EXISTENCE, UNIQUENESS AND REGULARITY OF SOLUTIONS OF LINEAR
D I F F E R E N T I A L EQUATIONS
By LARS HÖRMANDER
The aim of this lecture is to report some recent results concerning existence
of solutions of linear differential equations of general type. The main
emphasis will be placed on theorems concerning uniqueness and regularity
of the solutions of a differential equation, for such results imply existence
theorems for the adjoint equation (see section 3). We shall also exhibit
classes of differential equations which do not possess solutions.
1. Notations
Let O be a C°° manifold of dimension n which is countable at infinity,
and let P be a linear differential operator on O with C°° coefficients. This
means (Peetre [10]) that P is a linear operator in C°°(Q) which contracts
supports,
s u p p P / c s u p p / (fEG°°(ü)).
(1.1)
By continuity P can be extended to a linear operator, also denoted by P,
in the space D'(Q) of distributions on Q.
Let dÙ be a fixed positive C°° density in Q. The identity (x)
\(Pu)vdQ=
\uP*vdQ, when u, vEG00(Q) and supp u fi supp v<Q,
(1.2)
then defines a differential operator P*, called the formal adjoint of P
(with respect to the density dQ).
The operator P is of order <ra if and only if the polynomial
is of degree <ra for every cp E G0O(Q). The coefficient of rm is then for every
xEQ a form of degree m in grad <p(x), which we denote by Pm(x, grad ç?);
if the order of P is equal to m it is called the characteristic form of P .
If A c O we write
GS°(A) = {u; uEC°°(Q), supp
u<A},
and £ '(A) is defined similarly with C°°(Q) replaced by D'(Q). Finally, if
w€D'(Q) we denote by sing supp u the smallest closed subset of O such
that u is a C°° function in its complement.
(*) The notation A<LB means that A is relatively compact in B.
340
L. HÖRMANDER
2. Equations without solutions
Lewy [6] discovered that the differential equation
dujdxx + i dujdx2 + 2i(xx + ix2)duldxB = /
does not have a solution in any open subset of R3 for a suitable choice of
fEG°°(R3). This result can be generalized as follows (see Hörmander [3,
Chap. VI]):
THEOREM 2.1. Let P be of order <m, which implies that the commutator
C=P*P—PP* is of order <2m — 1. Assume that for some xEQ and some
real covariant vector | at x we have
Pm(x,i)=0, C^fefl+O.
(2.1)
Then one can choose fEC0C(Q.) such that the equation Pu=f does not have
a distribution solution in any neighborhood of x. If for every x in a dense
subset of O the hypothesis (2.1) is fulfilled for some real | , one can find fEC°° (Ù)
such that the equation Pu = f does not have a distribution solution in any open
subset of Q.
The main part of the proof consists in constructing functions which
satisfy the equation P*v = 0 except for a small error and which are strongly
concentrated at the point x. This is done by means of asymptotic expansions
closely related to those of diffraction theory.
3. Existence theorems
Results on regularity and uniqueness for the adjoint P * of P imply existence theorems for P. Indeed, we have the following two theorems.
THEOREM
3.1. Assume that
vES'(Q), P*vEC^(Q)->vECS3(Q).
Then the vector space
N(K) = {v;vECS>(K), P*v = 0}
is finite dimensional for every compact set KcQ.
f(v)=0
(A)
If /eD'(O) and
(vEN(K)),
(3.1)
there exists a distribution uEV(£i) such that Pu=f in the interior of K.(x)
To prove this theorem one first uses the closed graph theorem to give a
quantitative statement of the hypothesis (A). Then it follows immediately
that N(K) is finite dimensional and that P * defines an isomorphism of
C?(K)IN(K) into G?(K). By (3.1) the linear form
P*v^f(v)
(vECST(K)),
is therefore continuous for the topology in C°°(Q). Hence it can be extended
to a distribution üE£r (Q), and since û(P*v) =f(v), vECffiK), we have Pu =f
in the interior of K.
(l) The author owes this result to F. Treves.
SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS
341
3.2. Assume in addition to the condition (A) that to every compact
there is another compact set K'czQ, such that
THEOREM
set KcQ
vE£'(Q), supp P*v c K => supp v c K',
v E£'(Q), sing supp P*v cK=> sing supp v c K'.
(B)
(C)
TÄew £Äe vector space
N = {v;vEC%>(Q,),P*v = 0}
is finite dimensional and the equation Pu=f has a solution uEV'(Q) for every
/eD'(Q) suchthat
ï(v)=0 (vEN).
(3.2)
The Hahn-Banach theorem reduces the proof to showing that the linear
form
P*v^f(v)
(vEC?(Q)),
is continuous for the Schwartz topology in Co°(Q) when (3.2) is valid so
that the form is uniquely defined. The continuity can be established by a
slight modification of the proof of Theorem 3.6.4 in [3].
Simple examples show that the condition (A) is not necessary for the conclusion of Theorem 3.1 to hold; a weaker sufficient condition is that for
some integer k we have VECQ3^) if P*VECQ(D.) and v E (?o(Q). However,
(A) is fulfilled in most existence theorems known. Condition (B) in Theorem
3.2 is not necessary either, but if PV(Cl) => C°°(Q,) then (B) must hold at
least when v€<7o°(Q). When O c Rn and the coefficients of P are constant,
it follows that the full condition (B) is necessary in order that PD'(Q)
=>(7°°(0); conversely (B) then implies the existence of solutions of the
equation Pu=f when / is a distribution of finite order (Malgrange [7]).
When P has constant coefficients and D cz Rni condition (0) is also necessary
in order that PZ)'(Q)=D'(Q) (Hörmander [3]).
If P is a differential operator with constant coefficients in Rn, we have
if (supp v) = H(supp P*v),
üT(sing supp v) = H(siag supp P*v)
(v E £'), (3.3)
where H denotes convex hulls. The first identity is a special case of the
theorem of supports, and both have simple proofs via Laplace transforms.
(See John [5], Malgrange [8].) Hence P£>'(Q)=D'(Q) if Q is convex. This
result is due to Ehrenpreis [2] for Q = Rn and to Malgrange [8] for arbitrary
convex Q.
In the following paragraphs we shall give local regularity and uniqueness
theorems which lead to sufficient conditions for (A), (B) or (C) to be valid.
(One can also give local statements which are equivalent to (3.3); see John
[5], Malgrange [8].) The existence theorems thus obtained will not be stated
explicitly, but they go beyond those mentioned above even for differential
operators with constant coefficients.
4. Hypoelliptic operators
The operator P * is called hypoelliptic if
sing supp v=sing supp P*v,
vE D'(Q).
(D)
342
L. HöRMANDER
This local property of P * trivially implies both (A) and (C). It is classical
(Petrowsky [11]) that elliptic operators are hypoelliptic, and so are diffusion
operators. Algebraic conditions characterizing hypoelliptic differential
operators with constant coefficients in open sets O c= Rn are known (see
Hörmander [3]). The hypoellipticity of a related class of differential operators with variable coefficients, including elliptic operators, has been proved
by Hörmander and by Malgrange (see [3]). Weaker sufficient conditions for
hypoellipticity have been found by Treves [13] and by Hörmander [4],
but so far no sufficient algebraic condition for hypoellipticity has been
found which is invariant for coordinate transformations and is weaker than
ellipticity. (See Treves [14].)
It should be noted that condition (B) is not always fulfilled even for
elliptic operators. Indeed, Plis [12] has constructed an elliptic differential
equation which has a solution with compact support. This leads easily to
examples of elliptic operators which do not satisfy (B).
5. Holmgren's uniqueness theorem
From now on we always assume that O c= Rn, but since the results we shall
give are invariant for changes of variables, they are also applicable to
manifolds Q. We assume that P has a finite order which we denote by m
Then the classical uniqueness theorem of Holmgren is as follows.
THEOREM
5.1. Let <p be a real valued function in C^Q) and let x° be a point
in Q, where
Pm(xO,gr&d(p(x<>))*0,
(5.1)
that is, the surface (p(x)=(p(x°) is non-characteristic. If the coefficients of P
are real analytic, there exists a neighborhood Q'czß of x° such that every
+
UET)'(Q) satisfying the equation Pu =0 and vanishing in Çl = {x; xEQ,
<p(x)>(p(x0)} must also vanish in O'.
For a proof valid for distribution solutions we refer to Hörmander [3],
where the following supplementary result is also given.
5.2. Let (pbe a real valued function in G2(Q) and let P(x, D) be a
differential operator with analytic coefficients and with real coefficients in the
principal part. Let x° be a point in Q, where grad <p(x°)=N°3=0 and
Pm(x*,N*)=Obut
THEOREM
Ï
P^P%(x°,
N°)Ptf(x°, N°) + ÌP%\x°, N°) Pm, k(x°, N°) > 0.
j.k=lCXjCXjc
(5.2)
1
Here Pm denotes the principal part of P and we have used the notation
P%(x,Ç)=dPm(x£)ldÇj,
Pmtj(x,i)=dPJx,
{)ldz,.
(5.3)
Then we obtain the same conclusion as in Theorem 5.1.
On the other hand, Malgrange [9] has shown that there is non-uniqueness
if the inequality opposite to (5.2) holds. The meaning of (5.2) is perhaps
more clear if we note that the left-hand side is the second derivative of the
restriction of cp to the bicharacteristic with initial data x=x0,Ç = JV°.
SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS
343
6. Carleman estimates
In this paragraph we shall discuss a type of estimates first studied by
Carleman in connection with uniqueness theorems for equations with nonanalytic coefficients. These estimates are of the form
T (\Dm-1u\*#"dx<K1
[\Pu\2e2x(pdx
m-2
/•
+ K2 2 T2(m-;V1 \\D^u\2e2r(pdx, (uECSr(Q)).
(6.1)
Here T is a large positive parameter, <p is a fixed weight function in C2(Q)
and | Uu | 2 is the sum of the squares of the absolute values of the derivatives of u of order <j. Such estimates with K2 =0 have a particular interest
as we shall see in section 7. For proofs of all the following results we refer
to Hörmander [3, Chap. VIII].
THEOREM 6.1. Let A^=grad y(x) where xEQ, and let Ç=Ç+ioN with
| E Rn and 0 =t=o* E R± satisfy the characteristic equation
Pm(*,0=0-
(6-2)
If (6.1) is valid, it follows that
| f | «"-D -K%a\ | f | 2 + o T - * <2K1ip';(z,£,o)
(6.3)
when the left-hand side is positive. Here we have used the notation
<p'r(x, f, o) = i
jf%-F$(x,
C) P ^ C )
+ a'1 Im 2PmA*, 0 tfF(*. f).
(6-4)
1
Note that if K2 is a large positive constant we only obtain a condition
on <p'p(x,Ç,o) when Ç=Ç-\-ioN is nearly real.
The proof of Theorem 6.1 is achieved by applying (6.1) to functions of
the form
u(y) =exp(i x w(y)lo)y)((y-x)Vr)
wheTey)ECo>(Rn),wEC°°(Rn)a,iidw(y)=<y-x,Ç>
+
0(\y-x\2)wheny->x.
Letting T->°O leads to an inequality, involving %p and its first order derivatives, which is easily seen to be equivalent to (6.3).
When o->0 it is easy to see that (6.3) implies
C2m-i(x,Ì) = 0 iîxEQ,ÇERn
and Pm(x,C)=0
(6.5)
(cf. Theorem 2.1), îorcpp(x,^,a)—C2m_1(x,^)j2a is a polynomial in f and a.
To proceed we use a strengthened form of (6.5), but it is quite possible that
this additional assumption can be eliminated to a large extent.
Definition. P is principally normal if there is a differential operator Q
of order m - 1 with C1 coefficients such that
344
L. HÖRMANDER
C2m-i(x,è)=2RePm(xJ)Qm-i(x,Ç),
ÌERn.
(6.6)
It is obvious that P is principally normal if P m has real or constant
coefficients. When Pm(x,£ + ioN)=0 we now modify (6.4) as follows
(*, à cr) = 2 -^~
+ CT- 1
P§?(«,
C) PS?(«, f)
Im 2^m.fc(^, C) P(mfc)(s, C) - Re Pm(x, f ) Qm_i(z, C)
Since the right-hand side is a polynomial in | and cr by (6.6), it can be taken
as definition of (p'p(x,£,o) when or=0 and Pm(x,£) =0. If ÇERn and
Pm(x,i) = ÏP(l)(x,^)Nj
= 0,
(6.7)
i
we obtain in particular
?£(*,& 0)= 2 r ^ - P ^ V ^ ) ^ m V , l )
+ Re 2 (PS?. *(*, | ) Pg»(a:, | ) - P m . »(x, | ) PÄ»(ar, I)) JW>.
(6.8)
When P m has real coefficients, a simple computation shows that the righthand side is the second derivative of the restriction of q> to the bicharacteristic with initial data #,£. (The first derivative of q> along this curve
vanishes by (6.7).)
We can now state a partial converse of Theorem 6.1.
THEOREM 6.2. Let P be principally normal or elliptic, let grad q>(x) 4=0,
and let (pp(x,i,0)>0 if xEQ, 0=#|G-Bn and Pm(x,C)=0. If K is a compact
subset of O, we have for some constant C
"2T^""-- 0 " 1 f12^-uI*ea*^€fa< 0 f(|-P»|" + x**-!I vJ«)e8*^*!: (uEC?(K)).
/ / in addition (pp'(x, f, o) >0 for all xEQ,^ERn
fying the equation Pm(x,i- + ioN) =0, we obtain
(6.9)
and GERX with a+ 0. satis-
j ^ m - a - i [\&u\2e2r<pdx<C (\Pu\%e*"dx
(uEC$(K)).
(6.10)
In proving these estimates one first makes an integration by parts in
J \Pu\2e2r(pdx ("the energy integral method"), and the resulting expression
is then studied by means of Fourier transforms.
7. Uniqueness of the Cauchy problem
From the estimate (6.10) one obtains by standard arguments:
THEOREM
7.1. Let cp be a real valued function in C2(Q) and let P be a diffe-
SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS
345
rential operator which is either principally normal or elliptic. Let x° be a point
in Q where grad <p(x°) =N° 4=0, and assume that
(pp(x°,i,o)>0
(7.1)
if 0=¥£ = £ + iaN°, where £ERn and aERx, and £ is a solution of the equations
Pm(x°, 0 = 0,
2P%(x°, 0 N? = 0.
(7.2)
i
Then there is a neighborhood Q! of x° such that every uE*Df(0) satisfying the
equation Pu=0 in O and vanishing in 0 + = {x; xE Ù,<p(x) >(p(x°)}, must also
vanish in O'.
If Pm(x°,N°)=0, the condition (7.1) with £=N° and or=0 reduces to
(5.2), but there is a striking contrast between the strength of the hypotheses we have made in the case of analytic coefficients and in the case of
C°° coefficients.
Examples due to P. Cohen [1] show that the assumption (7.1) is vital at
least when o=0. Plis [12] has given examples of non-uniqueness even for
elliptic equations, but the role of the condition (7.1) when a 4=0 is not yet
completely clarified.
8. Unique continuation of singularities
In sections 5 and 7 we have stated results which lead to sufficient conditions for (B) to be valid. We shall now give another consequence of Theorem
6.2 which can be used to verify (A) or (C).
8.1. Let cpbea real valued function in C2(Q) and let P be principally normal. Let x° be a point in Q where grad tp(x°) =N° 4=0, and assume that
THEOREM
<pP\x°,C,0)>0
if0=¥ÇERn and
Pm(x°, | ) = 0,
fp^x0,
(8.1)
Ç) N? = 0.
(8.2)
i
Then there is a neighborhood O' of x° such that every w€D'(Q) for which
PuEC°"(Q) and uEC°°(Q+) is in C°°(Q').
Besults of Zerner [15] show that the convexity assumption (8.1) is essentially necessary for the conclusion to be valid.
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