long time, principally after the results of F. Tricomi [1]. The
works of M.V. KeldyS [2] and G. Fichera [3] were the beginning of a long series
of papers about the second order equations with non-negative characteristic form.
Numerous results have been obtained for a special class of equations (1), namely
for second order equations, which are elliptic in a domain and degenerate on its
boundary (see, for example, [4], [5]). General second order equations with nonnegative characteristic form are investigated in the papers [3], [6] — [18] and others.
Let us consider equation (1) in a domain £2 with boundary 2 and suppose for
simplicity that the coefficients of equation (1), the function / and the boundary
2 are sufficiently smooth. In the paper [3] the following boundary value problem
for equation (1) was posed. Let n = (nx ,. . . , nm) be the interior normal vector
to the boundary 2 of £2. We denote by 2° the set of points 2, where akinknj = 0.
At the points of 2° we consider the function
(2)
b = (bk -aV)nk
.
We denote by 2 0 , 2 , , 2 2 the sets of points of 2° where b = 0, b>0,
b<0
respectively. The set of points of 2 where ak}nknj > 0 is denoted by 2 3 .
The first boundary value problem is to find a function u(x) such that
(3)
L(u)=f
in
ft,
(4)
u = g on 2 2 U 2 3
where / and g are given functions.
The function u (x) is called a weak solution of the first boundary value problem
(3), (4) if for any v C C(2)(ft U 2), which is equal to zero on 2 t U 2 3 the integral
identity
(5)
f uL*(v) dx= f vfdx - f g |^- da + f bgvdo
dn
Ja
«'s, by
^E.
772
O.A. OLEINIK
D 10
b
..
b
holds where — = aK1 nk — , do is the area element on the surface 2 ,
L*(v) s a»vXkXi
+ ***v, fc + c*v,
*** = 2akJf - bk,
c* = «* Ä/ - bkXJç + c .
Suppose that in the neighbourhood of a point x which belongs to the boundary 2 of the domain ft, the set 2 is given by the equation
9(x1 ,.. . ,xm) = 0 , grad 9 # 0 , 9>0
in ft
and consider on 2 the function ß = L(W). Let F° be the boundary of 2 2 U 2 0
on 2 .
THEOREM 1. - Suppose that c(x)<
ternal points of
— c0 = const < 0 in ft, ß < 0 at the in-
22 U 20 ; / C £.(«)
and
g C ^ ( 2 2 U 2 3 ).
Then there exists a weak solution u(x) of problem (3), (4) which belongs to A,o(ft)
and satisfies the inequality (maximum principle) :
(6)
l f\
\u\ < max (sup — > sup \g\ j
/ / ZM addition c*(x) <-cx=
const < 0 in ft, /(x) C £„(ft), K p < «>, g = 0,
*7ze« f/zere existe a weak solution u(x) of the problem (3), (4) which belongs
to £ p (ft).
A weak solution u(x) of problem (3), (4) can be obtained by the régularisation
method which means that u (x) can be obtained as a limit as e -> 0 and N -* °° of
a sequence u%(x) of solutions of the elliptic equations
(7)
eAu + L(u)=fN ,
e = const > 0 ,
with the boundary condition u = gN on 2, where fN and gN are sequences of
smooth functions which converge to / and g as N -> «> in ft and on 2 2 U 2 3
respectively. One can find numerous applications of the régularisation method
in the books [19], [20].
THEOREM 2. - Suppose that c* < - cx = const < 0 in_ ft, ß* = L*(&) < 0
at the points of 2j U 2 0 and also in a neighbourhood of 2 3 n (2j U 2 0 ) o« 2.
Assume that the coefficients a * of equation (3) CU« òe extended to a neighbourhood of 2 2 wzz7z zVze .same smoothness as in ft û«d wzz7z fl*'£fc £;- > 0 /or
Ç £ Rm. Let T te the boundary of 2 2 OH 2 and T = T1 U T2, w/zere Tj û a
/zrate number of smooth (m — \)-dimensional manifolds and the area of a
8-neighbourhood of T2 on 2 has order 8q ,q> 2. Then a weak solution of
problem (3), (4) is unique in the class of functions which belong to ß2 in ft and
also to ß3 in a neighbourhood ofTl.
A weak solution u(x) of problem (3), (4) is unique in the class ^ ( f t ) , if the
area of the 6-neighbourhood of Y on 2 as order dq , q > 0.
Examples show that under the conditions of Theorem 2 the class of functions
ON LINEAR SECOND ORDER EQUATIONS
773
for the uniqueness of a weak solution of problem (3), (4) can not be enlarged.
A weak solution of problem (3), (4) can be non-unique in the class -£p(ft) for
p < 3, if Tj is not empty, and in the class ^ p (ft) for p < 2, if F2 is not empty.
One can construct such examples by considering of the heat equation ut = Au in
a domain such that its boundary coincides in a neighbourhood of the origin
with the surface t = \x\2+e, e = const > 0 and by taking g equal to a fundamental solution of this equation with a singularity at the origin.
In the paper [14] an example of problem (3), (4) is constructed which shows
that a weak solution may be non-unique in the class of bounded mesurable functions if the area of the 6-neighbourhood of T on 2 does not tend to zero as
ô -* 0.
Using the elliptic régularisation one can prove a uniqueness theorem similar to
that of R. Phillips and L. Sarason [14], who used methods of the theory of symmetrical systems.
The question arises, under which conditions a weak solution of problem (3),
(4) is a smooth function in the closed domain ft U 2. The same question can also
be asked with regard to smoothness in a neighbourhood of any given point of ft.
The last question is also connected with the problem of finding conditions
under which equation (1) is hypoelliptic.
For equations of the form
N
(8)
L(u) = - £ Xfu + iXQu + cu
=f,
b
where Xf(x, (0) = aj(x) (De, j = 0,1 ,. . . ,N ; (De = — i - — ; and coefficients
oxe
aej(x), c(x) are real functions in C°°(ft), sufficient conditions for thehypoellipticity
were given by L. Hörmander [15], using the Lie algebra theory. It is easy to
show that there exist equations (1) with real coefficients in C°° which can not
be written in the form (8). Using the theory of pseudodifferential operators one
can prove the following results for equations (8) and also for general equations (1).
Let us introduce some notations. Consider the system of operators
(XQ , Xj , . . . ,XN)
defined by equation (8). For any multi-index 3 = (ax , . . . , ak) where ae are
integers in the range 0 to N, we set
1*1 = S K
e=\
with \e = 1, if ae = 1 , . . . , N, and \e = 2, if ae = 0, We define the operator
X =adXai...adXakiXak
.
Here, as usually, ad AB = AB - BA for any operators A, B ; <&s is a space of
distributions in S'(Rm), for which
774
O.A. OLEINIK
ll«ll2=r
D 10
(l + l£l 2 /lw«)l 2 d£<°°,
sCR1 ,
JRm
«(£) is the Fourier transform of u(x). Let us denote by M a set of points which
is contained in CTI, where CT6 is a finite set of closed (m — l)-dimensional, smooth
manifolds and Olle ft.
THEOREM 3. — Suppose that for every point x0 C ft \M there exists an integer
R(x0) such that
(9)
I
\X,(xo,Ì)\>0
for
|È|=ÉO
|j|<Ä(x0)
w/zere Xa (x, £) is a symbol of the operator Xa. Suppose that at every
x0CM
(10)
£ |fli*-l + HWI^o
vv/zere &(x1 , . . . 9 xm)
point x0 with grad $
z/ Z,(w)CC°°(ft) û«d
and y>L(u) C 8CS for
inequality is valid :
(11)
point
= Q is an equation for Wl in a neighbourhood of the
¥= 0. Then equation (8) zs hypoelliptic, that is u C C°°(ft),
M C GO '(ft). /« addition, if u is a distribution in ö)'(ft)
any function <p C C^(ft), f/zew pu C 06^ awe? f/ze following
\\vu\\2<C{\\viLJu)\\1s
+ || V l n|Ç}
w/zere <p, <Pj C CQ (ft), </?, = 1 o« z7ze support of \p and either supp $C\M = 0or
<p = 1 on M ; y = const < s, C z's a constant dependant on y and <p, <Pj.
Let us write equation (1) in the form
(12)
L(u) = - ®j(akf(x) ®ku) + zßw + cu = /
where QM = (ò* — ax[)(Dku.
Suppose that
akK bk, c C C~(ft), L°(x , £) = A * ) É* t,, L0(f>, L°U)
for / = 1 , . . . , m are differential operators with the symbols bL°/b%j, bL°/bXj respectively ; ê_j is a pseudodifferential operator with the symbol <p(x) (1 + If l 2 )~ 1/2
where <p C C£(ft), <pQt) > 0, y(x) = 1 on a compact set K C ft.
Consider the system of operators{Q 0 , Qx,. . . , Q2m }where Q0 = Q, Q} = L°u)
for
for / = 1 ,. . . , m and Q7- = &_! ^\]-m)
i = m + I ,. . ., 2m. For any
multi-index 3 = (al ,. . . , ak) where ae are integers in the range 0 to 2m, we
set
w\ = i \
e=l
with Xc = 1, if ae = 1 , . . . , 2m, and \ e = 2, if a e = 0. For every 3 we define
the operator
Q3=adQax...
adQak xQak .
According to the theory of pseudodifferential operators, the operator Q3 has
the form
ON LINEAR SECOND ORDER EQUATIONS
QÖ=Q\+T
775
,
where the operator Tö has order less or equal to zero and Q° is a pseudodifferential
operator of the first order with a symbol q#(x, £).
THEOREM 4. - Suppose that for any compact set K C ft \ M there exists a
number R (K) and a positive constant C(K) such that the inequality
1+
2
\q°(x,0\2>C(K)(l
+ |£|2)
holds for all xC K and £ C Rm. Suppose that at every point xQCM
akj(x)$x*x+\L(*)\J=0.
(13)
K
J
Then equation (1) is hypoelliptic /«ft, that is u C C°°(ft), if L(u)C C°°(Çl)and
u Co)'(ft). In addition, if u is a distribution in a)'(ft) and ipL(u) C &£s for any
function \p C C£(ft), then yu C %ZS a«c? r/ze inequality (11) fto/cfr.
The proofs of theorems 1-4 are given in [21], (see also [11], [16], [17]).
We note that estimate (11) is also valid when the coefficients of equations (8)
and (12) are sufficiently smooth.
For equations (1) with analytic coefficients and with
m
£
(\akk\ + |Z>*|)^0
in ft,
k=l
the necessary and sufficient condition for the hypoellipticity is given in [25].
For equations of the form (8) such a theorem is proved by M. Derridj.
Results about the smoothness of weak solutions of problem (3), (4) in the
closed domain ftU2 are obtained in [10] - [13], [21]. We can not formulate
all these results here, but we note the following case. Let CM(ft) be the class of
functions with bounded derivatives in ft up to the order p. If we suppose that
the coefficients of equation (3), can be extended outside of ft with the same
smoothness and with the condition ak* %k fy > 0, then for the existence of a solution u(x) of problem (3), (4) in the class CM(ft), it is sufficient, for example,
to require that the coefficients of (3) and / belong to CM(ft), the boundary 2
and the function g are sufficiently smooth, the intersection of any two of the
sets 2 3 , 2 2 , 2 0 U 2! is empty, and an inequality between c(x) and bk(x), aki (x)
and their derivatives of the first and second orders is fulfiled at the points where
det ||flfc'|| = 0. (This inequality is satisfied, for example, if c < — c0 = const < 0
and c0 is sufficiently large, (see [10], [12]).
Examples show that all these conditions are essential. The solution u(x) in
this case can be obtained as a limit as e -• 0 of the solutions ue(x) of a boundary
value problem for elliptic equations of the form (7) in a domain which contains ft.
The case of an intersection of 2 3 and 2 2 is considered in [13]. The smoothness
of solutions of problem (3), (4) without the assumption about the extention of
the coefficients is investigated in [13] and [21].
776
0A
- OLEINIK
D 10
Second order equations with non-negative characteristic form appear in boundary
layer theory, the theory of filtration, in problems of Brownian motion, the theory
of probability and in other cases. Problem (3), (4) was also studied by the methods
of the theory of probability using K. Ito's stochastic equations (see, for example,
[22]).
For second order equations with non-negative characteristic form there are
many open problems. We mention some of them.
The structure of spectrum of problem (3), (4) has not been studied, also conditions for a finite index and for the Fredholm properties has not been found.
It is of interest to investigate in more detail conditions for smoothness and for
non-smoothness of weak solutions of problem (3), (4) and to find out what kind
of singularities exists for weak solutions of problem (3), (4) and under which
conditions they arise. (Let us notice that the smoothness of weak solutions of
problem (3), (4) has not been completely investigated even for the heat equation.
This question is discussed in detail in [13].
An open problem is to find the classes of equations (1) with analytic coefficients and with analytic functions / which have only analytic solutions in ft.
It is also of interest to describe well posed boundary value problems for equations (1).
It is very important to study quasilinear second order equations with nonnegative characteristic form. Such equations arise in boundary layer theory,
in gas dynamic problems and in other cases of physical importance (see [23], [24]).
REFERENCES
[1] TRICOMI F. — Sulle equazioni lineari alle derivate parziali di 2 ordinate di tipo
misto, Memorie della R. Accademia Nazionale dei Lincei, serie V, 14, 1923,
Fase. 7.
[2] KELDYS M.V. — On certain cases of degeneration of elliptic type equations on the
boundary of a domain, Dokl. Akad. Nauk S.S.S.R., 77, 1951, p. 181-183.
[3] FICHERA G. — Sulle equazioni differenziali lineari ellittico-paraboliche del
secondo ordine, Atti Accad. Naz. Lincei. Mem. CI. sci. fis., mat. Nat. ser.,
1 (8), 1956, p. 5.
[4] SMIRNOV M.M. — Degenerate elliptic and hyperbolic equations, Moscow, Nauka,
1966.
[5] VISHEK M.I., GRUSHIN V.V. — On a class of degenerate elliptic equations of higher
order, Matem. Sbornik, 79, 1969, No. 1, p. 3-36.
[6] FICHERA G. — On a unified theory of boundary value problem for elliptic-parabolic equations of second order, Madison, Wise, Univ. of Wisconsin Press,
1960, p. 97-120.
[7] ALEXANDROV A.D. — Investigation on maximum principle. I, Izv. vis. uchebn.
zaved., Matematika, No. 5, 1958, p. 126-157.
ON LINEAR SECOND ORDER EQUATIONS
777
[8] ALEXANDROV A.D. — Investigation on maximum principle, V. Izv. vis. uchebn.
zaved., Matematika, No. 5, 1960, p. 16-26.
[9] OLEINIK O.A. — On a problem of G. Fichera, Dokl. Akad. Nauk S.S.S.R., 157,
1964, p. 1297-1300, Soviet Mathem. Dokl, 5, 1964, p. 1129-1133.
[10] OLEINIK O.A. — On the smoothness of the solutions of degenerate elliptic and
parabolic equations, Dokl Akad. Nauk S.S.S.R., 163, 1965, p. 577-580,
Soviet Mathem. Dokl, 6, 1965, p. 972-976.
[11] OLEINIK O.A. — Linear equations of second order with non-negative characteristic form, Matem. Sbornik, 69, 1966, No. 1, p. 111-140.
[12] KOHN J.J., NIRENBERG L. — Non coercive boundary value problems, Communs.
Pure and Appi Mathem., 18, 1965, p. 443-492.
[13] KOHN J.J., NIRENBERG L. — Degenerate elliptic-parabolic equations of second
order, Communs. Pure and Appi Mathem., 20, 1967, No. 4, p. 797-872.
[14] PHILLIPS R.S., SARASON L. — Elliptic-parabolic equations of the second order, J.
of Mathem. and Mech., 17, 1968, No. 9, p. 891-918.
[15] HöRMANDER L. — Hypoelliptic second order differential equations, Acta Mathem.,
119, 1967, p. 147-171.
[16] RADKEVICH E.V. — On a theorem of L. Hörmander, Uspekhi Mathem. Nauk, 24,
1969, No. 2, p. 233-234.
[17] RADKEVICH E.V. — A priori estimates and hypoelliptic operators with multiple
characteristics, Dokl Akad. Nauk S.S.S.R., 187, 1969, No. 2, p. 271-274.
[18] BONY J.-M. — Principle du maximum, inégalité de Harnack et unicité du problem
de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier, 19,
1969, No. 1, p. 277-304.
[19] LIONS J.L. — Quelques méthodes de résolution des problèmes aux limites non
linéaires, Paris, Dunod, Gauthier-Villars, 1969.
[20] LATTES R., LIONS J.-L. — Méthode de quasi-reversibilité et applications, Paris,
Dunod, 1967.
[21] OLEINIK O.A., RADKEVICH E.V. — Second order equations with non-negative
characteristic form, Moscow, VINITI Acad. Nauk S.S.S.R., 1971.
[22] FREIDLIN M.I. — Markov processes and differential equations, In : « Theory of
Probability, Mathematical Statistics, Theoretical Cybernetics », Moscow,
VINITI Acad. Nauk S.S.S.R, 1967, p. 7-58.
[23] OLEINIK O.A. — Mathematical problems of boundary layer theory, Uspekhi
Mathem. Nauk, 23, 1968, No. 3, p. 3-65.
[24] VOLPERT A.I., HUDIAEV S.I. — On the Cauchy problem for quasilinear degenerate
parabolic second order equations, Matem. Sbornik, 78, 1969, No. 13, p.
374-396.
[251 OLEINIK O.A., RADKEVICH E.V. — On the local smoothness of weak solutions and
the hypoellipticity of differential second order equations, Uspekhi Matem.
Nauk, 26, 1971, W 2, p. 265-281.
Moscow University
Dept. of Mathematics,
Moscow V 234 (URSS)