ApPENDIX Witten's Proof of Morse Inequalities

o. Introduction In his paper "Supersymmetry and Morse Theory," E. Witten [Witl) pre• sented an analytic proof of Morse inequalities. It is the purpose of this appendix to introduce his proof. According to de Rham-Hodge Theory, the Betti numbers of a differential manifold M are related to the dimensions of harmonic forms. In the first section, we shall briefly review Hodge theory. The idea of Witten's proof is to introduce a perturbed elliptic complex for a given Morse function f as follows:

dp_l dp ••• ---+ AP-l(M) -----+ AP(M) -----+ AP+1(M) --+ ••.

e-t! 1 e-t ! 1 e-t! 1

P - 1 P d t d ••• ---+ AP-l(M) -----+ AP(M) ~ AP(M) --+ •.. with df = e-t ! dpetf, p = 0,1,2, ... ,n - 1, and to compute the perturbed Laplacian

We range the eigenvalues of D.f as follows

0::; Af(t) ::; A~(t) ::; ... ::; A~(t) ::; ....

The Hodge theory implies that there are j3p eigenvalues equal to 0, where j3p is the pth Betti number, p = 0, 1, ... ,n - 1. In local coordinates, 274 Witten's Proof of Morse Inequalities

Assume that x· is a critical point of f. We approximate ~f in a neighbor• hood of x· and obtain the approximate perturbed Laplacian:

where {J.'i} are the eigenvalues of the Hessian d? f(x·). If we put all these ~t,,,,' together (in the product space) for all critical 1 points {xi} as a new operator, and range all eigenvalues as follows:

o :::; tef :::; t~ :::; ... ,

the number of zero-eigenvalues is then proved to be the number of critical points with Morse indices p. The simple version of Morse inequalities

{3p:::; mp := #{xi I ind(f,xi) = p}, then will be proved if we have the following asymptotics:

lim A~(t) =~, k = 1,2, .... t->oo t k A revised elliptic complex is used to prove the final version of the Morse inequalities. The material of this appendix is based on [Witl), F. Annik [Annl), G. Henniart [Henl) and B. Helffer [Hell).

1. A Review of Hodge Theory Let (M, g) be a compact, connected, Coo, n-Riemannian manifold without boundary. T M denotes the tangent bundle; T",M denotes the tangent space at x EM; T· M denotes the cotangent bundle; T; M denotes the cotangent space at x EM; APT· M denotes the anti-symmetric tensor product of T· M. The section of APT· M is called a p-form over M. Coo (APT· M), the set of all Coo p-forms, is denoted by AP(M). V w E AP(M), in the local coordinates (Xl, ... ,xn), is expressed as follows:

W=

We write gii = 9 (a~i' a~,)' gii = g(dxi, dxi ), i :::; i,j :::; n. Then (gii) = (gii )-1 is positive definite. 1. Hodge Theory 275

We may also extend 9 to p-forms:

where (k1 , ••• ,kp) runs over (1, ... ,p), and is even ck,,,·k,, = ±1 if (k1, ... ,Pk ) 1, .. · , p is odd. The differential operator d: AP(M) ...... AP+l(M) is defined to be (1) a linear operator,

i i &a" ... ,p i it i " (2) d (a·., ....." (x)dx ' !I. ••• !I. dx ,,) = "'~L.Jl=1 &x) dx !I. dx !I. •.. !I. dx • From the definition, it is easily seen that (i) rP = O. (ii) d(w !I. B) = dw !I. B + (-I)Pw !I. dB, 'r/ wE AP(M), 'r/ BE Aq(M). The Hodge star operator *: AP(M) ...... An-p(M) is defined as follows: (1) *(a(x)w + b(x)8) = a(x) * w + b(x) * B, (2) * (dxi, !I. .•• !I. dxip) = '" .,. "...... gi,;>, ... gi"j>" dxi,,+, !I. L.Jk, .. ·k" "10, Jop ',,+1 'n .•. !I. dxin , where 1 ::; j1 < ... < jp ::; n, 1 ::; ip+l < ... < in ::; n,

{ill'" ,jp, ip+l .. , in} is a permutation of {I, ... n}, {k1, ... ,kp} is a per- mutation of {I, ... ,p}, 'fJl,."ln = Igj1/ 2cl, ..... In' Igl = det(gij), and 1,2, ... , n \ is even Cl,,,.ln = ±1 if ( ill i2,' .. , in) is odd. Then we have (i) *1 = 'fJ, *'fJ = 1 where 'fJ = Igl 1/2dx1 !I. ..• !I. dxn, (ii) * * w = (-I)p(n-p)w 'r/ w E AP(M), (iii) g(w, 8)'fJ = wll. (*8) 'r/ w, 8 E AP(M).

Claim. We only want to verify this identity for w = dxi, !I. ••• !I. dxi " , 8 = dxi1 !I. ... !I. dxip , with 1 ::; i1 < ... < ip ::; n, 1 ::; it < ... < jp ::; n.

LHS = L ck, ... k"gi';O, ... gi"io" . 'fJ, RHS= 276 Witten's Proof of Morse Inequalities

Since {i1, ... ,in} must be a permutation of {I, ... ,n}, and {t1···tp , ip+l'" in} is a permutation of {I, ... , n} with t1 < t2 < ... < tp, we have i1 = tt, ... , ip = tp. Therefore,

1 2 idk in RHS = IgI / ci''''in L ck,,,.kpg , ... gipjkPdxit /\ ... /\ dx k,,,.k p = Ig1 1/ 2 L ck,,,.kpgidkt ... gipjkpdx1 /\ ... /\ dxn k,,,.k p =LHS.

The scalar product on AP(M) is defined by

(w.8) = 1M g(w. 8)1/

= 1M w /\ (*8).

It is real, symmetric, bilinear and positive definite. The completion of AP(M) with respect to ( , ) is denoted by Ai2(M). It is a Hilbert space. The codi./Jerential operator dO: AP(M) -> AP-1(M) is defined to be the adjoint operator of d with respect to ( , ), i.e.,

(d*w, p) = (w, dp) v wE AP(M), V p E AP-1(M). Note. The scalar products on both sides are different! Basic properties of d*. (i) d* = (_1)n(p-1)+l * d*. Claim.

(d"w,p) = (dp,w) = (_I)p(n-p)(dp. ** w)

= 1M dp /\ (*w)

= (-I)P 1M P /\ (d * w) = (_I)p+(p-1)(n-p+l) 1M p /\ (* * d * w) = (_I)(p-1)(n-p)+1(p,*d*w)

(ii) dOdO = O. 1. Hodge Theory 277

The Laplacian. tl.P: AP(M) -. AP(M) is defined to be d*d + dd*. A p• form w satisfying tl.Pw = 0 is called a p-harmonic form. Denote HP(M) = ker(tl.P ). Example. p = O. V f E COO(M),

tl.°f = d*df = -lgl-1/ 2 ~ (lgI1/2gij~f) . E. . OXj OXi ',J This is the Laplace-Beltrami operator on (M, g). We have (i) Let D(tl.P), be the space of Wi-Sobolev sections of the vector bundle APT* M. Then tl.P is positive and self-adjoint. Claim. V O,w E AP(M), we have (tl.Pw,O) = (d*d + dd*)w, 0) = (dw, dO) + (d*w, d*O) = (w, (d*d + dd*)O) = (w, tl.PO).

Friedrich's extension provides the self adjointness. The positiveness is ob• vious. (ii) tl.P is an elliptic operator. See (vi) in the following paragraph. (iii) tl.P possesses only discrete spectrum, i.e., it has only eigenvalues O'(tl.P) = {Ai < A~ < ... }, with Ai 2: 0, A~ -. +00 as k -. 00, and each eigenvalue has only finite multiplicity. This follows from Riesz-Schauder theory. Exterior and interior product. V wE A1(M),

w/\: dxi1 /\ ... /\ dxip f-+ W /\ dX i1 /\ ... /\ dxip , AP(M) -. AP+1(M), P i",:dxi1/\ .. ·/\dxip f-+ E(-1)i+1g(w,dxij)dxil/\ j=l ... /\ dxij /\ ... /\ dxip , AP(M) -. AP-1(M),

are called the exterior and interior product with respect to w respectively. These products are extended to AP(M) linearly. (i) One has V 0 E AP-1(M), V t/J E AP(M),

(w /\ 0, t/J) = (0, i",t/J).

(ii) V f E AO(M), V 0 E AP(M),

d*(fO) = fd*O - idlB. 278 Witten's Proof of Morse Inequalities

Claim. V t/J E AP-1(M),

(t/J, d*(fO) = (dt/J,IO) = (fdt/J,O) = (d(ft/J) - dl 1\ t/J, 0) = (ft/J, d*O) - (t/J, idlO) = (t/J, Id*O - idfO).

Claim. We may verify this for W1 = dx1 and W2 = dx 1 or dx2 in suitable coordinates. (iv) The principal symbol of the differential operator d is O'Ld = if.l\, where f. = '£';=1 f.jdxj , (6, ... ,f.n) E T* M. Therefore

Therefore, O'Ldw = i '£ f.jdxj 1\ W, V wE AP(M). Note. For d, the symbol O'd = the principal symbol O'Ld. (v) O'Ld* = i . ie, where f. = '£ f.jdxj . Claim. Letting 8, W denote the Fourier transforms (in local coordinates) for 0 E AP(M) and t/J E AP+1(M) respectively,

(0, d*t/J) = (dO, t/J) = (O'Ld· 8,;j) = i(f.1\ (f,;j) = i(8, ie;j).

Therefore ((f,O'd*.;j) = i((f,ie;j). (vi) O'LI1 = -1f.12. Claim.

O'dd*d + dd*) = O'Ld*O'Ld + O'LdO'Ld* = - (f. 1\ ie + ie . f.1\) = -1f.12.

By choosing f. along an axis, say f. = (f.1>'" ,f.n) = 1f.le1> 1. Hodge Theory 279

Elliptic complex. Let Mn be a Riemannian manifold, and let E = {EiHi' be a family of vector bundles over M. Let d = {di } 0- 1 ,

i = 0,1, ... ,n - 1, be a family of pseudo differential operators (1/JDO) of order r, satisfying

(1) di+ 1di = 0, (2) V x E M, V ~ E T;M \ {9}, the sequence

is exact, where lTLd(x,~) is the principal symbol of the 1/JDO d. We say that (E, d) is an elliptic complex. Example (de Rham). We define (E, d) as follows:

n-l { } E= {N'T"M}np=o' d = dp p=o' where dp is the differential operator. This is an elliptic complex. Claim. We only want to verify the exactness of the sequence

°---+ A°T* M _u-=L....:do=---'l AIT* M

Since V W E AP(M), lTLd(x, ~)wx = i~ /\ Wx , where ~ = Ej=1 ~idxi' and it is easy to see that

(Choose ~ along an axis, say ~ = el>

if il = 1 if il > 1 Therefore,

ker lTLd(x,~) = Span{ei1 /\ ••• /\ eip 11 = il < ... < ip:-S: n} = 1m el/\ = 1m lTLd(x,~).)

Let (E, d) be an elliptic complex, define

as follows:

i = 0, 1, ... ,n - 1. 280 Witten's Proof of Morse Inequalities

We have (i) Di is symmetric (and it has a self-adjoint extension), and positive. The proof is quite similar to those for t1P • (ii) The 'l/JDO Di is elliptic, Le.,

Claim. Assume that for () E Ei, (ULDi)() = 0, then

9 ([(uLdi-l . (uLd;_l) + (uLdi) . (ULdi)](), ()) = 0, ~ g(uLd;_l)(), (ULdt_l)()) + g( (uLdi )(), (UL~)()) = 0, ~ (uLdt_l)8 = (ULdi)8 = o.

By the exactness of the sequence, (UL~)· () = 0 ~ 3'I/J E Ei-l such that 8 = uLdi-1'I/J, therefore

0= (uLdt_l)() = (uLdi_l)(ULdi-l)'I/J, ~ g(ULdi-l)'I/J, (uLdi-1)'I/J) = 0, ~ () = ULdi-l'I/J = o.

In the following, we use the same notations di (and di), representing the differential operators with domains COO(Ei) as well as their closed exten• sions in L2(Ei), i = 1,2, ... , n - 1. Hodge Theorem. Let (E, d) be an elliptic complex, and let

i = 0,1,2, ... , n - 1.

Then we have (i) L2(Ei) = N(Di) ffi R(di- 1) ffi R(di); (ii) N(di) = R(di-tl ffi N(Di); (iii) N(di_l) = R(d;) ffi N(Di);

(iv) Hi(E, d) = N(di )/R(d i - 1) defined to be the cohomology group of the elliptic complex. Then for each i = 0,1, ... , n - 1, the following iso• morphism holds: Hi(E, d) ~ N(Di), where we denote N(D) = ker(D), R(D) = Im(D) for each linear operator.

Proof. Di has a self-adjoint extension, which is denoted by the same notation. We have

(Because Di is elliptic, Di has closed range.) 2. The Witten Complex 281

By definition, however, implies that

(d;w, dBi_I) = 0, V wE COO(Ei+I)' VB E COO(Ei_I ), => R(dn ..1 R(di- I ) => R(Di) c R(di) EEl R(di- I ).

On the other hand,

R(d;) C N(di)l.,R(di_d C N(d;_l)l. => R(d;) EEl R(di-d C N(di)l. + N(d;_dl. C N(Di)l. = R(Di).

The last inclusion follows from

We obtain the first conclusion: R(Di) = R(d'i) EEl R(di- I ), and

For (ii), since N(di ) C R(di)l., we have

N(di ) C N(Di) EEl R(di-d·

Conversely, N(Di) C N(di ) is known, and

R(di-d C N(di ) follows from didi- I = O.

(ii) follows. (iii) is obtained in a similar manner. (iv) is a direct consequence of (ii).

Corollary. For the de Rham complex,

is defined to be the ith cohomology group of M, which is isomorphic to N(b.i ), i = 0, 1, ... , n - 1. The Betti numbers

f3i = dim Hi(M) = dim Hi(M)

= dimN(b.i ), 282 Witten's Proof of Morse Inequalities

i=O,l, ... ,n-l.

2. The Witten Complex

Let f: Mn ...... 1ml be a Coo-function. Xo E M is called a critical point of f if df(xo) = o. Let K be the set of critical points of f. A function f is called nondegenerate if rP f (x) is invertible for each xEK. For a given nondegenerate function f, we define a new complex (E,dd as follows: E= {AP(M) Ip=O,l, ... ,n}. V t 2: 0, let p=O,l, ... ,n-1, and let dt = {df I p = 0, 1, ... , n - I}, i.e., --+ .•• --+ °

...... 0.

It is easily verified that (E, dt ) is an elliptic complex, V t 2: 0. Claim. (1) dfdf-1 = e-tfdpdp_letf = 0, (2) O"ddf) = O"ddp ) = i~A, so that the sequence

is exact. Similarly, we define (d( w, 0) = (w, dfO) V w E AP+1, V 0 E AP; therefore

Then define AP - L..>.t - dP"dPt t + dP-1dP-1" t t • 2. The Witten Complex 283

By the Hodge theorem for elliptic complexes,

ker Ai ~ ker eft lim eft-I

~ ker dpjlm dp- I

=} {3p = dim ker Ai·

Claim. The second isomorphism holds, because a: W --+ e-t / W satisfies

(1) a Iker d p : ker dp --+ ker eft is an isomorphism, (2) a 1m dp - l ~ 1m eft-I. Next, we compute At. (1)

dtw = e-t/ d(et/ w) = e-t/ (tet/ df /\ W + etJ dw) =tdf /\w+dw. (2) d;w = et/ d* (e-t/ w) = et/ (e-t/ dOw - ide-tl w) = d* W - e t/·l_te-t1d/ W = dOw + tid/ W.

(3)

AtW = dtd;w + d;dtw = t df /\ d; w + d( d; w) + d* (dtw) + tid/ ( dtw ) = tdf /\ (d*w + tid/W) + d(d*w + tid/W) + d* (t df /\ W + dw) + tid/ (t df /\ W + dw) = Aw + t[df /\ dOw + d(id/W) + d*(df /\ w) + id/dw] +t2[dJ /\idfW + id/(df /\w)] = Aw + t 2g(df, df)w + t Pd/W, where Pd/W = id/dw + d(id/W) + d*(df /\ w) + df /\ dOw. Let us express Pdf explicitly in local coordinates. First we observe that

Pd/(cpW) = cpid/dw + id/(dcp /\ w) + cpd(idfw) + dcp /\ (id/W) + cpd* (df /\ w) - idcp (df /\ w) + cp df /\ d* W - df /\ idcpW = cp Pd/W + g( df, dcp)w - g( dcp, df)w = cpPd/W. 284 Witten's Proof of Morse Inequalities

Next, we assume that

K = {xi I j = 1,2, ... ,s}.

We may find coordinate charts {(Uj,'Pj) I j = 1,2, ... ,s} such that xi E Uj , Ui nUj = 0 if i =f j, 'Pj: Uj -+ jRn, with ipj(xj) = 9, and assign a special metric gj on Uj such that

j = 1,2, ... ,s, k, f = 1,2, ... ,n, where y = 'Pj (x). In this case, gj on Uj is fiat, so

where we use the notation

Let us introduce the commutator:

We obtain

Vw E AP(M).

In summary, for a suitable metric 9j, in a neighborhood of a critical point xi of I, we have 2. The Witten Complex 285

It is important to note that neither the Betti numbers i3p , p = 0, 1, ... , n, nor the Morse type numbers m p , p = 0,1, ... , n, are influ• enced by the changing of the Riemannian metric 9, so we could choose a suitable 9 to simplify our computations. First, by the Morse lemma, we find neighborhoods Uj of critical points xi of !, j = 1, ... ,8, as well as local charts CPj, such that Uj n Up = 0, if j i' j', cpj(xi) = (),

!(x) - !(xi) = ~ t~iy~, y = cPj(x) for x E Uj, k=l where d2!(xi) = diag(~{, ... , ~~).

Second, let V; be an open neighborhood of Uj , with V; n V;, = 0 if j i' j', j = 1, ... ,8, and let s Vo=M\UUj . j=l Then {V;}o is an open covering of M. We have a COO-partition of unity {1}j}o: 1 = 2:;=o1}j, where supp 1}j C V; and 1}j = 1 on U j , j = 1, ... ,8. Define s 9 = 1}09 + L, 1}j9j· j=l This is the metric we need. Provided by the new metric 9 on M, b.f equals the following operator in Uj :

P a 2 j2 2 j k . b.t ,.,; = ~n ((- aXj )2 +t ~k Xk+t~k[dx A,td.,k] ) .

It is an operator of separable variables. Notice that H t = _ (!) 2 +t2~2x2

is the Hermite operator in mathematical physics (harmonic oscillation). It has eigenvalues t I~I (1 + 2N), with eigenfunctions

where HN are the hermitian polynomials, N = 0,1,2, .... 286 Witten's Proof of Morse Inequalities

Denote k . ( H ,3 = _ __8 )2 + t2 11.3~ X 2 t 8X k rk k, and

k = 1,2, ... ,no We have

n AP - t u.t,x~ - "'(Hk,i~ t + J-lkiKk) . 3 k=l

Since where if k E I if k ¢ I, obviously Kk is a scalar operator on AP(JRn). Thus the operator 6.f,x~ is J self-adjoint, with eigenvalues

n t L [(1 + 2 Nt)lJ-lkl + cf J-lk] k=l and eigenvectors (orthonormal)

where Ni = (Ni, ... ,ND runs over Nn, and Ii = (i{, ... ,i~) runs over NP with i{ < ... < i~, and j = 1,2, ... ,so We define the direct sum space

H = E9At2(JRn), (s-copies), i=l and a self-adjoint operator Af

We range the eigenvalues of Af as follows:

o :::; tei:::; te~ :::; ... :::; te~ :::; . .. . 3. The Weak Morse Inequalities 287

Theorem. dim ker(Af) = mp:= #{xi E K I ind(f,xi) = pl. Proof. By definition, ~f.x; !p~j ,Ij = 0 iff

n 2:[(1 + 2Nk>lttil + c,j ttiJ = 0

k = 1, ... ,n

if k tJ. Ij => ind(f, xi) = p. if k E Ij

Therefore, each xi with Morse index p has a one-dimensional contribu• tion to the null space, but if ind(f, xi) f= p, there is no contribution and therefore dimker(Af) = mp.

3. The Weak Morse Inequalities We shall prove the following inequalities:

p = 0,1, ... ,no

If we compare with the two operators ~f and Af, we see that

mp = dimker(Af) and (3p = dimker(~f).

We range the eigenvalues of the operator ~f as follows:

The weak Morse inequalities hold, if we can prove «*»

First, let us pull back the eigenvector in H

!p~ = {!p~j ,Ij }S , 3=1 onto the differential manifold M. We have charts (Uj,!pj), where Uj is a neighborhood of xi, !Pj is a coordinate such that the Morse lemma holds, and, on (Uj,!pj), the metric 9 is Euclidean, j = 1, ... ,s. 288 Witten's Proof of Morse Inequalities

Define a cut-off function p E coo(~n), such that 0 ::; p::; 1, and

Iyl ::; 1 p(y) = { ~ Iyl ~ 2. And define s 'IjJ~ = L p(t2/5<,Oj(x»)(<,O~)j 0 <,OJ (x), j=l where (<,O~)j is the j-component of <,O~, which is a vector in AP(M). For t > 0 large, the support of 'IjJ~ is concentrated in Uj=l Uj • These vectors are considered to be "approximate eigenvectors" for the operator ~f. In order to prove (*), the following Rayleigh-Ritz principle is needed. Theorem [Rayleigh-Ritz]. Assume that A is a self-adjoint opemtor bounded below on a Hilbert space H. If A only possesses discrete spectrum, consisting of eigenvalues with finite multiplicities, Al ::; A2 ::; ... ::; An ::; "', then (Ax, x) An = sup inf -112 • 'P', ... ,'Pn_,EH xED(A) IIx xEspan{ 'P, , ... ,'Pn_,}-L Proof. According to the spectral decomposition theorem,

where Xi = (x, ei), and ei is the ortho-normal eigenvector corresponding to Ai, i = 1,2, .... Therefore,

inf (Ax, x) xED(A) Tx1f2 xEspan{e" ... ,en_,}-L (Ax, x) < sup inf - 'PI,. .. ,'Pn-l EH xED(A) IIxll 2 • xEspan{ 'PI, ... ,'Pn_,}-L

On the other hand, V {<,Ol, ... ,<,On-I} 1= {el, ... ,en-I}, we may choose Xo = E;~ll xjej such that Xo ..L {<'o1>' .. ,<'on-I} and

This proves the equality in the theorem. Let us make some computations: (i) ('IjJ~,'IjJ~) = 00./3 + O(exp(-atl / 5 » as t -+ +00, where a > 0 is a constant. 3. The Weak Morse Inequalities 289

Claim. Noticing that (rp~,rp~) = Ocx{3 in the space H, we have

(1/J~, 1/J~) = ~ hn p2(t2/5y)( rp~)j (y)(rp~)j (y)dy

= (rp~, rp~) - ~ hn [1 - p2(t2/5y)]( rp~)j (y)(rp~)j (y)dy

= ocx{3 + I.

Since 0 ~ p ~ 1, p(y) = 1 for Iyl ~ 1. Letting z = Vty, we have

I ~ 1 t(rp~)j(z)(rpb)j(z)dz, Izl~tl/IO j=1

~ 1 P(z)e-1zI2 dx = O(exp( _at1/ 5 )), IzL~:t'/'O where the explicit expansions for Hermitian functions are used, P(z) is a polynomial of z, and a is any positive number less than 1. Before computing (1/J~, flf1/J~), we need (ii) V hE COO(M),

Claim. V w E AP(M),

[h, [h, flP])w = (h2 flP - 2hflPh + flPh2 )w = h2dd*w - 2hdd*(hw) + dd*(h2w) + h2d*dw - 2hd*d(hw) + d*d(h2w) = h2dd*w - 2h2dd*w + h2dd*w - 2h dh 1\ d* w + 2h dh 1\ d* w + 2hdidhW - 2hdidhW - 2dh 1\ idhW + h2d*dw - 2h2d*dw + h2d*dw + 2hidhdw - 2hidhdw - 2hd*(dh 1\ w) + 2hd*(dh 1\ w) - 2idh(dh 1\ w) = -2(dh 1\ idhw + idh(dh 1\ w)) = -2('Vh)2w. 290 Witten's Proof of Morse Inequalities

(iii) (1fJ~, ilf1fJ~) = ~(e~ + e~)(1fJ~, 1fJ~) + o (exp( _atl / 5)) as t -+ +00, where e~ and ~ are the eigenvalues associated with cP~ and cp1 respectively, and a > 0 is a constant. Claim. (1fJ~, ilf1fJ1) - ~(e~ + ~)(1fJ~, 1fJ1)

= ~ [(p(t2/5 y)( cp~)j, ilf.x;p(t2/5y)(cp~)j) A" (lIIn)

Ip t·2 t' 12 t·p t· ] - 2" (ilt •x; (CPc.)3,p . (cp,a)J)A"(Jlln) - 2"(p . (CPa)J,ilt.x;(cp,a)J)AP(Jlln)

= ~ t ((cp~)j, (2Pilf.x;p - ilf.x;p2 - p2 ilf.x; ) (cp~)j) A,,(Jlln) 3=1

= -~ t (( cp~)j, [p, [p, ilf.x; 1]( cp~)j) A,,(Jlln) j=l

8 = '"L.,..((CPa)J, t' ( \1p(t 2/5)2y) (cp,a)J)AP(Jlln),t . j=1 because ilf.x~ = ilP+ terms without differentials, which commute with p. J Again, we see

(\1p(t2/5 y))2 = t4 / 51 (\1p)(t 2/5y) 12 , which is equal to zero outside It2/ 5yl ~ 2, and therefore

((CPa)J,t . ( \1p(t 2/5)y) 2 (cp,a)1)t . A" (Jlln)

= r t4/ 5P(z)e-1zI2 dz = O(exp( _at1/ 5 )), Jlzl?2tl/lO where P(z) is another polynomial of z. Now, we turn to the first half of our conclusion:

Proof. We range {1/J~ I k = 1, 2, ... } in such a way that cP~ corresponds to the eigenvalue e1, k = 1,2, .... By the Gram-Schmidt procedure we obtain -1 k-l 1fJ~ - L c;k1fJj j=1 3. The Weak Morse Inequalities 291 where k-l L C}k(1/I;, 1/ID = (1/It,1/ID, i = 1, ... , k - 1. j=1 Therefore, C;k = O(exp( _at1/5» as t -+ +00. It follows that

j, k = 1,2, ... , and that

{.;pt I k = 1,2, ... } is an orthonormal basis.

By the Rayleigh-Ritz principle,

1 --~w = sup inf (1/1, -b..f1/l) t 'h, ... ,,pk_,EAP(M) ,pED(~n t 1I,p1l=1,,pESpan{1/l,, ... ,,pk_d.L < sup inf (PV1/1) b..f Pv 1/I) ,p" ... ,,pk_,EAP(M) ,pEV .L t 1I,p11=1,,pEspan{,p,, ... ,,pk-d where V = span{.;pi, ... , .;pD, and Pv is the orthogonal projection on V. Therefore

A1(t) ::::: sup inf (Pv 1/I, !b..fPv 1/I) t ,pI , ... ,,pk-' EAP(M) ,pEV .L t 1I,p1l=1,,pEspan{Pv,p,, ... ,Pv,pk-d . 1 sup mf (1/1, -b..f1/l) ,p, , ... ,,pk-' EV ,pEV t 1I,p1l=1,,pEspan{ ,p" ... ,,pk_d.L 1 = sup (cp, -b..fcp) 'PEV t lI'PII=1 ::::: e1 + O(exp(-at1/ 5 ») as t -+ +00.

This proves

The rest of this section is devoted to proving the second half of our con• clusion, Le., lim A1(t) > cl'. t---+oo t - k 292 Witten's Proof of Morse Inequalities

On the manifold M, we define a cut-off function

J.(x)t = { 0 t ~ 1, j = 1,2, ... ,s J p(t2/5

and let 8 (J8)2 = 1- 2)Jj)2. j=1 Then we have (iv) Af = L:j=o J} Af J} - L:j=o (V' J})2. Claim. Substituting h = J} in (ii), we obtain

(Jj)2 Af - 2JjAf Jj + Af(Jj)2 = (Jj, [Jj, Am = (Jj, [Jj, APJ] = -2(V' Jj)2.

Since 8 L(Jj)2 = 1, j=O it follows that 8 8 Af = LJjAPj - L(V'Jj)2. j=O j=O

Lemma. Suppose that e~ < r < e~+1' Then for large t > 0, there is a finite mnk opemtor Fk(t): A~2 (M) --+ A~2 (M) with dim 1m Fk(t) ~ k, such that

Proof. Since

8 8 Af = J8AP8 + LJjAPj - L(V'J:,)2, j=1 j=O the operator of the second term acts as the same as the operator Af together with a cut-off function. Let Pk be the orthogonal projection onto the sub• space spanned by the first k eigenvectors, corresponding to the eigenvalues e~, ... ,e~. Then the operator • Fk(t) = L JjPkAf hJj j=1 3. The Weak Morse Inequalities 293

(A stands for the pull back of Pk on AP(M» is of finite rank, with dim 1m Fk(t) :::; k. We have V'I/J E AP(M), (i)

(J&~f J&'I/J, 'I/J) = (~f J&'I/J, J&'I/J) = (~P J&'I/J, J&'I/J) + t21Y' fI211J&'l/JII2 + t(PdfJ&'I/J, J&'I/J) = Tl +T2 +T3 , where Tl ~ o. As for T2 , 3co > 0 such that

8 1Y'1I2 ~ co, for x E Vi> = M\ U Uj. j=l

and therefore

As for T3 , Pdf is a bounded operator, which commutes with the multi• plications of a function, and therefore

T3 ~ -MtllJ&'l/JII2, for some constant M > o.

In summary, (J&~f J&'I/J, 'I/J) ~ te~+1I1J&'l/JII2 for t large,

s (ii) L(Jj~PN,'I/J) = (Af'I/Jt,'l/Jt), j=l where 'l/Jt E H equals the element {p(t2/5 y)'I/J( rpj l(y»}j=1. And, according to the orthogonal decomposition,

(Af'I/Jt, 'l/Jt) = (Af(I - Pk)'l/Jt, (I - Pk)'l/Jt) + (iA(t)'I/J, 'I/J) ~ te~+111'I/Jtll2 + (Fk(t)'I/J,'I/J) s = te~+1 L ((Jj)2'I/J, 'I/J) + (Fk(t)'I/J, 'I/J), j=l 294 Witten's Proof of Morse Inequalities

(iii) We know that

('\7 Jj(X))2 = 0 if x (j Uj

('\7Jj(X))2 = ~ ((a~J p(t2/5~i(X))) 2

= O(t4/5) ~ ((:::) (t2/5~j(X))) 2

= O(t4/ 5 ) if x E Uj , j = 1,2, ... ,so And

1/2 ( ) J5(X) = 1 - ~(Jj(X))2 , so that

8 ('\7 J5(X)? = ° if x E Vo = M \ UU j , j=l

('\7J5(X))2 = t (~~~)2 k=l k

= t4/5 ~ [(::J (t2/5~j(X)). p(t2/5~j(X))r

1[1- p(t2/5~j(x))2]

= O(t4/ 5 ) if x E Uj •

Then, finally, we obtain V1/1 E AP(M),

8 (~f1/1, 1/1) 2 te1+1 ((J5)21/1, 1/1) + te1+1 L((Jj?1/1, 1/1) j=l

+ (Fk(t)1/1,1/1) + O(t4/5 ) 111/1112

= t(e1:+1 + O(C1/ 5 ))1I1/1112 + (Fk(t)1/1, 1/1).

If e1: < r < e1:+l> then for large t > 0, we have

Now we are going to prove lim At?) 2 e1:. The proof is divided into two t->+oo cases. 4. Morse Inequalities 295

(1) ek-l < ek. We choose e > 0 such that

Then we have Fk-l(t) (a bounded operator with rank S k -1) such that

6f ~ teet - c)ld + Fk_l(t) for t> 0 large.

According to the Rayleigh-Ritz principle,

>'~(t) = t

~ e~ - e for t > 0 large, provided we take '1/Jl, ... ,'1/Jk-1 as a basis of the subspace 1m Fk_l(t). Since c > 0 is arbitrary, we have

(2) eLl = e~. We may assume that e~ > 0, and then 3 d > 1 such that eLd < ~-d+1 = ... = e~. According to case (1), we have

This proves our conclusion.

Theorem. Suppose that M is a compact, connected, orientable coo_ manifold. Then there exists a Riemannian metric 9 such that

lim >'~(t) = e~. t---++oo t

4. Morse Inequalities

We have defined f3p , m p , p = 0,1, ... ,n in Sections 1 and 2. Now we are going to prove the following inequalities:

mo ~ f3o, ml - mo ~ f31 - f30 296 Witten's Proof of Morse Inequalities

or, in a compact form, letting

pM (t) = E t1p tP, Mf (t) = E mptP,

we have Mf (t) = pM (t) + (1 + t)Q(t), where Q(t) is a formal power series with nonnegative coefficients. Let 0 < c: < Min{e~p+1 I p = 0,1, ... , n}. Fixing t large enough, we define a new cohomology complex as follows:

XP = Xf = {w E AP(M) I it is an eigenvector of ~f,

with eigenvalue A~(t) such that A~(t) < c:}.

According to the theorem in Section 3, we see that

p = 0,1, ... ,n, and we have (i) df:XP ---+ Xp+l, df-I:Xp ---+ Xp-l.

Claim. V wE XP, we have ~fw = A~(t)W with A~(t) < c:t. Therefore

P+1' P+1 ~p+ldPwt t - (d t d t + dPdP')dPwtt t = dfdf dfw = df(df df + df-Idf-I')w = df~fw = A~(t)dfw.

This implies that dfw E Xp+l. Similarly, one proves df- I ' w E Xp-I, so we obtain a smaller cohomology complex, d: d;'-' O -----+ X o ----td~ Xl ---+ . ., -----+ xn ----+.0

(ii) dim N(df)/ R(df-I) = t1p' Warning. This is different from the property stated in Section 2 because the complex is different.

Claim. We see easily that (1) N(~n c XP n N(df). (2) V w E XP n N(df) n N(~f)J.., we have

~fw = A~(t)W where A~(t) =1= 0, 4. Morse Inequalities 297 and ~fw = (elf df + df-1df-I·)w = df-1df-l• w.

Since df-l· w E X p - l , we see

d P-1dP- I • t t w (JP-I) W = A~(t) E R at , i.e., those p-forms in XpnN(df), which have contributions in N(df)/ R(df-l ), are just ~f harmonic forms. Therefore,

in the smaller cohomology complex. Theorem. Suppose that M is a compact, connected, orientable Coo manifold and that f: M -+ ~l is a nondegenerate Coo function. Then the Morse inequalities hold. Proof. We start with the following cohomology complex:

O --+ X o --+

We have shown that

(i) dim XP = m p , and (ii) dim N(df)/ R(df-l) = (3p. Since dimXP = dimN(df) + dim R(df) , and dimN(df) = dimR(df-l) + {3p, we obtain mp = (3p + dimR(df) + dimR(df-I), p = 1, ... ,n, where we assume elf = O. It follows that

m ~)-l)m-P(mp - (3p) = dimR(df') ~ 0, p=o for m = 0,1,2, ... ,n. And for the last one, it is an equality:

n n L(-1)n-Pmp = L(-l)n-p{3p. p=o p=o REFERENCES

[Ama1] Amann, H., Fixed point equations and nonlinear eigenvalue prob• lems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. [Ama2] Amann, H., Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), 122-166. [Ama3] Amann, H., A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982), 591-597. [AmH1] Amann, H. and Hess, P., A multiplicity result for a class elliptic boundary value problems, Proc. Roy. Soc. Edinburgh, Section A 84 (1979), 145-151. [AmZ1] Amann, H. and Zehnder, E., Nontrivial solutions for a class of non• resonance problems and applications to nonlinear differential equations, Annali Scuola Norm. Sup. Pisa 7 (1980), 539-603. [AmZ2] Amann, H. and Zehnder, E., Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math. 32 (1980), 149-189. [AmbI] Ambrosetti, A., Differential equations with multiple solutions and nonlinear functional analysis, Equadiff.1982; LN in Math.l017 (1983), 1-22. [Amb2] Ambrosetti, A., Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. (Madras), (1984), 1-10. [AmC1] Ambrosetti, A. and Coti-Zelati, V., Morse theory and periodic solutions of Hamiltonian systems, ISAS Trieste YEAR? [ACE1] Ambrosetti, A., Coti-Zelati, V., and Ekeland, I., Symmetry break• ing in Hamiltonian systems, JDE 67 (1987), 165-184. [AmP 1] Ambrosetti, A. and Prodi, G., On the inversion of some differen• tiable mappings with singularities between Banach spaces, Ann. Mat. Pum Appl. 93 (1973), 231-247. [AmR1] Ambrosetti, A. and Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. FUnct. Anal. 14 (1973), 349-381. [Ann1] Annik, Franc, Demonstration analytique des inegalites de Morse, Univ. Libre de Bruxelles, 1984-85. [AmI] Arnold, V.I., Problems in present day mathematics, XX: Fixed points of symplectic diffeomorphisms, Mathematical Developments Arising from Hilbert Problems (F.E. Browder, ed.) Proc. Symp. Pure Math., Vol. 28, AMS Providence 1976, p. 66. References 299

[Arn2] Arnold, V.I., Mathematical Methods of Classical Mechanics, Springer, 1978. [AtB1] Atiyah, M. and Bott, R., On the Yang-Mills equations over Rie• mann surfaces, Phil. Trans. R. Soc. London A 308 (1982), 523-615. [Bah1] Bahri, A., Une methode perturbative en theorie de Morse, These, d'Etat, Univ. Paris 6, (1981). [BaB1] Bahri, A. and Berestycki, H., A perturbation method in critical point theory and applications, TAMS 267 (1981), 1-32. [BaB2] Bahri, A. and Berestycki, H., Forced vibrations of superquadratic Hamiltonian systems, Acta Math. 152 3-4, (1984), 143-197. [BaB3] Bahri, A. and Berestycki, H., Existence of forced oscillations for some nonlinear differential equations, Comm. Pure Appl. Math. 37 (1984), 403-442. [BaC1] Bahri, A. and Coron, J.M., On a nonlinear elliptic equation in• volving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41(1988), 253-294. [BaLl] Bahri, A. and Lions, P.L., Morse index of some Min-Max critical points, I. Applications to multiplicity results, Comm. Pure and Appl. XLI8 (1988), 1027-1038. [BaL2] Bahri, A. and Lions, P.L., Remarques sur la theorie variationnelle des points critiques et applications, CR Acad. Sci. Paris 301 (1985), 145-147. [BaR1] Bahri, A. and Rabinowitz, P., A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal. 82 (1989), 412-428. [BaR2] Bahri, A. and Rabinowitz, P., Periodic solutions of Hamiltonian systems of 3-body type, CMS Report #90-8, (1989), Univ. Wisconsin. [BaC1] Bartsch, T. and Clapp, M., Bifurcation theory for symmetric po• tential operators, preprint. [BCP1] Bartsch, T., Clapp, M., and Puppe, D., A mountain pass theorem for actions of compact Lie groups, preprint. [Ben 1] Benci, V., A new approach to Morse theory, Recent Advances in Hamiltonian systems (G. Dell' Antonli, ed.), World Sci. Publ. [Ben2] Benci, V., Some applications of the generalized Morse-Conley index, Con/. Semin. Mat. Univ. Bari 218 (1987), 1-32. [Ben3] Benci, V., A geometrical index for the group 8 1 and some applica• tions to the periodic solutions of ordinary differential equations, Comm. Pure Appl. Math. 34 (1981), 393-432. [Ben4] Benci, V., On the critical point theory for indefinite functionals in the presence of symmetries, TMAS 274 (1982), 533-572. [BeC1] Benci, V. and Coron, J.M., The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere, Analyse Nonlineaire 2 (1985), 119-141. 300 References

[BeP1] Benci, V. and Pacella, F., Morse theory for symmetric functionals in the sphere and application to a bifurcation problem, Nonlinear Anal. 9 (1985), 763-773. [BeR1] Benci, V. and Rabinowitz, P.H., Critical point theorems for indefi• nite functionals, Inv. Math. 52 (1979), 241-273. [BeLl] Berestycki, H. and Lions, P.L., Sharp existence results for a class of semilinear elliptic problems, Bol. Bmsil. Mat. 12 (1981), 9-20. [Berl] Berger, M.S., Nonlinearity and Functional Analysis, Acad. Press, 1977. [BeP1] Berger, M.S. and Podolak, E., On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1975), 837-846. [Bir1] Birkhoff, G.D., Dynamical systems with two degrees of freedom, TAMS 18 (1917), 199-300. [BoF1] Bonic, R. and Frampton, J., Smooth functions on Banach manifolds, J. Math. and Mech. 15 (1966), 877-898. [Botl] Bott, R., Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248-261. [Bot2] Bott, R., Lectures on Morse theory, old and new, Bull. AMS 7 (1982), 331-358. [BreI] Brezis, H., On a characterization of flow invariant sets, Comm. Pure Appl. Math. 23 (1970), 261-263. [BrC1] Brezis, H. and Coron, J.M., Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), 203-215. [BrC2] Brezis, H. and Coron, J.M., Multiple solutions of H-systems and Rellich's conjecture, Comm. Pure Appl. Math. 37 (1984), 147-187. [BrB1] Brown, K.J. and Budin, H., On the existence of positive solutions for a class of semilinear elliptic BVP, SIAM J. Math. Anal. 10 (1979), 875-883. [Cam1] Cambini, A., Sui lemma di Morse, Boll. Un. Mat. Ital. 10 (1974), 713-723. [CaLl] Castro, A. and Lazer, A.C., Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pum Appl. 70 (1979), 113-137. [Cha1] Chang, K.C., Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), 693-712. [Cha2] Chang, K.C., Morse theory on Banach spaces and its applications, Chinese Ann. Math. Ser. b4 (1983), 381-399. [Cha3] Chang, K.C., A variant mountain pass lemma, Sci. Sinica, Ser A 26 (1983), 1241-1255. [Cha4] Chang, K.C., Variational method and the sub- and super- solutions, Sci. Sinica, Ser A 26 (1983), 1256-1265. [Cha5] Chang, K.C., Applications of homology theory to some problems in differential equations, Nonlinear Functional Analysis (F.E. Browder, ed.), Proc. Symp. Pure Math., AMS, 1986, References 301

[Cha6] Chang, K.C., On a bifurcation theorem due to Rabinowitz, J. Syst. Sci. Math. Sci. 4 (1984), 191-195. [Cha7] Chang, K.C., On the mountain pass lemma, Equadiff. 6, LN Math. 1192 (1986), 203-207. [ChaB] Chang, K.C., An extension on the minimax principle, Proc. Symp. DD3 (Chern, S.S., etc. ed.), Science Press, Beijing, 1986. [Cha9] Chang, K.C., On the periodic nonlinearity and the multiplicity of solutions, Nonlinear Analysis TMA 13 (1989), 527-537. [Cha1O] Chang, K.C., Heat flow and boundary value problem for harmonic maps, Analyse Nonlineaire 6 (1989), 363-396. [Cha11] Chang, K.C., Morse theory for harmonic maps, Variational Meth• ods, Pmc. of a Conf. Paris, June 1988, Berestycki, Coron, Ekeland, eds, Birkhiiuser (1990), 431-446. [Cha12] Chang, K.C., Infinite dimensional Morse theory and its applica• tions, Univ. de Montreal 97 (1985). [Cha13] Chang, K.C., Critical point theory and its applications, Shanghai Sci. Techn. Press (1986), (in Chinese). [Cha14] Chang, K.C., Variational methods for nondifferentiable functionals and its applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. [ChE1] Chang, K.C. and Eells, J., Unstable minimal surface coboundaries, Acta Math. Sinica, New ser. 2 (1986), 233-247. [ChJ] Chang, K.C. and Jiang, M.Y., The Lagrange intersections for (lCpn, m.pn) , Manuscripta Math. 68 (1990), 89-100. [ChL1] Chang, K.C. and Liu, J.Q., Morse theory under general boundary conditions, J. System Sci. (3 Math. Sci. 4(1991), 78-83. [ChL2] Chang, K.C. and Liu, J.Q., A strong resonance problem, Chinese Ann. Math. 11, B.2. (1990), 191-210. [CLZ1] Chang, K.C., Long, Y., and Zehnder, E., Forced oscillations for the triple pendulum, Analysis et cetera, Rabinowitz, Zehnder, eds, Aca• demic Press, 1990. [CWL1] Chang, K.C., Wu, S.P., and Li, S., Multiple periodic solutions for an asymptotically linear wave equation, Indiana Math. J. 31 (1982), 721-73l. [Cla1] Clark, D.C., A variant of Ljusternik-Schnirelman theory, Indiana Math. J. 22 (1972), 65-74. [Cofl] Coffman, C.V., Ljusternik-Schnirelman theory, Nodal properties and Morse index, Nonlinear diffusion equations and their equilibrium states 1, Springer, (1988), 245-266. [Cof2] Coffman, C.V., Ljusternik-Schnirelman theory: complementary prin• ciples and the Morse index, Nonlinear Analysis, TAM, to appear. [ConI] Conley, C.C., Isolated invariant sets and the Morse index, CBMS Regional Conference Series 38, AMS, 1978. [CoZ1] Conley, C.C. and Zehnder, E., The Birkhoff-Lewis fixed point the• orem and a conjecture of V. Arnold, Invent. Math. 73 (1983), 33-49. 302 References

[CoZ2) Conley, C.C. and Zehnder, E., Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207-253. [CoWl) Costa, D. and Willem, W., Points critique multiples de fonction• nelles invariantes, CR Acad. Sci. Paris 298 (1984),381-384. [CotI) Coti-Zelati, V., Morse theory and periodic solutions of Hamiltonian systems, Ph.D. thesis, Trieste, 1987. [Cou1) Courant, R., Dirichlet Principle, Conformal Mapping and Minimal Surfaces, Wiley, Interscience, 1950. [Dan 1) Dancer, E.N., Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1-22. [Die1) Dieck, T. Tom., Tro.nsformation Groups, Walter de Gruyter, 1987. [Din1] Ding, W.Y., Ljusternik-Schnirelman theory for harmonic maps, Acta Math. Sinica 2 (1986), 105-122. [DiLl) Ding, Y.H. and Liu, J.Q., Periodic solutions of asymptotically linear Hamiltonian systems, J. Sys. Sci. & Math. Sci. 9 (1989), 30-39. [DoLl] Dong, G.C. and Li, S., On the infinitely many solutions of the Dirichlet problems for some nonlinear elliptic equations, Scient. Sinica (1982). [Dou1] Douglas, J., Some new results in the problem of Plateau, J. Math. Phys. 15 (1936), 55-64. [Du1] Dumford, N. and Schwartz, J. T., Linear Opemtors, vol. 2, Wiley: Interscience, 1962. [DuLl) Duvaut, G. and Lions, J.L., Les inequations en mecanique et en physique, Dunod, 1972. [EeLI] Eells, J. and Lemaire, L., A report on harmonic maps, Bull. London Math. Soc. 16 (1978), 1-68. [EeL2) Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. [EeS1] Eells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds, JAMS 86 (1964), 109-160. [Eke1] Ekeland, I., Une thoorie de Morse pour les systemes hamiltoniens convexes, Analyse Nonlineaire 1, (1984), 19-78. [Eke2) Ekeland, I., Periodic solutions of Hamiltonian equation and a theo• rem of P. Rabinowitz, J. DiJJ. Equa. 34 (1979), 523-534. [EkL1] Ekeland, I. and Lasry, J.M., On the number of closed trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math. 112 (1980), 283-319. [Fad1] Fadell, E.R., Cohomological method in non-free G-space with appli• cations to general Borsuk-Ulam theorems and critical point theorems for invariant functionals, Nonlinear Funct. Anal. and Its Appl. 1-45 (Singh, S.P., ed.), Reidel Publ. Co. (1986). [FHR1] Fadell, E.R., Husseini, S.Y. and Rabinowitz, P.H., Borsuk-Ulam theorems for arbitrary 8 1 actions and applications, TAMS 274 (1982), 345-360. References 303

[FaR11 Fadell, E.R. and Rabinowitz, P.H., Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal. 26 (1977), 48--67. [FaR21 Fadell, E.R. and Rabinowitz, P.H., Generalized cohomological in• dex theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174. [FellI Felmer, P.L., Periodic solutions of spatially periodic Hamiltonian systems, CMS Report #90-3, Univ. of Wisconsin, (1990). [Flo11 Floer, A., A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Erg. Theo. and Dynamic Sys. 7 (1987), 93-103. [Flo21 Floer, A., Proof of the Arnold conjecture for surfaces and general• ization to certain Kahler manifolds, Duke Math. J. 53 (1986), 1-32. [Flo31 Floer, A., A Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988), 513-547. [Fl041 Floer, A., A cuplength estimate for lagrangian intersections, Comm. Pure Appl. Math, XLII (1989), 335-357. [FIZ11 Floer, A. and Zehnder, E., The equivariant Conley index and bifur• cations of periodic solutions of Hamiltonian systems, Erg. and Dynam. Syst. (1987). [FoM11 Fonda, A. and Mawhin, J., Multiple periodic solutions of conserva• tive systems with periodic nonlinearity, preprint, Univ. Catholique de Louvain, 1988. [ForI] Fortune, C., A symplectic fixed point theorem for cpn, Invent. Math. 81 (1985), 29-46. [FoWl] Fournier, G. and Willem, M., Multiple solutions of the forced dou• ble pendulum equations, Analyse nonlineaire, supplement au vol. 6, (1989), 259-281. [FoW2] Fournier, G. and Willem, M., Relative category and the calculus of variations, preprint, 1988. [Fra1] Franks, J., Generalizations of the Poincare-Birkhoff theorem, pre• print. [Fuel] FuCik, S., Remarks on a result by A. Ambrosetti and G. Prodi, Boll. Un. Mat. Ital. 11 (1975), 259-267. [Gho1] Ghoussoub, N., Location, multiplicity, and Morse indices of Min• Max critical points, Z. Reine und Angew. Math., in press. [GiT1] Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equa• tions of Second Order, Springer-Verlag, 1977. [Gorl] Gordon, W.B., Conservative dynamical systems involving strong forces, TAMS 204 (1975), 113-135. [Gre1] Greenberg, M.J., Lectures on Algebraic Topology, Benjamin, 1967. [Grm1] Gromoll, D. and Meyer, W., On differentiable functions with iso• lated critical points, Topology 8 (1969), 361-369. [Ham1] Hamilton, R., Harmonic maps of manifolds with boundary, Lecture Notes 471, Springer, 1975. 304 References

[Hell] Helffer, B., Etude du laplacien de Witten associe a une fonction de Morse degeneree, (1985) preprint. [Hen1] Henniart, G., Les inegalites de Morse, Seminaire Bourbaki, 1983/ 84,617,43-60. [Hes1] Hess, P., On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type, Boll. Un. Mat. Ital. (5) (1980), 17-A, 187- 192. [Hes2] Hess, P., On multiple positive solutions of nonlinear elliptic eigen• value problems, Comm. PDE 6 (1981), 951-961. [HeK1] Hess, P. and Kato, T., On some linear and nonlinear eigenvalue problems with indefinite weight functions, Comm. Partial Diff. Equa• tions 5 (1980), 999-1030. [Hin1] Hingston, N., Equivariant Morse theory and closed geodesics, J. Diff. Geometry 19 (1984), 85-116. [Hon] Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493-514. [Hof2] Hofer, H., A note on the topological degree at a critical point of mountain pass type, PAMS 90 (1984), 309-315. [Hof3] Hofer, H., Ljusternik-Schnirelman theory for Lagrangian intersec• tions, Analyse nonlineaire 5 (1988), 465-500. [Hof4] Hofer, H., A geometric description of the neighbourhood of a critical point given by the mountain pass theorem, J. London Math. Soc. 31 (1985), 566-570. [Hus1] Husemoller, D., Fiber Bundles, McGraw-Hill, 1966. [JiW1] Ji, M. and Wang, G.Y., Minimal surfaces in Riemannian manifolds, (1988) preprint. [Jia1] Jiang, M.Y., An existence result for periodic solutions of a class of Hamiltonian systems, Kexue Tongbao 33 (1988), 1679-1681. [Jia2] Jiang, M.Y., A remark on periodic solutions of singular Hamiltonian systems, Acta Math. Sinica, in press. [Jos1] Jost, J., The Dirichlet problem for harmonic maps from a surface with boundary onto a 2-sphere with nonconstant boundary values, J. Diff. Geometry 19 (1984), 393-401. [Jos2] Jost, J., Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds, J. Reine Angew. Math. 359 (1985), 37-54. [JoS1] Jost, J. and Struwe, M., Morse Conley theory for minimal surfaces of varying topological types, Invent. Math. 102 (1990), 465-499. [KaW1] Kazdan, J.L. and Warner, F.W., Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597. [Kell] Kelley, J.L., Geneml Topology, D. Van Nostrand, 1955. [Klil] Klingenberg, W., Lectures on Closed Geodesics, Springer, 1978. [Kra1] Krasnoselskii, M.A., Topological Methods in the Theory of Nonlinear Integml Equations, Pergamon, 1964. [Kuil] Kuiper, N., CI-equivalence of functions near isolated critical points, Symp. on Infinite-dimensional topology, Annals of Math. Studies 69, References 305

Princeton Univ. Press, 1972. [Lazl] Lazer, A.C., Introduction to multiplicity theory for boundary value problems with asymmetric nonlinearities, LN in Math. 1324 (1988). [LaMl] Lazer, A.C. and McKenna, P.J., On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282-294. [LaM2] Lazer, A.C. and McKenna, P.J., Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. PDE 10 (1985), 107-150. [LaSl] Lazer, A.C. and Solimini, S., Nontrivial solutions of operator equa• tions and Morse indices of critical points of Minmax type, Nonlinear Analysis TMA, 12.8 (1988), 761-775. [Leml] Lemaire, L., Boundary value problems for harmonic and minimal maps of surfaces into manifolds, Ann. Scuola Norm. Sup. Pisa 9 (1982), 91-103. [Lil] Li, S., Multiple critical points of periodic functional and some appli• cations, ICTP IC-86-191. [LiLl] Li, S. and Liu, J.Q., Morse theory and asymptotically linear Hamil- tonian systems, JDE 78 (1989), 53-73. [Liul] Liu, J.Q., A Morse index of a saddle point, preprint. [Liu2] Liu, J.Q., A generalized saddle point theorem. J. Diff. Eq. (1989). [Liu3] Liu, J.Q., Doctoral thesis, Academy of Science, Beijing, 1983. [Liu4] Liu, J.Q., A Hamiltonian system of second order, ICTP preprint. [Llol] Lloyd, N.G., Degree Theory, Cambridge Univ. Press, Cambridge, 1977. [Lon1] Long, Y., Maslov index, degenerate critical points and asymptoti• cally linear Hamilton systems, preprint. [Lon2] Long, Y., The structure ofthe singular symplectic matrix set, preprint. [Lon3] Long, Y., Multiple periodic solutions of perturbed superquadratic second order Hamiltonian systems, TMAS 311 (1989), 749-780. [LoZ1] Long, Y. and Zehnder, E., Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, preprint. [LuS1] Lupo, D. and Solimini, S., A note on a resonance problem, Proc. Royal Soc. of Edinburgh 102A (1986), 1-7. [Marl] Martin, R.H., Nonlinear Opemtors and Differential Equations in Banach Spaces, John Wiley & Sons, 1976. [MaP1] Marino, A. and Prodi, G., La teoria di Morse per spazi di Hilbert, Rend. Sem. Mat. Univ. Padova 41 (1968), 43-68. [MaP2] Marino, A. and Prodi, G., Metodi perturbativi nella teoria di Morse, Boll. Un. Math. Ital. Suppl. Fasc. 3 (1975), 1-32. [Maw1] Mawhin, J., Problemes de Dirichlet, variationnels nonlineaires, Univ. de Montreal, 104 (1987). [Maw2] Mawhin, J., Forced second order conservation systems with periodic nonlinearity, Analyse nonlineaire, Suppl. au vol. 6 (1989), 415-434. [MaW1] Mawhin, J. and Willem, M., Multiple solutions of the periodic BVP for some forced pendulum-type equations, J. Diff. Equa. 52 306 References

(1984), 264-287. [MaW2] Mawhin, J. and Willem, M., Critical point theory and Hamiltonian systems, Appl. Math. Sci. 14 Springer-Verlag, 1989. [Meyl] Meyer, W., Kritische Mannigflatigkeiten in Hilbertmannigflatig• keiten, Math. Ann. 110 (1967),45-66. [Mill] Milnor, J., Morse Theory, Princeton Univ. Press, Princeton, 1963. [MiI2] Milnor, J., Topology from the Differential Viewpoint, Univ. Press of Virginia, Charlottesville, 1969. [Morl] Morse, M., Relations between the critical points of a real function of n independent variables, TAMS 21 (1925), 345-396. [Mor2] Morse, M., The calculus of variations in the large, Amer. Math. Soc. CoIl. Pub. No. 18, Providence, 1934. [MoCl] Morse, M. and Cairns, S.S., Critical Point Theory in Global Anal• ysis and Differential Topology, Academic Press, New York, 1969. [MoTl] Morse, M. and Tompkins, C., The existence of minimal surfaces of general critical types, Ann. of Math. 40 (1939),443-472. [MoT2] Morse, M. and Tompkins, C., Unstable minimal surfaces of higher structure, Duke Math. J. 8 (1941), 350-375. [Nil] Ni, W.M., Some minimax principles and their applications in nonlin• ear elliptic equations, J. d'Analyse Math. 31 (1980), 248-275. [Nikl] Nikol'ski, S.M., Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, 1975. [Nirl] Nirenberg, L., Topics in nonlinear functional analysis, Courant In• stitute Lecture Notes, New York, 1974. [Nir2] Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. AMS 3 (1981), 267-302. [Nir3] Nirenberg, L., Comments on nonlinear problems, Le Matematische 16 (1981). [Nir4] Nirenberg, L., Variational Methods in Nonlinear Problems, LN in Math. 1365, 1989, Springer-Verlag, 100-119. [Ossl] Osserman, R., A Survey of Minimal Surfaces, Van Nostrand, 1969. [Pall] Palais, R.S., Morse theory on Hilbert manifolds, Topology 2 (1963), 299-340. [PaI2] Palais, R.S., Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. [Pa13] Palais, R.S., Ljusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. [PaI4] Palais, R.S., Critical point theory and the minimax principle, Global Analysis, Proc. Symp. Pure Math. 15 (ed. Chern, S.S.), AMS, Provi• dence (1970), 185-202. [PaSl] Palais, R.S. and Smale, S., A generalized Morse theory, BAMS 10 (1964),165-171. [Pitl] Pitcher, E., Inequalities of critical point theory, BAMS 64 (1958), 1-30. References 307

[PuS1] Pucci, P. and Serrin, J., The structure of the critical set in the mountain pass theorem, TAMS 91 (1987), 115-132. [PuS2] Pucci, P. and Serrin, J., Extensions of the mountain pass theorem, J. FUnct. Anal. 59 (1984), 185-210. [Qil] Qi, G.J., Extension of mountain pass lemma, Kexue Tongbao 32 (1987). [Rabl] Rabinowitz, P.H., Variational methods for nonlinear eigenvalue prob• lems, Eigenvalues of Nonlinear Problems, Ed. Cremonese, Roma (1974), 141-195. [Rab2] Rabinowitz, P.H., A bifurcation theorem for potential operators, J. FUnct. Anal. 25 (1977), 412-424. [Rab3] Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. [Rab4] Rabinowitz, P.H., Multiple critical points of perturbed symmetric functionals, TAMS 272 (1982), 753-770. [Rab5] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65 AMS (1986). [Rab6] On a class of functionals invariant under a zn action, CMS Report #88-1, Univ. of Wisconsin, Madison (1987). [Reel] Reeken, M., Stability of critical points under small perturbations, Part 2, analytic theory, Manuscripta Math. 8 (1973), 69-92. [Rotl] Rothe, E., Morse theory in Hilbert space, Rocky Mountain J. Math. 3 (1973), 251-274. [Rot2] Rothe, E., Critical point theory in Hilbert space under regular boundary conditions, J. Math. Anal. Appl. 36 (1971), 377-431. [Rot3] Rothe, E., On the connection between critical point theory and Leray Schauder degree, J. Math. Anal. Appl. 88 (1982), 265-269. [Rot4] Rothe, E., Critical point theory in Hilbert space under general boundary conditions, J. Math. Anal. Appl. 2 (1965), 357-409. [Ryb1] Rybakowski, KP., The Homotopy Index and Partial Differential Equations, Springer-Verlag, 1987. [SaUl] Sacks, J. and Uhlenbeck, K, The existence of minimal immersions of 2 spheres, Annales of Math. 113 (1981), 1-24. [SaU2] Sacks, J. and Uhlenbeck, K, Minimal immersions of closed Riemann surfaces, TAMS 271 (1982), 639-652. [SaZ1] Salamon, D. and Zehnder, E., Floer homology, the Morse index, and periodic orbits of Hamiltonian equations, preprint. [ScJ1] Schwartz, J.T., Nonlinear FUnctional Analysis, Gordon and Breach, 1969. [ScR1] Schoen, R., Conformal deformation of a Riemannian metric to con• stant scalar curvature, J. Diff. Geom. 20 (1984), 479-495. [Sch1] Schoen, R. and Yau, S.T., Existence of incompressible minimal sur• faces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Annales of Math. 110 (1979), 127-142. 308 References

[Shill Shiffman, M., The Plateau problem for minimal surfaces of arbitrary topological structure, Amer. J. Math. 61 (1939), 853-882. [Sik1] Sikarov, J.C., Points fixes d'un symplectomorphisme homologue d'identite. J. Dill. Geom. 22 (1985), 49--79. [SoIl] Solimini, S., Existence of a third solution for a class of BVP with jumping nonlinearities, Nonlinear Analysis TMA, (1983), 917-927. [SoI2] Solimini, S., Morse index estimates in Min-Max theorems, Manuscripta Math. 63 (1989), 421-454. [SoI3] Solimini, S., On the solvability of some elliptic PDE with linear part at resonance, JMAA 117 (1986), 138-152. [Spa1] Spanier, E.H., Algebraic Topology, McGraw-Hili, 1966. [Str1] Struwe, M., Infinitely many critical points for functionals which are not even and applications to nonlinear BVP, Manuscripta Math. 32 (1982), 753-770. [Str2] Struwe, M., On the critical point theory for minimal surfaces span• ning a wire in Rn, J. Reine Angew. Math. 349 (1984), 1-23. [Str3] Struwe, M., A Morse theory for annulus type minimal surfaces, J. Reine Angew. Math. 368 (1986), 1-27. [Str4] Struwe, M., On the evolution of harmonic mappings, Commet. Math. Helvetici 60 (1985), 558-58I. [Str5] Struwe, M., Plateau Problem and the Calculus of Variations, Prince• ton Univ. Press, 1988. [Szu1] Szulkin, A., A relative category and applications to critical point theory for strongly indefinite functionals, preprint. [Tau 1] Taubes, C.H., A framework for Morse theory for the Yang-Mills functional, preprint. [Tau2] Taubes, C.H., Minimax theory for the Yang-Mills-Higgs equations, Comm. Math. Phys. 97 (1985), 473-540 [Tia1] Tian, G., On the mountain pass theorem, Kexue Tongbao 29 (1984), 1150-1154. [Tro1] Tromba, A., On Plateau problem for minimal surfaces of high genus in Rn, preprint. [Tr02] Tromba, A., Degree theory on oriented infinite dimensional varieties and the Morse number of minimal surfaces spanning a curve in Rn , 1, TAMS 290 (1985), 385-413; 2, Manuscripta Math. 48 (1984), 139-16I. [Tr03] Tromba, A., A general approach to Morse theory, J. Diff. Geom. 12 (1977), 47-85. [Uhll] Uhlenbeck, K., Morse theory on Banach manifolds, J. Funct. Anal. 10 (1972), 430-445. [Uh12] Uhlenbeck, K., Morse theory by perturbation methods with appli• cations to harmonic maps, TAMS 267 (1981), 569--583. [Viti] Viterbo, C., Indice de Morse des points critiques obtenus par mini• max, Analyse nonlineaire 5 (1988), 221-226. [WaT1] Wang, T.Q., Ljusternik-Schnirelman category theory on closed subsets of Banach manifolds, preprint. References 309

[WaZ1) Wang, Z.Q., Equivariant Morse theory for isolated critical or• bits and its applications to nonlinear problems, LN in Math. 1306, Springer, 1988, 202-221. [WaZ2) Wang, Z.Q., On a superlinear elliptic equation, Analyse nonlineaire 8 (1991),43-58. [WaZ3) Wang, Z.Q., Multiple solutions for indefinite functionals and appli• cations to asymptotically linear problems, Math. Sinica, New Series 5 (1989), 101-113. [WaZ4) Wang, Z.Q., A note on the deformation theorem, Acta Math. Sinica 30 (1987), 106-110. [Warl) Ward, J.R., A boundary value problem with periodic nonlinearity, Nonlinear Analysis, TMA 10 (1986), 207-213. [Was 1) Wasserman, A.G., Equivariant differential topology, Topology 8 (1969), 127-150. [Weil) Weinstein, A., Bifurcation and Hamilton's principle, MZ 159 (1978), 235-248. [Wei2] Weinstein, A., Critical point theory, symplectic geometry and Hamil• tonian systems, Proc. 1983 Beijing Symp. on DD4, Gordon Breach, (1986), 261-288. [Wit1) Witten, E., Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982), 661-692. [Wu1) Wu, S.P., The nontrivial solution for a class of quasilinear equations, Applied Math. A.J. of Chinese Univ. 3 (1988), 339-345. [YaS1) Yakubovich, V.A., Starzhinskii, V.M., Linear Differential Equations with Periodic Coefficients, John Wiley & Sons, 1975. [Yan1) Yang, X.F., The Morse critical groups of minimax theorem, pre• print. INDEX OF NOTATION

df differential of f 19 fa level set of f, not above the level a 20 K critical set 19 Ka critical set with critical value a 21 (PS) Palais-Smale condition 20 exp(·) exponential map 72 Ll Laplacian operator 142 LlM Laplace-Beltrami operator 230 Ll tension operator 230 'V gradient operator 141,229 mes(-) measure 235 IAI measure of A 175 EI1 direct sum 180 WT transpose of the matrix W 182 1\1A loop space on A 204

~A cardinal number of the set A 216 Fix(·) fixed point set 216 1\ exterior product 277 iw interior product 277 Id identity operator 99 INDEX

Arnold conjecture Euler characteristic, 6 on fixed points, 216 on Lagrangian intersections, 217 Finsler manifold, 18 Finsler structure, 15 Fredholm operator, 47, 97 Banach manifold, 14 Betti number, 3 bifurcation, 129, 161 G-action, 66 blow up analysis, 232 G-cohomology, 75 Bott 79, 206 G-critical group, 76 G-equivariant, 66 G-space,66 cap product, 9 Galerkin approximation, 111 category, 105 general boundary condition, 55 relative category, 109 genus, 96 conformal group, 360 cogenus,96 convex set, 60 gradient flow, 19 locally, 60 Gromoll-Meyer pair, 48 Courant Lebesgue lemma, 268 Gromoll-Meyer theory, 43 critical group, 32 critical manifold, 69 critical orbit, 67 Hamiltonian system, 179 critical point, 18 handle body theorem, 38 w.r.t. a locally convex closed set, harmonic map, 229 62 harmonic oscillation, 285 critical set, 18 heat flow, 229 critical value, 18 Hilbert Riemannian manifold, 19 cuplength, 9 Hilbert vector bundle, 70 cup product, 9 Hodge theory, 274 homology group, 3 relative, 3 Deformation lemma, 21 homotopy group, 12 Deformation retract, 20 relative, 12 strong, 21 Hurewicz isomorphism theorem, 13 Deformation theorem hyperbolic operator, 41 first, 29 equivariant first, 67 second,23 invariant function, 111 equivariant second, 68 isolated critical manifold, 69 degenerate critical point, 43 isolated critical orbit, 74 non, 33, 41 isolated critical point, 43 312 Index

Jacobi operator, 251 jumping nonlinearity, 164 Poincare-Hopf theorem, 99 projective space real, 6, 11 Kiinneth formula, 5, complex, 6, 111 pseudo gradient vector field, 19 Landesman-Lazer condition, 153 Leray-Schauder degree, 99 link regular point, 18 homological, 84 regular set, 18 homotopical, 83 regular value, 18 Ljusternik-Schnirelman theorem, 105 locally convex set, 60 saddle point reduction, 188 shifting theorem, 50 Marino-Prodi theorem, 53 Sobolev embedding, 141 G-equivariant, 80 Sobolev space, 141, 231 Maslov index, 183 splitting theorem, 44 maximum principle, 143 strong resonance, 156 strong, 143 subordinate classes, 10 minimal surface, 260 subsolution, 145 minimax principle, 87 supersolution, 145 Morse decomposition, 250 symplectic form, 215 Morse index, 33 symplectic matrix, 183 Morse inequality, 36, 79 Morse lemma, 33 Morse-Tompkins-Shiffman theorem, tangent bundle, 15 271 cotangent bundle, 15 Morse type number, 35 mountain pass point, 90 variational inequality, 65, 177 vector bundle, 15

Nemytcki operator, 141 normal bundle, 70 Witten complex, 282

Palais-Smale condition, 20 w.r.t. a convex set, 62 (PS)*,117 Palais theorem, 14 pendulum, 209 periodic solution, 179 perturbation on critical manifold, 131 Uhlenbeck's method, 136 Plateau problem, 260 Progress in Nonlinear Differential Equations and Their Applications

Editor Haim Brezis Departement de MatMmatiques Universite P. et M. Curie 4, Place Jussieu 75252 Paris Cedex 05 France and Department of Mathematics Rutgers University New Brunswick, NJ 08903 U.S.A. Progress in Nonlinear Differential Equations and Their Applications is a book series that lies at the interface of pure and applied mathematics. Many differential equations are motivated by problems arising in such diversified fields as Mechanics, Physics, Differential Geometry, Engineering, Control Theory, Biology, and Economics. This series is open to both the theoretical and applied aspects, hopefully stimulating a fruitful interaction between the two sides. It will publish monographs, polished notes arising from lectures and seminars, graduate level texts, and proceedings of focused and refereed conferences.

We encourage preparation of manuscripts in some form ofTeX for delivery in camera-ready copy, which leads to rapid publication, or in electronic form for interfacing with laser printers or typesetters.

Proposals should be sent directly to the editor or to: Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139

PNLDE I Partial Differential Equations and the Calculus of Variations, Volume I Essays in Honor of Ennio De Giorgi F. Colombini, A. Marino, L Modica, and S. Spagnolo, editors PNLDE2 Partial Differential Equations and the Calculus of Variations, Volume II Essays in Honor of Ennio De Giorgi F. Colombini, A. Marino, L Modica, and S. Spagnolo, editors PNLDE3 Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems Michael Beals PNLDE4 Variational Methods Henri Berestycki, lean-Michel Coron, and [var Ekeland, editors PNLDE5 Composite Media and Homogenization Theory Gianni Dal Maso and Gian Fausto Dell'Antonio, editors PNLDE6 Infinite Dimensional Morse Theory and Multiple Solution Problems Kung-ching Chang PNLDE7 Nonlinear Differential Equations and their Equilibrium States, 3 N.G. lloyd, W.M. Ni, LA. Peletier, l. Serrin, editors