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p-Sylow group of @ and if the center ofZ contains an element P of order pa, thenB cancontain characters of height X only if

X <d -a

For p = 2 and X > 0, we even have

X < d-a- 1.1 Brauer, R., these PROCEEDINGS, 30, 109-114 (1944); 32, 182-186 (1944); and 32, 215-219

(1946). Cf. also Brauer, R. Math. Zeitschr., 63, 406- 444 (1956).2 See, for instance, Brauer, R., Ann. Math., 57, 357-377 (1953).3 Brauer, R., Ann. Math., 42, 936-958 (1941).4Cf. the paper quoted in reference 3.

ESTIMATES AND EXISTENCE THEOREMS FOR ELLIPTIC BOUNDARYVALUE PROBLEMS*

BY FELIX E. BROWDER

DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY

Communicated by Einar Hille, January 12, 1959

Let A be a linear elliptic differential operator of order 2m (m > 1) with complex-valued coefficients defined on an open subset G of Euclidean n-space En. It is ourpurpose in the present note to discuss the existence and regularity of solutions of ageneral family of boundary-value problems of local type for A on G. An impor-tant aspect of our discussion is that it assumes no regularity conditions on thecoefficients of A except continuity, makes no semiboundedness assumption on theboundary-value problem, and yields information on the nature of A realized as anoperator in LP(G) under null boundary conditions for unbounded as well as boundeddomains G. The key tool of our discussion is a family of a priori estimates forthe derivatives of solutions in LP(G) (as well as in Holder norm, provided the coeffi-cients are Holder continuous). Since the existence results are obtained underextremely weak assumptions upon the regularity of the coefficients of A, they area useful step toward the construction of a theory of linear equations sufficient forthe attack upon boundary value problems for nonlinear elliptic equations. Theresults which are discussed below are immediately extendable to the general classof elliptic systems under analogous hypotheses. In a note to follow, we shall dis-cuss the stronger conclusions that may be obtained under similar hypotheses for theDirichlet problem for strongly elliptic equations and systems (as well as other semi-bounded problems) and derive from these, existence theorems for mixed initial-boundary value problems for temporally inhomogeneous equations of propagation.We shall use the notation Dj = i-1 a/axj, while for an n-tuple a of nonnegative

(al, . . . , an), we set De' = Dial ... D:R, a = Zjaj. Similarly, for a complexn-vector r = (id, ... ., n), ra = W" . .. Rnn. In terms of this notation, the differ-ential operator A may be written as E la <2m aa(x)Da, and its characteristic formis, a(x, I) = E la I=2maa(x).a If B1, ... X Bm are m differential operators of order<2m defined on a neighborhood of the boundary of G, each Bj may be written as

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E I <2m bj, e(x) Dfl and its characteristic form as bj(x, l)= =rbi, p(x)r, whererj is the order of B1.The domain G with boundary r is said to be uniformly regular of class C'(s > 1)

if there exists an integer r > 0 and sequences {NJ of open subsets of E and { Jd ofhomeomorphisms of Nj on the unit ball in En such that:

(1) Any (r + 1) distinct sets Nj have an empty intersection;(2) 4j(Nj n G) ={: y I < 1,y n >O}1Xj(Nj n r) = {y: IJI < 1, Yn = °};(3) If N1' = o(tYI <'/2), U1N1' contains the (1/r)-neighborhood of r;(4) If 4Jk and Ark are the kth coordinates of 0j and 4,pl, respectively, then for

xin Nj,y in 0j(Nj),I(Dajk1) (I) < r, I(Da#PJ) (y) < r, for Ia I < 8.

We shall assume that G is uniformly regular of class C2a and consider boundaryvalue problems for Au = f.Forj > 0, 1 < p < a), we introduce the Banach space WJ' P(G) = {f: Daf e L"(G)

for la j, derivatives taken in the distribution sense}, with the norm IffIifi, =El l<jJIIDafIILV(GAP'I Under ourassumption on G, the family CC(En) of in-

finitely differentiable functions with compact support in En forms a dense subsetof W2m. V(G) for every p > 1 if each function is restricted to G. If we use normal-ized tangential derivatives on r and integrate with respect to the (n - 1)-dimen-sional area element induced on r by En, we may similarly define the spaces WJ V(r).For each # with I# I< 2m, we define a linear mapping of of C. (En) intow2mnl-I- 1. P(r),byT(f) = Doff Ir.

If G is uniformly regular of class C2m, the mapping ars is a bounded linear mappingfrom the dense subset C,'(En) of W2M' P(G) into W2m-l-1t ' (F) and hence may beextended by continuity to a mapping defined on all of W2M, (G). If we composethis extended mapping xr, with the injection mapping of W2nl--o1I (r) intoLP(r), we obtain a continuous mapping of WM" P(G) into LP(r) which we shalldenote by DO Ir

Similarly if, for j a nonnegative integer, 0 < h < 1, we define the Banach spaceCJ '(G) as {f: f e >(G), Ha(f) = supe, EGI(Daf) (T)-(Daf) (y) * Ix-y Ih <o for a = j}, with the norm I f c.kioet() = E < j sup, aGIDaf(x) + E la I =jHa(f), then art may be defined for each IB with I# < 2m - 1 as the restriction ofDYf to r. In this case or, is a bounded linear mapping of C2M' h(G) intoC2I- "b̂ (r), where the latter is defined analogously using the normalized tangentialderivatives on r, and DI Ir is the corresponding mapping of C2mn "(G) into C' (1(r).Suppose that we are given m partial differential operators B1, . . . , Bm with coeffi-

cients bj, e(x) uniformly continuous and bounded on the (1/r)-neighborhood of I.We may define corresponding mappings B1, . . . X Bm of W2M P(G) into LP(r) andof C2M "(G) into Co, "(r), by Bjf = 61 < 2mbj, ,(x) DYf Ir. Each of the mappingsB. is continuous in the norm topologies of the domain and range spaces. Moregenerally, we may localize the definition of the operators Bj by considering B1 asacting on functions f which lie in WM, P(N) for some neighborhood N of each pointof the boundary r.Our basic a priori estimates are the following:(W),p Suppose that u is a function lying in W2e" Pi (N) for a neighborhood N in

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G of each point of the closure of G, (pr> 1), and satisfies the conditions Bju = 0 1 <j < m, Au E LP(G), (1 < p < cx). Then u lies in W2I", P(G) and there exists aconstant c, independent of pi or u, such that

IIUIJ2m,p< cp IIAuIILP(G) + JUlIO. PvI(C)h If in addition, Au e Co h(G) for some h with 0 < h < 1, then u lies in

C2m. (G). There exists a constant ch, independent of u or pi, such that

IIUIIC2m.h(G) < Ch{ IAUIJCO h(G) + IjUIICOB h(G)} .Let us formulate the precise hypotheses under which these estimates hold. We

shall assume that A is uniformly elliptic on G, i.e., there exists a constant cl > 0such that a(x, t) > cl for x in G and all real n-vectors t with = 1. At aboundary point x of G, let n be a unit normal vector to r and let t denote any unittangent vector to r.

(E1) We assume that the polynomial a(x,t + Xn) in the complex variable Xhas exactly m roots with positive imaginary part (as is always the case for n > 2),and that if pi, k(X, t) = fc, X'-1bk(x, t + Xn) {a(x, t + Xn)} -1 dA, where Ct is acurve in the upper X half-plane surrounding the zeroes of a(x, t + Xn), (1 < j <mY 1 < k < m), then there exists c2> 0 such that IDet(pjk(x, t)) > C2 for all tangen-tial vectors t to r at x with It = 1, and all x in r.THEOREM 1.1 Suppose that the coefficients aa of A are bounded measurable on G

with a. uniformly continuous on G for a = 2m. Suppose further that G ts uni-formly regular of class C2m, the coefficients bj, , of Bj are uniformly bounded and con-tinuous in the (r-1)-neighborhood of r, that A is uniformly elliptic on G, and that Elholds. Then the a priori estimate (W), is valid for all p with 1 < p < oo.THEOREM 2. Suppose that A and the Bj satisfy the conditions of Theorem 1, with

the additional hypothesis that the coefficients aa and bj , satisfy uniform HNoder con-ditions with exponent h on G. Suppose also that G is regular of class C2M+1. Thenthe a priori estimate (C)h is valid.The proofs of Theorems 1 and 2 which are lengthy and technical will appear

elsewhere. We remark that the condition (E1) for ellipticity of the boundary valueproblem is essentially due to Lopatinski.2 It is essentially equivalent to the uni-form form of the following condition which has been considered by a number ofrecent writers:'

(E2) Let x be a boundary point, t a unit tangent vector to r at x, n a unit normalvector to r at x. Let q(x, t, X) be the product of all linear factors of a(x, t + Xn)with roots having positive imaginary parts, where a(x, t + Xn) is considered as apolynomial in the complex variable X. Then the polynomials in X, b1(x, t + Xn),... , bm(x, t + Xn) form a linearly independent set modulo q(x, t, X) for every t.LEMMA 1. (E2) holds if and only if IDet(pjk(x, t)) > C2(X) > 0 for all tangent

vectors t to r at x with It = 1.Proof of Lemma 2: If the bj(x, t + Xn) are linearly dependent modulo q(x, t, X)

for some unit tangent vector t to r at x i.e., Ekckbk(X, t + Xn) = r(X)q(x, t, X) forsome polynomial r(X), (ck 0 ) it follows that

EkCkpjk(x, t) = Ct X1 (ZkCkbk(x, t + Xn)) a(x, t +± n) 1 dX =fe, XJ-1 r(X)q(x, t, X)a(x, t + Xn)-' dX = 0

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for everyj since the integrand is analytic inside Cj.To establish the converse, it suffices by continuity to consider the case where

Det(pjk(x, t)) = 0. Then there exists a non-null sequence { ck such that EkCkPjk(X,t) = 0 for allj. If we set Q(X) = Ekckbk(x, t + Xn), we have fic X-''Q(X)a(x, t +Xn)-I dX = 0 for 1 < j < m. If h(X) is the greatest common divisor of Q(X) andq(x, t, X), either h(X) is of lower degree than q(x, t, X) or q(x, t, X) divides Q(X). Inthe second case (E2) does not hold. To complete the argument, it suffices to showthat the first case is impossible. For that case, there exist polynomials s(X) andsl(X) such that s(X)Q(X) + s8(X)q(x, t, X) = h(X). Multiplying by the polynomialX.a(x, t + Xn) -{q(x, t, X)} -1, where r = (m - 1) - order(h), we obtain poly-nomials S(X) and S1(X) such that S(X)Q(X) + S1(X) q(x, t, X) = X1h(X) a(x, t +Xn) {q(x, t, X)}-1. Multiplying by Ia(x, t + Xn)}-' and integrating around Ct,we obtain 0 = fJCt XAh(X) {q(x, t, X) }-1 dX. But the integrand has a zero of orderone at o and poles only inside C:, so that the integral is nonzero, yielding a contra-diction.The remainder of our discussion is concerned with deducing consequences of

Theorem 1. Analogous arguments may be given in the spaces CM. Q(G) on thebasis of Theorem 2, but we shall not consider them explicitly here.THEOREM 3. Suppose that (W), holds for the boundary value problem (A, B1,...,

Bi). Consider A as an operator with dense domain D(A) = {u: u e W2"' P(G),Bju = Ofor 1 < j < m} in L"(G) and range in LP(G). Then:

(a) A is a closed operator in LP(G). If in addition, G is a bounded domain, then:(b) The null-space ofA isfinite dimensional.(c) R(A), the range of A, is a closed subspace of LP(G).Proof of (a): Let uk be an infinite sequence from D(A) with uk -* u, Auk-- vo

in LP(G). By (W)p, {uk} is a Cauchy sequence in WI"' P(G) and converges to anelement of Wv, P(G) which must coincide with uo. Since BJuk = 0 for every kand since Bj is a continuous mapping of W2rn P(G) into LP(G), Bjuo = 0. Thusuo e D(A). But A is a continuous mapping of W2rn, P(G) into LP(G), so that u1uo in W2rn P(G) implies that Auk -. Auo in LP(G), i.e., vo = Auo and A is a closedoperator.

Proof of (b): For u in N(A), the null-space of A, we have by (W)p, JuJJ2m, p <cp J1ulio, . However, if G is bounded and uniformly regular, every bounded setin W2mP (G) is pre-compact in LP(G). Thus the unit ball in N(A), considered asa closed subspace of LP(G), is pre-compact, and N(A) must be finite-dimensional.

Proof of (c): Since N(A) is finite dimensional, there exists a bounded projectionP of LP(G) on N(A). IfH = (I - P) (LP(G)), D(A) = N + {D(A) n H}. ThusA is an injective mapping of D(A) n H onto R(A), and H is closed. Since A is aclosed operator by (b) and A is one-to-one on D(A) n H, to show that R(A) isclosed is equivalent to showing that there exists a constant c3 > 0 such that JIAuIlo P. C3 1lullo, for all u in D(A) n H. Suppose the contrary. Then there mustexist a sequence {Iu} from D(A) n H such that I1u,1|o, p = 1, while Auk -O 0 inLP(G). By (W)p, the sequence I uk} is bounded in W2M. (G), and this latter spacebeing separable and reflexive (since it is uniformly convex) there exists a subse-quence, which we may identify with the original sequence, converging weaklyto a function u0 in W2m, P(G). Weak convergence in W2M P(G) together with thecompactness of the imbedding mapping of W2m' P(G) in LP(G), however, imply

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thatUk -usuo strongly in LV(G) and hence that Iluolfo, , = 1. Since H is closed inLP(G), uo must lie in H. Since Bjuk = 0 for each j and k, and since Bj is a weaklycontinuous map of W2, P(G) into LP(r), Bjuo = 0, i.e., uo e D(A). Since A is acontinuous and hence weakly continuous mapping of W2m, P(G) into LP(G), Aukconverges weakly to Auo in LP(G). Since Auk -O 0 strongly in LP(G), Auo = 0.Thus uo is a non-null element of N(A) n {ID(A) n H}, which is a contradiction.THEOREM 4. Suppose that the boundary value problem (A, B1, . .. , B.) and the

domain G satisfy the hypotheses of Theorem 1, and that in addition G is bounded.Then the codimension ofR(A) in LP(G) isfinite.Proof of Theorem 4:. We sketch the basic idea of the proof for the Dirichlet

problem. For the case of more general boundary operators {Bj}, the localizationand the treatment of the variable boundary terms involve more precise estimatesof the type (W), for solutions of Au = f with inhomogeneous boundary conditions.We may find a finite open covering {N,} of the closure of G of the type described

in the definition of uniform regularity such that (1) for each Nj not intersecting r,there exists a bounded linear mapping Tj of LP(Nj) such that Tju is locally inW2m P in the interior of Nj while A Tju = u; (2) For each N, which intersects r,there exists a bounded linear mapping Tj of LP(Nj n G) such that Tju lies inW2' P on a neighborhood of each point of Nj n cl(G), ATju = in Nj n G, whileD1'Tju = 0 on N, n r for 10 < m. Let { 4,} be an infinitely differentiable parti-tion of unity corresponding to the covering {Nj}. For each u in LP(G), let Pju bethe restriction of u to Nj n G. Finally set Uu = EjTj(Pju).U is obviously a bounded linear transformation of LP(G). By the construction

of the Tj for the sets Nj intersecting r, Uu satisfies the null Dirichlet conditionslocally at each point of the boundary. Since Tj(Pju) is in W~M P on a neighbor-hood of every point of N, n cl(G) and since the support of he in G is a pre-compactsubset of that set, Uu lies in W2n" P(G). Finally

A(Uu) = ZJ,4A TJ(Pju) + E 1,.I <2mi-jb,t', D Tj(Pju) = u + Su,where Su = J1j,,0A DiT,(Pju).

Since T, maps LP(N, n G) into functions locally in W2n PV by the closed graphtheorem, DIT,(Pju) is a bounded mapping into W1 P (K) for any compact sub-domain K of N, n (cl(G)) and hence a compact mapping into LP(K) ifK is smoothlybounded. Since the support of il',, is such a domain, S is a compact linear trans-formation of LP(G). But R(A) contains R(A U) = R(I + S), and by the Riesztheory of compact operators, R(A) has finite co-dimension.

Since A is a closed operator in LP(G), it has a densely defined adjoint operatorAt in L2(G), where q = p/(p - 1). If G is bounded, the range of A is closed byTheorem 3, and therefore so is the range of A*. Further, R(A) is the orthogonalcomplement in the obvious sense of N(A*), and by Theorem 4, N(A*) is finitedimensional.The remainder of our discussion is devoted to giving a more concrete character-

ization of A * under more restrictive hypotheses on A, namely the continuity of thecoefficients of its formal adjoint. The adjoint differential operator A' to A may bedefined by A'u = Ea1 2mDa(Ja(d2x)u), with differentiation in the sense of the theoryof distributions, and may be written in the form E la < 2maa'(x)Da, where the aa'are distributions on G. We shall assume henceforward that A' satisfies the same

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regularity conditions as the original differential operator A. We suppose alsothat we are given a family of m differential operators (B1',. . ., B.') of the sametype as the family {By}. The family {B'} is said to be formally adjoint to thefamily {Bj} with respect to A if the following condition is satisfied:A function v in W2m, 2(G) satisfies the equation (Au, v) = (u, A'v) for all u in

W2m P(G) such that B1u = 0, 1 < j < m, if and only if Bj'v = 0 for 1 < j < m.The Dirichlet boundary conditions are self-adjoint in the above sense for every

elliptic differential operator A.The pair of boundary value problems (A, B1, . . , Bi) and (A', B1t, . .' , B.)

are said to be regularizable if there exist a pair of sequences (Ak, B,, k, * * , Bm)and (Ak', B1', . . . , Bm , k)where Ak, for each k, is an elliptic differential operator,Ak' is its adjoint, the two boundary problems are formally adjoint for each k, thecoefficients of As, Ak', Bj k, and Bj, k' are locally of class C2M in the neighborhoodof each point of cl(G), while each of the coefficients converges uniformly on G ask -o o to the corresponding coefficient of A, A', Bj, or Bj'. The Dirichlet prob-lem for an arbitrary elliptic operator A is regularizable, since we may keep theboundary conditions fixed and take the convolution of all the coefficients of A andA' with a sequence of functions in C,'(En) converging to the Dirac B. Any pairof formally adjoint problems with coefficients of class C2m are obviously regulariz-able.THEOREM 5. Let A be an elliptic differential operator on G, A' its formal adjoint.

Suppose that the pair of boundary value problems (A, B1, ... , B.) and (A', B1',....Bi') are formally adjoint and regularizable, and suppose that each of the two problemssatisfies the conditions of Theorem 1. Suppose also that G is locally of class C4m aswell as being uniformly regular of class C2m. Let A and A' be defined with D(A) ={u: u e W2r 7(G), Bju = Ofor 1 < j < m}, D(A') = {u: u e W2m (G), Bj'u =Ofor I < j < m},p + qg = 1. Then if A is considered as an operator in LP(G),A' inLQ(G), we haveA* = A', (A')* = A.4COROLLARY TO THEOREM 5. If we suppose also that G is bounded, then Au =

f has a solution u in D(A) for a given f if and only if (f, v) = 0 for all solutions v inD(A') of the equation A'v = O. The family of such v, N(A"), is offinite dimension.5

Proof of Theorem 5: Because of (W)P holding for (A', Bj'), it suffices to showthat if v lies in LO(G) and satisfies the equation (Au, v) = (u, f) for an f in L"(G) andall u in D(A), then v is locally in W2m, Pi for every point of the closure of G for somePi > 1. Further if A* = A', it follows from the reflexivity of LP(G), since A is aclosed operator, that (A')* = ((A)*)* = A. Since the pair of problems is inter-changeable, we may assume that p < 2, q> 2. It follows for bounded domainsthat the theorem for general p follows from the result for p = 2. Indeed on abounded domain G, the domain of A in L2(G) lies in the corresponding domainin LP(G) for p < 2, while every f in LO(G) also lies in L2(G). A similar but moreinvolved argument based on localization gives the same result for unboundeddomains. Thus we need to consider only the case p = 2.

Next, we observe that we may restrict our consideration to problems of the typeof (A*,{B) 4I) and (A*', {B11'}) where the coefficients are locally of class C2M.Indeed, suppose that (A1) * = A*' for each of the approximating regularized prob-lems. Then for each k which is sufficiently large, (W)2 is valid for A1 and Ak' witha constant c2 independent of k. For each such k, Ak'Ak + I which is self-adjoint

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in L2(G) and . I, has an inverse Rk defined on all of L2(G) with ffRkf II 1. For agivenfin L2(G), let Uk = Rkf, Wk = Akuk. By its definition, Wk lies in D(Ak') whileA wkW= -Uk + f Further, IfUkIf <. ffff, and ffwk112 = Akuk ff2 = (i, Uk) - ffuff2 <fAfI2. It follows from (W)2 that f|Ukff2m, 2 + ffWkff2m, 2 < 2c2 11ff Ifor all k. By choos-ing a subsequence, we may assume that Uk -_ uo, wk wo weakly in W~M 2(G).But weak convergence of uk and wk in W~M 2(G) implies weak convergence ofDMur and D"wk in L2(r) for f << 2m - 1, and indeed local strong convergence. Itfollows that uo e D(A), wo e D(Ao'), since the coefficients of By, k and By, k' convergeuniformly to those of Bj and B,'. Moreover, Ak and Ak' considered as mappingswith domain W2?nl 2(G) and range in L2(G) are convergent in norm to A and A'respectively as k c. It follows that f = Ak'wk converges weakly in L2(G) toA'wo, i.e., A'wo = f, and that Akuk = Wk converges weakly in L2(G) to Auo. Thusuoliesin D(AA') and AA'u + u = f. Thus AA' + I, which is contained in (A')*A'+ I, must be equal to the latter, since both are onto and the bigger has a boundedinverse. It follows that AA' = (A') *A', and similarly, A 'A = A *A. In par-ticular, A' coincides with A* on R(A). But, since AA' + I is onto, every v inL2(G) may be written in form AA 'u + u, where AA 'u lies in R(A) and u lies inD(A'). If v e D(A*), v -u e D(A*) n R(A) c D(A'), and v = u + (v - u) eD(A'). Therefore, A* = A', and the conclusion of the theorem follows for A.To complete the proof, we assume that A, A', Bj, and B/' all have coefficients

locally of class Cm. Suppose u lies in D(A*A). Then for all v in D(A), (Au, Av) =(f, v) where f = A *Av e L2(G). It follows from (W)2 that u is a solution of thevariational boundary value problem associated with the coercive form [u, v] =(Au, Av) on the subspace D(A) of W2m 2(G). Since all the coefficients are locallyof class C2rn, it follows from the regularity theory of solutions of coercive variationalboundary value problems that u is locally of class W4m, 2 in the neighborhood ofevery point of the closure of G.6 Thus if w = Au, w is locally of class W~M 2,A*w = A'w, and (w, Au) = (A'w, u) for all v in D(A). It follows that w e D(A').Thus Au e D(A'), u e D(A'A), and A'Au = A *A. Similarly, AA' = (A')*A '. Itfollows that A' and A * coincide on D(A *) n R(A). But since AA' is self-adjointand positive, (AA' + I) maps onto L2(G). Hence, as in the preceding paragraph,D(A*) c {D(A*) n R(A)} + D(A') c D(A'). Therefore, A* = A' and thetheorem is established.

* The main results of this note were presented by the writer in a 15-minute talk at the Interna-tional Congress of Mathematicians in Edinburgh, August, 1958. They were obtained while thewriter held a National Science Foundation Senior Postdoctoral Fellowship in 1957-1958. Thewriting of this note has been supported by the Office of Scientific Research, U.S. Air Force, undercontract SAR/AF(638)224.

l Estimates in W2m, 2 for the Dirichlet problem for strongly elliptic operators A on boundeddomains were obtained by Guseva, 0. V., Dokladi Akad. Nauk S.S.S.R. (n.s.), 102, 1068-1072(1955) and Browder, F. E., Comm. Pure App. Math., 9, 351-361 (1956). (See also Nirenberg, L.,Comm. Pure App. Math., 8, 648-674 (1955), where a slightly weaker result was obtained.) W2". 2estimates in the Dirichlet problem for the general elliptic operator were obtained by Schechter,M., Am. J. Math., 79, 431-443 (1957). More general variational boundary value problems forstrongly elliptic operators have been treated by the writer (loc. cit.), N. Aronszajn, and K. T.Smith (unpublished), and Agmon, S., Journal d'Analyse Math., 6 (1958-1959) (to appear). W2m.estimates for the Dirichlet problem for strongly elliptic operators on unbounded domains havebeen applied by the writer [Compt. Rend. Acad. Sci., 246, 526-528, 1363-1365 (1958)] to the studyof the spectral theory of these operators.

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2 Lopatinski, Ya. B., Ukranian Mat. J., 5, 123-151 (1953). The proofs of Theorems 1 and 2are based upon constructions of integral representations of solutions using techniques developedin Lopatinski's paper. In the case of Theorem 1, we also utilize the methods and results of theCalderon-Zygmund theory of singular integral operators in LP spaces.

3 See Agmon (loc. cit.) and L. Hormander, Acta. Math., 99, 225-264 (1958), where he refers tounpublished results of Agmon, Douglis, and Nirenberg and of Schechter. Hormander showsfor operators and boundary operators with constant coefficients on a half-space, under homogeneityconditions, that (E2) is a n.a.s.c. for all solutions of the boundary value problem in C2- to be in-finitely differentiable up to the boundary.

4 For unbounded G, Theorem 5 contains an extension of a result of the writer for the Dirichletproblem for strongly elliptic operators (see the Compt. Rend. notes cited in reference 3).

6 The Corollary to Theorem 5 contains a strengthened form of the theorem of Schechter, M.,Bull. A.M.S., 64, 371-372 (1958), which states essentially that for the Dirichlet problem on abounded domain and with C coefficients, if N(A') = {O}, then R(A) D L2(G) n C'(G).

The conclusion follows from the results of Aronszajn, N., and K. T. Smith, which have been re-produced in the Tata Institute Lectures on Elliptic Differential Equations of J. L. Lions, Bombay,1958.

ON THE HOMOLOGY AND HOMOTOPY DECOMPOSITION OFCONTINUOUS MAPS

BY B. ECKMANN AND P. J. HILTON*ABTEILUNG FOYR MATHEMATIK, E.T.H., ZUJRICH, SWITZERLAND, AND DEPARTMENT OF

MATHEMATICS, CORNELL UNIVERSITY, ITHACA, NEW YORK

Communicated by Saunders Mac Lane, December 31, 1968

1. Introduction.-In this note we discuss two decompositions of a continuous mapof 1-connected spaces which stand in a heuristic duality to each other. The homol-ogy (Moore) decomposition' of the mapf: X -, Y reduces to a homology decomposi-tion2 of Y when X is reduced to a point, while the homotopy (Postnikov) decompo-sition of f reduces to a Postnikov system for X when Y is reduced to a point.

Considerable use is made here of the notions of homology, cohomology, andhomotopy groups of maps.3 '4 By framing our definitions in terms of the mapitself, we avoid resorting to replacements by mapping cylinders or equivalentfibrations, and, further, no transgression theorems are required. The resultingsimplicity, together with the duality, are the features of the treatment.

2. Basic Facts.-We collect together in this section the basic results we require.We consider maps and homotopies in the category of 1-connected spaces with basepoints o. There is a Hurewicz homomorphism 0: 7rm(f) Hm(f)f

PROPOSITION 2.1. If H,(f) = 0, r < n, then 0: 7rn(f) Hn(f).There are coefficient theorems in homotopy5 and cohomology; namely, for an

arbitrary abelian group G,

PROPOSITION 2.2. (a) The sequence

0 -O Ext(G, 7r±+,(f)) 7r1(G;f)f Hom(G, 7rn(f)) -> 0

is exact;(b) The sequence

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