A DUALITY THEOREM FOR SOLUTIONS OF ELLIPTIC EQUATIONS

Let L be a second order linear partial differential operator of elliptic type on a domain fl of IRm with coefficients in C(R)(fl). We consider the linear space of all solutions of %he equation Lu 0 on fl with the topology of uniform convergence on compact subsets and describe the topological dual of this space. It turns out that this dual may be identified with the space of solutions of an adjoint equation "near the boundary" modulo the solutions of this adjoint equation on the entire domain.

;,;=1 i=1 We assume that L is of elliptic type in f.This means that m i, j=l aij(x)i > O, (1.2) for each (l,...,m) E {m\{0} and for each x e .From (1.2) we see that for every compact subset K of f/, there exists a constant A A(K) > 0 such that for' each e [R m and for each x e K.
Ox (1.5) , The operator L defined by (1.5) is called the adjoint of L. , We denote by L(f) the set of all solutions of Lu 0 on f and by L (f/) the set of all solutions of L u 0 on f/.The set L(f/) is a locally convex linear subspace of the locally convex topological linear space C0(f/).We endow L(f/) with the topology of uniform convergence on compact subsets of f/ and we denote by L(f/)' the linear space of all continuous linear functionals on L(t) We wish to describe L(f/)' In the next sections we shall identify this dual space with a .certain quotient space by using a formula which relates the two operators L and L In the remainder of this section we state this formula and refer the reader to Miranda [3] for additional information on this.
Let S be a C 1 hypersurfacein 12.If (x)= (nl(x),...,nm(X)) is a continuous vector field normal to S, then by setting m m I--1 j=l we define a conormal to S continuous components, as being a vector field (x)= (Ul(X),...,m(X)) on with m aij(x)nj(x);i l,...m, j=l and whose direction is never, by (1.2), tangent to S.
The directional derivative in the direction of a conormal to S will be, m =-.V = aij(x)nj(x) o and we shall call the differential operator 0/o L a(x)O/Ou a conormal derivative.aij=ij then a=l and u=n.
When GREEN'S IDENTITY.Let D be a subdomain with compact closure in 12 and with a C 1 boundary 0D If u or v in (1.6) has compact support then we obtain a formula which justifies the , terminology used for L 2.

SPACES OF GERMS AND STATEMENT OF THE DUALITY THEOREM.
In order to describe our dual space L(fl)' we need to introduce now some terminology and to make some remarks.
We shall say that a subdomain D of f, relatively compact and having a C (R) boundary, is a normal subdomain when f\D does not have any compact connected component.An open subset U of 12 will be called a (punctured) neighborhood of infinity if there exists a normal , subdomain D of f such that 12\13 U.If U is a neighborhood of infinity and if u e L (U) , then we say that (u,U) (or simply u) is a solution at infinity of L u 0.
Let U 1 and U 2 be two neighborhoods of infinity and let (Ul,U1) and (u2,U2) be two .
solutions at infinity of L u 0. Then we say that u 1 and u 2 are equivalent if there exists a neighborhood of infinity in U Iq U 2 on which u I and u 2 coincide.The equivalence classes so obtained are called germs of solutions at infinity and we denote by L ((R)) the linear space of .
germs of solutions at infinity of L u 0.Here the symbol (R) (R)f2 represents the point such that 12 13 {(R)} is the Alexandroff compactification of 12. Therefore a germ [u] belongs to L ((R)) if L u 0 on a (punctured) neighborhood of (R) in fl.
Uniqueness properties for solutions of elliptic equations were studied in a large number of works and one can obtain several references about this in Miranda [3].The statement that follows comes from Agmon [4, page 151].
UNIQUENESS THEOREM.Let L be a second order linear partial differential operator of elliptic type on a domain 12 of Im with coefficients in C(R)(t2).If a C 2 solution u of Lu 0 has a zero of infinite order at a point in 12 then u 0.
In view of the uniqueness theorem we may identify a solution u e L (12) with the germ * , , [u] E L ((R)).The mapping from L (fl) to L ((R)) which maps each u to its equivalence class , [u] is indeed injective because if [Ul] [u2] with u 1 and u 2 in L (12) then u 1 u2.vanishes on a neighborhood of infinity and, therefore, identically on 12 by the uniqueness theorem.
Let W 1 and W 2 be two normal subdomains of 12 and let u and v be two functions of class C 2 satisfying Lu =0 and L v=0 on a neighborhood of 12\Wj, j= 1,2.Then by applying Green's identity (1.6) on W0\W where W 0 is a normal subdomain such that W LI W 2 c W 0 we obtain, for W 0 Wj F Formula (2.1) means that the integral | Iv,u] does not depend on the choice of the , normal subdomain W such that u e L(12\W) and v e L (tg\W).We can therefore define the integral on the ideal boundary of fl (obtained by compactifying fl in the sense of Alexandroff) as follows, I Iv,u] wf/lima Iv,u], (2.2) , whenever u and v are solutions at infinity of Lu 0 and L v 0.
Finally, by choosing an equivalence class e in L ((R)) and an element v E e we see that the continuous linear functional defined by, L(f/) 9 u J Iv,u]   (2.3) does not depend on the choice of v e e.
We can now state our result.DUALITY THEOREM.There is a vector-space isomorphism L(fl)' L ((R))/L (fl) (2.4) , where the action of a germ in L ((R)) on a solution in L(f/) is given by (,u) I [,u]. (2.5) A representation similar to (2.5) was obtained by M. Nakai, L. Sario [1] for harmonic functionals on noncompact Riemann surfaces.When the operator L is self adjoint, as it is in the harmonic case, then the isomorphism (2.4) reduces to the one of Nakai, Sario. 3.

SOME TOOLS.
In order to make easier the reading of the proof of the duality theorem, we present in this section, some of the results that will be used in the next sections.
The first result is an extension, to solutions of second order linear partial differential equations of elliptic type, of the classical Runge approximation property for harmonic functions.
The statement that follows comes from Lax [5, page 760].
LAX'S EQUIVALENCE THEOREM.Let L be a second order linear partial differential operator of elliptic type with C (R) coefficients in a domain t of Rm and let D be a normal subdomain of f.Every C 2 solution of Lu 0 in D is the uniform limit on compact subsets of D of a sequence of solutions of Lu 0 in ft if and only if every solution of the adjoint , equation L u 0 vanishing on a hypersurface S and whose conormal derivative vanishes on S is identically zero.
The existence and the properties of fundamental solutions for elliptic operators will also be used.We now recall the main facts connected with them and, as in section 1, we refer the reader to Miranda [3] for additional information on the remainder of this section.
Let (aij) be the matrix of coefficients of the principal part of L. We denote by Aij the elements of the inverse matrix of (aij) and by A the determinant of (aij If r Ixyl then in every relatively compact subdomain of l'l we have by (1.3), o"X. 0x.
A function E(x,y) will be called a fundamental solution of the equation Lu 0 (L u 0) on a domain W if it satisfies the following properties" i) E(x,y) and all its partial derivatives of first and second order with respect to the coordinates x of the variable x are continuous on W W outside the diagonal.
ii) E(x,y) and all its partial derivatives of first and second order with respect to the coordinates x of the variable x satisfy, for some A > 0, the following bounds,
uniformly on every relatively compact subdomain of W.
Any fundamental solution E(x,y) on a domain W together with all its partial derivatives of first order with respect to the coordinates x are locally integrable in each variable and also on WxW.Moreover if E,(x,y) is a fundamental solution of the equation L u 0 then by [3, page   19, (9.3)] we have LxE(x,y y.Therefore E is a fundamental solution in the sense of distribution theory.GIRAUD'S THEOREM.[3, page 66].Let fl be a domain in [m and let y be a point of Each C 2 solution u of Lu 0 in f\{y} satisfying, u(x) o(H(x,y)) when x-y, extends to a solution of Lu o on the entire domain fl.
PROOF OF FORMULA 2.5.We set, F(o)(u) I Iv,u]   and obtain a linear mapping, F" L ((R)) L(fl)' (4.1) Given ! e L(fl)' we look for o { L ((R)) such that F(o) t.We first extend t to a continuous linear ffmctional on C0(fl), [6, page 10S].Then, [1, page 154], there exists a signed regular Borel measure # with compact support in fl such that, l (u) I ud# (4.2) for every u e L(fl) We denote by S the intersection of all normal subdomains of 12 containing the support of #.Then S# is a compact subset of fl and contains the support of #.Let {Wj}j:I be an exhaustion of f by normal subdomains such that S# {: W 1 For each integer _> 1, let Ej(x,y) be a fundamental solution of the equation L v 0 on Wj.Since L has its coefficients in C(R)( 12) and since Wj has compact closure in fl, the existence of Ej(x,y) is guaranteed, [3, page For each >_ 1, we define a continuous function vj(x) by setting, vj(x) Ej(x,y)dy), for every x E Wj\S# with # given by (4.2).
Let to E Cc(Wj\S#).Since Ej(x,y) is integrable, we find by using Fubini's theorem and Green's identity on a regular domain G such that supp o c G Wj\S#, vj(x)Lx)dx O, Wj S# and we conclude by Weyl's lemma that vj E C(R)(W.I'\S/z and that L vj(x) 0 on Wj\S#.
In order to complete the proof we wish to replace v e L (D\S#) in (4.6) by an element We fix a positive integer and consider for each positive integer k, functions defined by Vk(X Vt+k+l(X)-Vt+k(X on Wt+k\S # By Giraud's theorem, the two fundamental solutions Et+k+l(x,y and Et+k(x,y differ at most on Wt+k by a function gk(x,y) of class C 2 in the variable x satisfying Lxgk(x,y)= 0 on Wt+k for each y e Wt+k We claim that gk(x,y) is continuous on Wt+k Wt+k To prove this we write L as, m m , L aij(x) Ox.Ox. + bi(x)ax o .
i,j=l i=l + b(x), where bi(x and b(x) are of class C Since gk(x,y) is continuous on Wt+k Wt+k when x y, it suffices to prove the continuity of gk(x,y) at those points on the diagonal.
We show that gk(x,y) is continuous on B B where B is any small ball in Wt+k Let us consider such a ball B sufficiently small to guaranty the existence, in a neighborhood w of B, of a positive C 2 solution w(x) of L w(x) =-1, [3, page 65 and 66].
For each x E w and each y E Wt+k we set, gk(x,y) gk(x'Y) w(x) -w(x) g(x,y)w(x), and we show that g(x,y) is continuous on B B. Let us remark that g(x,y)satisfies an elliptic equation Axg(x,y 0 on w where m A= aij(x)  By the continuity of g(x,Yl) at the point x 1 we see that the second term in the right member of inequality (4.7) can be made arbitrarily small when x is close to x 1 By the maximum principle [3, page 7], gk(x,Yl) < )l, (4.8) for each x and y in B. But by the continuity of g(x,y)-g(x,Yl) outside the diagonal, the right member of (4.8) and therefore those of (4.7) can be made arbitrarily small when y approaches Yl and (x,y) approaches (xl,Yl).This proves the continuity of g(x,y) at (xl,Yl) and therefore in B B. Thus gk(x,y) is continuous on B B and therefore on Wt+k Wt+k This proves the claim.
Using the claim, we extend continuously Vk(X by setting, Vk(X) I gk(x'y)d#(Y) for each x e Wt+k Using Fubini's theorem and Green's identity as above we obtain,

Wt+k
for every , e C(Wt+k) and we conclude by Weyl's lemma that V k e L (Wt+k).
By the uniqueness theorem we may apply Lax's equivalence theorem in order to obtain, for .
each positive integer k, a global solution h k e L (f) such that, sup{]Vk(X -hk(X)l.xeWt+k_l} < 1/2 k (4.9) Estimate (4.9) shows that the series (V k hk) converges uniformly on compact subsets of k=l Let us consider the series defined on Wt+I\S# by, v= vt+1+ (V k-hk), k=l , and whose sum belongs to L (Wt+l\S#).Since vt+/+Vt= vt+i+ on Wt+/\S # l= 1,...,k-1 and each k_2,weseethat This shows that, for each positive integer k, the function defined on Wt+k\S # Vt+k -(h +...4-hk_l) + (V-h), =k extends v to an element of L (Wt+k\S/).Such an extension being possible for every k, we , see that v extends to an element, still denoted by v, of L (fl\S#).
Thus by applying Green's identity on a subdomain D such that S# c D C I3 C W to an element u e L(fl) and by using (4.6) we have, for every u e L(fl).Since v e L (fl\S#) we obtain by (2.1) the desired representation, t (u) I [(x),(x)], and the existence of an element o Iv] E L ((R)) such that F(o) t.
Let us denote by ker F the kernel of the mapping (4.1).In order to complete the proof of , the duality theorem, we have to show.that ker F L fl).
It is clear that ker F3L (fl).Indeed if vEL (fl) then for every uL(fl) we have, F()(u) I Iv'u] Conversely let us take an element v (v,U) o e ker F and a normal subdomain W such that fl\W c U. We extend v to an element in C2(fl) and we consider g e C2(W) such that Lg=0 in W and g=0 on 0W.
By Green's identity we have, v ds=-gL dx.
(5.1 OW W Let " be the function on fl\OW which coincide with v on fl\W and with h on W Then by the preceding remark " extends continuously to the entire domain ll and satisfies the following relation, (x)L()dx 0, (.) for every o E C(12) To prove (5.5) let o be in C() and let D be a domain with compact closure in 12 containing the support of o such that the two subdomains D 1 D N(I\N), D 2=DNw have a boundary of class C By using the fact that v-h and its conormal derivative vanish on OW, we see that the integral in (5.5) is equal to * I * (x)dx x)L v(x)dx + x)L h D 1 D 2 and (5.5).follows.Therefore, by Weyl's lemma ker F c L (l).This proves the duality theorem.ACKNOWLEDGMENT.The author would like to thank Paul M. Gauthier and Thomas Bagby for their interest.This work was supported by NSERC-Canada.