Math. Nachr. 192 (1998), 125 - 157
The Newton Polygon and Elliptic Problems with Parameter
By ROBERTDENK and REINHARDMENNICKEN of Regensburg, and LEONIDVOLEVICH of Moscow
(Received November 27, 1996) (Revised Version September 30, 1997)
Abstract. In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well-known Agmon - Agranovich - Vishik condition of ellipticity with parameter which guarantees the existence of a ray of minimal growth of the resolvent. The paper is devoted to the investigation of the same problem in the case of systems which are elliptic in the sense of Douglis- Nirenberg. We look for algebraic conditions on the symbol providing the existence of the resolvent set containing a ray on the complex plane. We approach the problem using the Newton polyhedron method. The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon. On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found. One of the equivalent conditions can be formulated in the following form: all the upper left minors of the symbol satisfy ellipticity conditions. This subclass of systems elliptic in the sense of Douglis- Nirenberg was introduced by A. KOZHEVNIKOV[K2].
1. Introduction We consider an operator on a manifold M without boundary acting in the cor- responding LZ-space and being defined by an elliptic matrix operator with smooth coefficients. In the study of spectral properties of this operator a natural question is the non-emptiness of the resolvent set, i.e., of the set A c Q: such that the operator A(z,D)- XI has a bounded inverse for X E A. In the case of a scalar elliptic operator A(%,0) the existence of a nontrivial resolvent set is provided by the so-called condition of ellipticity with parameter. This condition 1991 Mathematics Subject Classification. Primary: 47B25; Secondary: 47Al1, 47A56. Keywords and phmses. Newton polygon, systems elliptic in the sense of Douglis- Nirenberg, systems elliptic with parameter. 126 Math. Nachr. 192 (1998) is due to AGMONand AGRANOVICH-VISHIK(see [Agm] and [AV]) and means that there exists a ray on the complex plane
(1.1) C = {A E G : X = pexp (id), p > 0) such that
(1.2) A0(4- # 0 ((WE L: x R",It1 + 1x1 > 0, z E M)1 where Ao(z,[) denotes the principal symbol of the operator A(z,D). The condition above can be naturally extended to elliptic matrix operators of con- stant order [AV]. In this case the principal symbol Ao(z,[) is a matrix, whose elements are homogeneous polynomials in 4 of constant order, and (1.2) must be replaced by
(1.3) det (Ao(z,[) - XI) # 0 ((A,[) E C x IR", 141 + 1x1 > 0, x E M). Note that the left-hand side of (1.3) is a quasihomogeneous polynomial of [ and A, and (1.3) means quasiellipticity of this polynomial. It should be pointed out that Conditions (1.2), (1.3) permit us to use the calculus of pseudodifferential operators with parameter and to seek the resolvent as a series of such operators, where the norms of the remainders can be estimated by a constant times a negative power of 1x1 (the parametrix method). On this way it is possible not only prove that the equation
( 1.4) A(z,D)u(z)- XU(Z) = f(~) on a compact manifold M without boundary has a unique solution u E L2 for an arbitrary right -hand side f E L2, but also to obtain for this solution a two-sided a priori estimate in norms depending on the parameter A. Difficulties of principal character arise when one tries to extend the definition above to the case of matrix operators of variable order, i.e., to the systems elliptic in the sense of Douglis - Nirenberg. In this case the left-hand side of (1.3) is not a quasiho- mogeneous symbol and (1.3) must be replaced by a more adequate condition.') In his study of spectral asymptotics of systems elliptic in the sense of Douglis- Nirenberg A. KOZHEVNIKOV(see [Kl], [K2]) introduced an algebraic condition on the symbol (called parameter - ellipticity condition) which permitted him to prove the similarity of the system satisfying this condition to an almost diagonal system up to a symbol of order -00. The questions of solvability of (1.4)(i.e., the existence of the resolvent) and the validity of two-sided a priori estimates were not treated by him. In connection with this M. AGRANOVICHposed the problem of finding conditions on the symbol of a Dough-Nirenberg elliptic system which make it possible to extend to these systems the above mentioned theory of elliptic systems of constant order with parameter (a priori estimates, solvability for large 1x1, estimates of the norms of remainders in parametrix series by negative powers of the parameter). The present paper is devoted to the solution of this problem.
')In the case of systems elliptic in the sense of Dough-Nirenberg and having elements of the same order on the main diagonal the determinant (1.3) is also quasihomogeneous. This case was treated by AGRANOVICH[A]. Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 127
For heuristic arguments, we consider a diagonal system
(1.5) Aii(z, D)u~- XU~= fi (i = 1,.. . ,q) of elliptic operators Aii of various orders Ti (i = 1, . . . ,q). For simplicity here we con- sider this system in the whole space R". We choose the indexing of equations (1.5) such that r1 2 7-2 1 2 rq. It follows from [AV] that the existence of a ray (1.1) and a po > 0 such that for X E L, 1x1 > PO, the equations (1.5) are uniquely solv- able is equivalent to conditions (1.2) with Ao(z, <) replaced by the principal symbols A:i(z,t), i = 1,.. . ,q. Because of the quasihomogeneity of A:i(z, <) - X we can find positive constants 6i, di such that for X E L:
&(l 4 4 (1.6) 6 n ( l (1.7) (0,O), (0, q) , (Tl, q - 1) , (T1 + T2, q - 2) , . . . 1 (T1 + . . . + T4,0) , and set (1.8) %(P)( (1.9) 60%(P)(<,X) IIPO(.,<,X>l IdOEN(P)(<,X). Therefore there exists a po > 0 such that (1.10) 61sN(P)(<,X) 5 lP(z,<,A)l 5 dlEN(P)( d,(P) = max (la1 + rk : bak(z) f 0). Denote by Pp(z,<,A) the r-principal part of P: This polynomial is (1, r)-homogeneous in (c,A) of degree d, (P),i. e., we have P,(z,tE, tTX) = tdp(P)P,(z,(, A) . A natural modification of the results stated in [GV] leads to the necessary and sufficient conditions for the estimate (1.10) to take place: for every r > 0 (1.12) Pr(z,E,X)# 0 ((X,E) E L x R”,IEI > 0, 1x1 > 0, x E M). Formally (1.12) contains an infinite number of conditions on the symbol. However (as it will be seen below), only a finite number of values of T (defined by the Newton polygon) are essential. Conditions (1.12) can be reformulated in terms of the matrix symbol A(z,C) - XI. We assign to the variable X a “weight” r and define the r - principal part (A(z,S) - XI), (in the sense of Leray-Douglis-Nirenberg) of the symbol A(z,c) - XI. Then (1.13) det (A(z,E) - XI), = Pv-(z,(9 A) . Comparing (1.13) with (1.12) and with the explicit form of the symbol (A(z,<)-XI),, we find that the condition of ellipticity with parameter formulated above is equivalent to the condition of KOZHEVNIKOV(see [K2]). We show that the equivalent conditions of ellipticity with parameter also include sharp estimates for the elements of (A($,<) - XI)-’. Combining these estimates with the standard calculus of pseudodifferential operators with parameter (see [GI and [Sh]), we prove the unique solvability of (1.4) in the function spaces defined by the structure of the system. These spaces are endowed with norms which depend on the parameter A. Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 129 The plan of presentation is the following. In Section 2 all the needed facts about the Newton polygon of a polynomial are presented and N - ellipticity with parameter for polynomials is treated. Section 3 is devoted to the definition of ellipticity with param- eter for polynomial matrices. We prove the equivalence of our and KOZHEVNIKOV'S definitions. In Section 4 the main theorem on the unique solvability of the system (1.4) is proved in R" in the case of constant coefficients. The cases of variable coefficients in R" and of a compact manifold without boundary are treated in Section 5. 2. The Newton polygon of a polynomial. N - ellipticity with parameter for polynomials 2.1. Definition of N - elliptic polynomials with parameter Consider a polynomial in the variables 6 = ( where a = (al,.. . ,an)and ta = t,"l . . . k Definition 2.1. The polynomial (2.2) is called N -elliptic with parameter if there exists a ray (1.1) and positive constants d, PO such that the estimate holds. The polynomials (2.2) satisfying (2.4) for X belonging to some lower half-plane {A E Q: : ImX 5 PO} were studied in detail in [GV, Chapter 21. They were called 130 Math. Nachr. 192 (1998) r: I'g i i Fig. la) Fig. 1b) N-stable correct (if N(P)is as indicated in Fig. 1b)) and N-parabolic (if N(P)is as indicated in Fig. 1 a)). We now introduce some simple notions connected with the Newton polygon. 2.2. Leading and lower points of the Newton polygon Denote by r; = (ij,kj), j = 0,. . . , J + 1, the vertexes of N(P) indexed in the clockwise direction beginning with the vertex I?; = (0,O).Denote by I?:, j = 0,. . . , J, the side joining the vertexes I?? and ry+l and by the side joining l?;+l and I'g. The vertex I'y belongs to the vertical axis and belongs to the horizontal one. We distinguish two cases, corresponding to Fig. 1 a) and 1b): In the first case the exterior normals to the sides of N(P)not belonging to the coor- dinate axes have positive components. In the case (2.6) the side I': is vertical. An integral point (i,k) E N(P)is called a lower point if there exists a point (i', k') E N(P) (not necessarily integral) such that i' 2 i, k' 2 k and one of the inequalities is strict. Otherwise, the integral point is called a leading point. Accordingly, the monomials uaktaXk in (2.1) are classified as leading or lower monomials. The vertex is called leading if it is a leading point of N(P). Accordingly, the side I'j is called leading if it joins two leading vertexes. All the points belonging to a leading side are leading points of N(P). Note that in the case (2.5) (cf. Fig. 1 a)) all the vertexes I?:, . . . , and the sides I':, . . . , joining them are leading. In the case (2.6) (cf. Fig 1b)) the side I'> and the vertex are not leading. Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 131 2.3. Principal parts of a polynomial connected with the Newton polygon Denote by dj), j = 1,.. . , J, the vectors of the exterior normals to the sides I?:, j = 1,.. . , J. Since the components of v(j)are defined up to a positive constant we can suppose that (2-7) = (1,rj) , rj = (ij+l - ij)/(kj - kj+1) From obvious geometrical arguments it follows that (2.8) r1 > > TJ > 0 in the case (2.5), (2.9) TI > > rj-1 > rJ = 0 in the case (2.6). For j = 2,. . . , J denote by Vj the open angle formed by the directions of v(j-l) and dj). Then a vector v = (1,r) belongs to I$ if and only if rj-1 > r > rj. The angle Vl corresponding to the vertex I?(: is formed by the directions (-1,O) and (1, TI). With each vertex I'g and each side I?: we connect the corresponding principal part of the polynomial (2.10) In the Introduction we defined the r-degree d,(P) as max (la1 + rk) extended over all monomials of P. We set (2.11) It is easy to check that There exists a close relation between the principal parts of P in the sense of (2.10) and (2.11). Namely, if the numbers rj are as in (2.7), then (2.13) Pr; (t,A) = Prj (t, 9 (2.14) Pr; ((7 A) = Pr(t,A) ((1,r) E I$) . 2.4. Main theorem Now we can formulate the main result of this section. Theorem 2.2. For the polynomial (2.2) the following conditions are equivalent. (I) The polynomial P(<,A) is N -elliptic with parameter, a. e., there exists a ray L and constants d > 0, po > 0 such that the estimate (2.4) holds. (11) For arbitrary r > 0, (2.15) Pr(t,X) # 0 ((A70 EL: x nn,It1 > 0, 1x1 > 0) * 132 Math. Nachr. 192 (1998) (111) For each leading vertex rg and each leading side I'j Remark 2.3. The connection between estimates from below of polynomials by means of their monomials and "quasiellipticity" of all principal parts was found by MIKHAILOV.A number of results of such type can be found in [GV]. The above theorem is a generalization of [GV, Theorem 2.4.21. The proof we present is based on the geometrical construction used in [GV, Chapter 41. Proof of Theorem 2.2. (I) j(11). For j = 1,.. . ,J and K. = 0,l we set From the definition of the r-degree d,(P) and (2.13), (2.14) it follows that (2.17) ~(p<,p'~) = pdr(P)Pr;(<, A) + ,(pd,(")) and (2.18) :N(p) (p<, p'~)= pdr(P)zp (<, A) + ~(p~r"')) hold for r = rj if K. = 1 and for (1,r) E vj if K. = 0. To prove (2.15), for T > 0 and p 2 1 we replace (<,A) in (2.4) by (p[,p'X). If we divide both sides of the resulting estimate by pdr(P)and use (2.17), (2.18), then for p + 00 we come to (2.15). (11) + (111) trivially follows from (2.13) and (2.14). (111) (I). We fix E > 0. For each leading vertex rg = (ij,kj) we choose an open subdomain Uj" of the (1<1,IXl) -plane with the following property: for each integral point (i, k) E N(P)\I'g and for each (1[1,IXl) E Uj" we have (2.19) I (2.20) where the sum on the right-hand side is extended over all integral points of the segment I?!. Now we use the fact that for each E > 0 the subdomains Uj" and U: can be chosen in a way such that for some po = PO(€) we have Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 133 The proof of this result can be found in [GV, Subsections 4.2.3 - 4.2.51. Due to this fact, it is sufficient to prove the estimate (2.4) separately for ((,A) E 6; (K = 0, l), where we have set 6; = {((,A) E R” x L : (I([, 1x1) E v;} (K = 0,l) I Estimate in 6:: According to the definition (2.10), Pr;(t,A) = C aa,kj IPr; ((7 8 I 2 6; IClij lAlkj. Taking into account (2.19), we obtain IP(C,A>I L IPr;(t, A)I - IP(c, A> - Pr; (6,~11 2 6; I 2 [6j“ - K&(K(P)- l)] I = T;(()AkJ + . * * + T;+l(<) Pj+l = .j0(()A”+lTjl((, A), where T; (<, A) is a (1,rj) -homogeneous function of degree ij+l - ij. It follows from (2.16) and the (1,rj)-homogeneity that T$((,A) can be estimated from below by const(l(lij+1+ + IAlbj-kj+1 ). Hence 134 Math. Nachr. 192 (1998) Using this inequality we can literally repeat the argument in the proof of (2.4) for 6; and establish the desired inequality in 6;. Thus the theorem is proved. 0 3. Polynomial matrices elliptic with parameter 3.1. Polynomial matrices elliptic in the sense of Douglis- Nirenberg Consider a q x q matrix (3.1) A(() = (Aij(C))i,j=l,,..,,9 whose elements Aij are polynomials of degree aij: (3.2) aij = degAij(6) (i, j = 1,.. . , q) . As the degree of a product of two polynomials is the sum of their degrees, it is natural to set aij = -ain the case Aij(6) = 0. For a given permutation ?r of the numbers 1, . . . ,q we define R(n) = %7r(l) + * *. + (Ygn(q) and consider the maximum of these numbers over all permutations: R = maxR(?r) . (3-3) x Proposition 3.1. [Vl, V2]. Let {aij : i,j = 1,.. . , q} be a given set of integers and R be defined by (3.3). Then there exists a system of integers ml,.. . ,m,, 11,. . . ,1, such that (3.5) i=l The numbers ml, . . . ,1, defined above permit us to define the Leray - Douglis - Niren- berg principal part of the polynomial matrix (3.1). Denote by ATj(() the homogeneous part of Aij (6) of order li +mj, and set 0 in the case aij < li + mj. The matrix is called the principal part of (3.1). Definition 3.2. [Vl, V2]. The polynomial matrix (3.1) is called nondegenerate if R = degdet A(<). Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 135 This definition means that in the calculation of det A(<) = fAIA(l)(<) * ' ' AQA(Q)(<) A the leading terms do not cancel. Note that in the case R > degdet A(<)the above defined principal part is of no use. It follows from (3.4), (3.5) that for a nondegenerate matrix (3.1) (3.7) det A'(<) = principal part of det A(<). Definition 3.3. The matrix (3.1) is called elliptic in the sense of Douglis - Nirenberg if the following conditions are satisfied. (I) A(<)is a nondegenerate matrix. (11) P(<)= det A(<)is an elliptic polynomial, i. e., (3.8) Po(<) # 0 for I E Rn\{O}, where PO(<)is the principal part of P(<). The numbers ml, . . . ,1, in Proposition 3.1 are defined up to the transformation (3.9) {rnl,. . . ,m,, 11,. . . ,Zq} t-) {rnl + K,. . .,mq + K, 21 - K,. . . ,1, - K}, so without loss of generality we can suppose that the numbers in one group, say, rnl, ...,m, are nonnegative. In general, the other numbers 11 ,...,1, can be either positive or negative. If some of the numbers 11,. , . ,I, are negative, we have difficul- ties to consider the so-called L2 -realization of the corresponding matrix differential operator with the domain (see [GG]): DA = {U E L2 : A(D)u E Lz} . To obtain a definition of an elliptic matrix with nonnegative numbers in (3.4), (3.5) we define (3.10) and (3.11) R' = max (aiA(l)+ . + a:,(,,) . According to Proposition 3.1, we can define integers ml , . . . ,1, satisfying the relations (3.12) (~ij5 Zi +mj (i, j = 1,.. .,n), (3.13) i=l We can choose the constant K in (3.9) such that minli = Zi, = 0. 136 Math. Nachr. 192 (1998) Then inequalities (3.12) with i = io show that mj 2 2 0 for j = 1,.. . ,q. Definition 3.4. The matrix (3.1) is called positively nondegenerate if R' = degdet A(<). If the condition of Definition 3.4 is fulfilled, we can suppose that all numbers ml, . . . ,1, are nonnegative. Moreover, by changing the indexing of the rows and the columns of the matrix (3.1) we can suppose that and therefore (3.15) TI 2 r2 2 ... 2 T, 2 0, rj = lj +mi, (j= 1,..., q). In the following, we assume without additional stipulation that (3.14) and, conse- quently, (3.15) hold. 3.2. Conditions of ellipticity with parameter for polynomial matrices Now we consider matrices of the form (3.16) A(<)- XI1 where A(<)satisfies the conditions of the preceding subsection. As the definition of ellipticity, the definition of the ellipticity with parameter will contain a condition on the determinant of (3.16) and an analogue of the nondegeneracy condition of Definition 3.3. Consider the determinant of (3.16), Q (3.17) P(<,A) = det (A(<)- XI) = (-A), + Pj(<)(-X),-j . j=l From the rule of calculation of the determinant and (3.15) it follows that (3.18) degPj(<) 5 rl+...+rj (j=l, ...,q), Definition 3.5. The matrix (3.16) is called elliptic with parameter if the conditions below are satisfied: (i) the Newton polygon N(P)of the polynomial (3.17) contains the points (3.19) (0,O) , (0,q) , (TI,q - 1) 3 (r1 + 7-2, q - 2) 7 . . . , (r1 + . . . + Tq, 0); (ii) the determinant (3.17) is N-elliptic with parameter (cf. Definition 2.1), i.e., there exists a ray (1.1) and positive constants d, PO such that (3.20) dZP(<,X) L IP(<,X)I for (A,<) E L x R", 1x1 2 Po * Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 137 Remark 3.6. According to (3.18), the polygon N(P)is contained in the convex hull of the points (3.19). Hence, condition (i) of the definition above can be reformulated as (i') N(P)coincides with the convex hull of the points (3.19). If the conditions of Definition 3.5 are satisfied, then the estimate 0 (3.21) di n(l Note that in view of (3.15) we have r,, + + rsj 5 1'1 + + rj, and the points (rsl+. *+rSj,q-j) belong to N(P).Hence the terms (J(r81+"'+Taj IA('7-j are contained in the sum Zp(<,A) (see (2.3)), and (3.21) follows from (3.20). Proposition 3.7. Conditions (i), (ii) of Definition 3.5 hold if and only if there exist a ray (1.1) and positive dl, po such that (3.21) is valid. We have already proved the necessity. The proof of the sufficiency is based on two elementary lemmas. Lemma 3.8. (cf. [GV, Lemma 1.2.11). Let the point (a,b) belong to the convex hull of the nonnegative points (aj,bj), j = 1,.. , , J. Then J j=1 Proof. According to the definition of the convex hull of a finite set, there exist nonnegativenumbersrcl, ...,rcJ, K~+...+IcJ = 1,such that a= xrcjaj andb= C rcjbj. Then, cf. [H, (2.5.2)], J J j=l j=1 Lemma 3.9. (cf. [GV, Lemma 1.2.21). Let P(J,A)be a polynomial of the form (2.2), and assume that for some (io,ko), io 20, ko 10, d, and PO the inequality (3.22) ~l Proof. If (io,ko) does not belong to the polygon N(P),then there exists a vector (q1,qz) with q1, qz 1 0 such that for some K > 0 we have qli + qzk 4 K for (i,k) E N(P) and qlio + q2ko > K. Replacing [ and X in (3.22) by [tql, Xtqz and passing to the limit as t + +oo, we arrive at a contradiction to (3.22). 0 P r o o f of Proposition 3.7. Sufficiency. Suppose that (3.21) holds. According to Lemma 3.9 all the points (3.19) belong to N(P),and hence (see Remark 3.6) N(P) coincides with the convex hull of (3.19). Using Lemma 3.8 we estimate the left-hand side of (3.20) by a constant times lXlq + 2 I[lrl+...+rj p1q-j j=1 then Definition 3.5 can also be reformulated using inequalities for the elements of the matrix (3.23). Proposition 3.10. For a matrix (3.1) satisfying (3.2), (3.4), (3.5) and (3.14) the conditions of Definition 3.5 hold if and only if for X E C with suficiently large 1x1 the estimates (3.24) lGij([,X)l 5 const (1 + /[l)'i+m~(l[lri + ~X~)-l(~[~rJ+ IXl)-' (i # j), (3.25) lGii([, A)( 5 const (![Iri + lXl)-' hold. Proof. Necessity. According to the definition of the inverse matrix, Gij([,A) = PL1(<,A) det (A(<)- XI)(j*i), where the (q - 1) x (q - 1) matrix (A([)- XI)(jli) is obtained by canceling the j-th row and the i- th column in (3.16). The determinant of this matrix is a polynomial in [ of degree not greater than and a polynomial in X of degree not greater than q - 2 (if i # j) and q - 1 (if i = j). Replacing l[lri by I[lri + 1x1 we estimate the determinant of (A- XI)(jti)by const (1 + ~ (3.26) ~G~~(<,x)I5 const(ltlri + IXI)-~(~~IQ+ IX~)-? . This estimate will have sense also in the case when either ri = 0 or rj = 0 if we formally set = 1 or = 1, respectively. In the case T, = rj = 0 the estimates rj (3.24), (3.25) should be replaced by (3.27) lGij(t,X)I 5 const IXI-’. 3.3. The Kozehvnikov conditions According to Theorem 2.2, the estimate (3.20) holds if and only if conditions (2.16) are satisfied. Now we reformulate these conditions in terms of determinants of “prin- cipal parts” of A - XI. On this way we come to the KOZHEVNIKOVconditions [K2]. For simplicity we begin our analysis with the case of strict inequalities in (3.15), i. e., (3.28) r1 > r2 > > rq 2 0. If (3.28) holds, all the points (3.19) are vertexes of N(P), and the segments that connect them are the sides. If, in addition, rq > 0, then all the points (3.19) (except the origin) are leading vertexes, and the conditions of Theorem 2.2 can be written in the form (3.29) P,.,((,X) # 0 ((X,E)ECxlRn, IXI>O, Id>o, ~=l,...,q), (3.30) pp(t,X)# 0 ((A,() E L: x R”, 1x1 > 0, Id > 0) , where in (3.30) we consecutively take the values of T satisfying the following inequalities here we formally set r0 = 00 and T~+~= 0. 140 Math. Nachr. 192 (1998) Following KOZHEVNIKOV,for given IC = 1,.. . ,q we denote by A(&)(<)the block (3.32) A(IC)(O = (Aij(o)z,j=l,.,,,,,7 and denote by AO(IC)(<) the corresponding principal part (defined by means of the numbers 11,. . . , l,, ml,. . . ,m,,) (3.33) Lemma 3.11. Let (3.28) and (3.31) be satisfied. Then the relation (3.34) Pr(<,A) = det A'(.)(<) (T,, > r > T,,+I) holds for IC = 1,.. . ,q. In the case rq = 0, relation (3.34) is true for IC 5 q - 1. Proof. According to the definition of the T - principal part we calculate P, attaining the weight r to the variable X and take in (2.2) the monomials which (1, r)-order equal to dr(P)= TI + . + r,, + (q - n)r. Now we want to construct a matrix whose determinant equals to the left-hand side of (3.34). We set aij if i#j or i=j= 1, ... , K, a&) = { T if i = j = n+l, ... ,q. To obtain the principal part of A(<) - XI corresponding to the degrees aij(~),we consider the system of relations (3.35) 14.) + mj(4 2 Qij(4 (i,j= L...,q), II (3.36) C (Zi(.) + mi(.)) = TI + * * + rK+ r(q - IC) . i=l It is easy to verify that the numbers l&) = li (i=1,...,IC), mi(tc) = mi (i = 1, ...,K), li(~~)= mi(.) = T/Z (i > IC) satisfy (3.35), (3.36). The principal part of (3.16) corresponding to these numbers is a block diagonal matrix (3.37) where I is the (q - IC) x (q - IC) identity matrix. Taking the determinant of (3.37) we obtain (3.34). 0 Lemma 3.12. Let (3.28) be satisfied. Then the relations (3.38) pr, (<, A) = (-A)'-" det (Ao(~)(t)- A&) , Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 141 hold, where E, is the K x K matrix whose element with indices (K, K) equals 1 and the other elements are zero. Proof. Attaining the weight r, to X and repeating the argument of Lemma 3.11 we obtain that the principal part of (3.16) corresponding to this weight is equal to (3.39) Taking the determinant of this matrix we obtain (3.38). 0 As a consequence of the relations (3.29), (3.30), (3.34) and (3.38) we obtain that under the validity of (3.28) the definition of ellipticity with parameter is equivalent to the following conditions: (3.40) det AO(.)(<) # 0 (I (3.41) det (A0(4(0 - WC) # 0 (14 # 0, 1x1 # 0, X E .c) * Now we consider the general case where some inequalities in (3.15) may be nonstrict. More precisely, suppose that (3.42) r1 = = ~t~ > rtl+l = ..- = rt2 > > rtq-l+l = ... = TtQ 1 0, and set - - (3.43) TI = rtl > P2 = rta , . . . , rQ = rtq . We introduce a block structure in A(<).For a,/3 = 1,.. . ,Q we define the blocks A(.,P)(O = (Ad<))i=te-l+l, ...,tP . j=tp-l+l, ...,tp Accordingly, we write A(<)in the form Replacing Aij by A:j we obtain the principal parts of the blocks which we denote by AO(%PI. We preserve the notation E, for the block matrix whose right - corner block is the identity matrix and the other blocks are zero. Definition 3.13. (KOZHEVNIKOV,see [K2]).We say that the matrix (3.16) has elliptic with parameter principal minors if for K = 1, . . . ,Q det AO(.,.>(<) # 0 (IS1 # 0) 7 and if there exists a ray (1.1) such that 142 Math. Nachr. 192 (1998) Replacing in the arguments above the elements of A by the corresponding blocks we obtain the main result of this section. Theorem 3.14. Suppose that for the matrix (3.1) the conditions (3.2), (3.4), (3.5), and (3.42) hold. Then the following statements are equivalent: (I) The matrix is elliptic with parameter (Definition 3.5). (11) Inequality (3.21) holds (Proposition 3.7). (111) The elements of the inverse matrix (3.23) satisfy the estimates (3.24), (3.25) (Proposition 3.10). (IV) The KOZHEVNIKOVconditions hold (Definition 3.13). 4. Ellipticity with parameter for systems in IR". The case of constant coefficients. In this section we consider the system of partial differential equations in IR" (4.1) A(D)'LL(Z)- WZ) = f(.) , where u = (211, . . . ,u,) and f = (fi, . . . , f,) belong to the function spaces which will be constructed below, and the symbol A(<)satisfies the equivalent conditions of Theorem 3.14. Moreover, to simplify the presentation, we will suppose that the strong condition (3.28)holds. The unessential technical details connected with the replacement of scalar elements by square block matrices are left to the reader. To formulate the statement we introduce the function spaces corresponding to the left-hand and right-hand side of (4.1). Set Q u = U(IR") = nH-j(IRn), j=1 (4.2) where H"(IR") designates the standard Sobolev space. In these spaces we introduce parameter-dependent norms, and we denote by U(X) and F(X) the spaces U and F, respectively, endowed with these norms. We set (4.3) where 11 . I( denotes the norm in L2(IEtn). As in Section 3 we formally set mi/ri = 1 for ~i = 0. The corresponding term on the right-hand side of (4.3) is equal to (1 + IX12)IIuiI12. Defining the norm in F we shall distinguish the cases r, > 0 and rq = 0. In the first case we set (4.4) Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 143 In the general case we suppose that Then we set Theorem 4.1. For the matriz A(() satisfying (3.2), (3.4), (3.5) and (3.15) the following statements are equivalent: (A) The matrix A(()- XI satisfies the equivalent conditions of Theorem 3.14. (B) There ezists a ray (1.1) and a PO > 0 such that for A E L, 1x1 2 PO, and for arbitrary f E F(X) the equation (4.1) has a unique solution u E V(A), and the inequality (4.7) IIu,V(X>ll I KIlf,F(NII holds with K independent of A. (Bo) There exists a ray (1.1) and a PO > 0 such that for uj E Hm(IRn),j = 1,.. . ,q, the a priori estimate holds with K independent of A. P roof . (A) + (B). Applying the Fourier transform to both sides of (4.1) for large 1x1, we obtain .^(O = G(wm, where G(() and f(()denote the Fourier transform of u and of f, respectively. At first, we suppose that all rj are strictly positive. Then, according to the Cauchy inequality, j=1 j=1 If we multiply both sides by (1(12" + IA12)ni'ri and take into account inequalities (3.26), we obtain the estimate Integrating both sides with respect to (, we obtain (4.7). In the case (4.5) the above argument should be slightly modified. For i 5 q' in- equalities (4.9) have sense because (as it was mentioned in Section 3) estimates (3.26) can be extended to the case when one of the numbers Ti, rj is zero but the other is 144 Math. Nachr. 192 (1998) It follows from (3.26) and (3.27) that the first, bracket on the right-hand side can be estimated by a constant independent of t and A. Summing up, we have proved the solvability of (4.1) for 1x1 2 po and the validity of (4.7). (B) + (Bo) =$ (A). The first implication is trivial, the second follows from the lemmas below. Lemma 4.2. Suppose that (4.8) holds for X E L: with 1x1 2 PO. Then for = 1,.. . ,q, rn > 0 we have (4.10) uj E Hm(Rn), j = 1, ... , IE, uj E Hw(IRn), j = 1, ... ,., x E L, 1x1 > 0. Lemma 4.3. (i) Suppose that (4.10) holds. Then (ii) Suppose that (4.11) holds. Then det (Ao(.)(() - XE,) # 0 ((A,<) E L X R",1x1 > 0 , > 0) Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 145 Proof of Lemma 4.2. We begin with deriving (4.10). We replace X and u(z) in (4.8) by Xp' and (4.12) up(.) = (UlP(X),. . . ,ugp(z)),ujp(z) = p':-Pj(%( 3 P x 1, where the numbers r and pj(~)are chosen in the following way (4.13) rn > r > rn+l, pj(rc) = mj, j = 1, ... , IC, (4.14) r pj(lc) = -mj, j 2 ~+l. rj Put Summing up we see that for p + +oo Now we consider 146 Math. Nachr. 192 (1998) Note that Ajk(P0 = P'j+mb (A;k(E) 4- O(P-')) For j = 1,. . . , K. the limit of Jjp as p -+ +oo is equal to For j > K the limit of J;,, as p + +oo is equal to I XpJ/'j I IUj 112. Thus, we have proved the inequality j=1 j=n+l Setting un+l(z) = . . - = uq(z) 0 we obtain (4.10). To prove (4.11) we make the same change in (4.12) with (4.13) replaced by (4.15) Tn = T. Repeating the above arguments almost literally we prove (4.11). 0 P roof of Lemma 4.3. (i) To simplify the notation we suppose that IE = q. Assume that (i) does not hold, i. e., there exists a to# 0 such that det A'([') = 0. Then there exists a vector h = (hl, . . . , hq) E C9, Ihl # 0, such that (4.16) Ao(to)h = 0. Now we take up(.) = (ulp(z),. . . ,ugp(z)), where (4.17) ujP(x) = $(z)exp(ipEo -~)p-~jhj(j = 1, ...,q), and 4(z) E D(IRn). We substitute (4.17) in (4.10) and show that 9 (4.18) c II IDlrnj%jl12 = O(1) (P + +oo) 9 j=1 Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 147 Since (4.18), (4.19) contradict (4.10), we prove our statement. Note that Spj(() = p-m' $(( - p(O) hj . Then as p + +oo j=1 Since IpI # 0 and C lhj12 # 0, we obtain (4.18). Now we calculate the limit of the left-hand side of (4.19) as p + +oo. We have = J ... + J ... l€l>w IEICEP JYp = 1 1&()121(o +p_'(l-21' c cap-ia1(" d(. 148 Math. Nachr. 192 (1998) For < EP we have if E is small enough. Then (ii) As above we assume that n = q. We suppose that there exist to E Rn,1to1 # 0, Xo E L, [Ao\ # 0 and h E Cq, Ihl # 0, such that (4.20) (Ao((') -XoEq)h = 0. We replace X and uj(z) in (4.11) by Xop"- and ujp(z),respectively. Repeating the above arguments we obtain the contradiction which proves our statement. 0 5. Ellipticity with parameter for systems with variable coeffi - cient s 5.1. Plan of the proof of the solvability in Rn We begin with the investigation of unique solvability (for large X E C) of the system where the coefficients are C" and are constant at infinity. More precisely, As in Section 3 we define the numbers 11,. . . ,l,, ml, . . . ,m, such that the relations (3.4), (3.5) are satisfied. We suppose, in addition, that we can choose nonnegative numbers 11, ... ,mq, and they are indexed in such way that inequalities (3.14) and (3.15) hold. To simplify the notation we suppose that (3.28) holds. Using the numbers 11, . . . ,m, we define the principal part of the symbol (5.2) in the standard way: Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 149 Definition 5.1. The symbol (5.2) is called elliptic with parameter if the following conditions are satisfied: (i) For each xo E R" the Newton polygon N(P(xo))of the polynomial P(xo,I, A), where (5.4) Pb,r, 4 = det Ma,r> - XI) 7 contains the points (3.19). (ii) There exists a ray (1.1) and positive d, PO such that the estimate (5.5) d=P(r,A) 5 IP(x,E,A)I for (x,r) E R" x R", A E f., 1x1 I Po , holds. Remark 5.2. (Cf. Remark 3.6). For each xo E R" the polygon N(P(zO))be- longs to the convex hull of the points (3.19). Thus, according to (i) all the polygons N(P(xO))coincide with the convex hull of the points (3.19) and are independent of zo. Due to this fact the notation on the left-hand side (5.5) is correct. Theorem 5.3. For the system (5.1) with coeficients of the form (5.3) the following statements are equivalent: (A) The symbol A(z,()- XI satisfies the conditions (i), (ii) of Definition 5.1. (B) There exists a ray (1.1) and a number PO > 0 such that for A E f. with 1x1 1 po and for arbitrary f E F(A) the equation (5.1) has a unique solution u E U(A),and the estimate (5.6) 11% WVII 5 K Ilf,F(A)ll holds with K independent of A. (Bo) There exists a ray (1.1) and PO > 0 such that for uj E HW(Rn),j = 1,.. . , q, the a priori estimate (5.7) holds with K independent of A. Proof. (A) =+ (B). Set and consider the pseudodifferential matrix operator G(x,D, A) with this symbol. The proof of the solvability of '(5.1) is traditional and is based on Proposition 5.4. Suppose that the conditions of Definition 5.1 are satisfied and the matrix G(x,D, A) is given by (5.8). (i) The operator (5.9) F(4 - W),f t-) G(GD,Nf, is continuous for 1x1 2 PO, A E L, and its nofm is uniformly bounded for such A. 150 Math. Nachr. 192 (1998) tends to zero as X E L, 1x1 + 00. (iii) The norm of the operator (5.11) U(X) + U(X), u + G(x,D,X)(A(x,D) - XI)u - U, tends to zero as X E L, 1x1 + 00. Assuming that the proposition is proved we complete the derivation of the implica- tion (A) + (B). Solvability of Equation (5.1). We seek the solution in the form u = G(x,D, A) g with g E F(X).According to Proposition 5.4 (i), u E U(X).Inserting in (5.1) the expression for u, we obtain an equation for g given by (5.12) g+[(A-XI)G-I]g = f. According to Proposition 5.4 (ii), there exists p1 > 0 such that the norm of the operator in the square brackets is less than 1 for X E t,1x1 2 p1. Then equation (4.11) has a unique solution g E F(X). Proof of the estimate (5.6). Applying the operator G(z,D, A) to both sides of (5.1), we obtain u+[G(A-XI)-I]U = Gf. Therefore, taking into account Proposition 5.4, we obtain According to Proposition 5.4 (iii), &(A) + 0 (1x1 + 00). Hence there exists a p1 such that &(A) c 1 as X E L, 1x1 > p1, and we obtain (5.6). The uniqueness for (5.1) follows from this estimate. (B) =+ (Bo) + (A). The first statement is trivial. The proof of the second implication is a slight modification of the analogous proof in Theorem 4.1 and is based on Proposition 5.5. Suppose that (5.7) holds. Let xo E Rnbe arbitrary and Ao(D)= Ao(xO,D).Then the estimates (4.10), (4.11) are valid for K = 1,.. . ,q (r, > 0) with constant K independent of xo. It follows from the Proposition and Lemma 4.3 that the operator A(xo,D) satisfies the equivalent conditions of Theorem 4.1. Hence (5.5) holds for each xo E Rnwith d = d(zo). Since d(xo) depends only on the constant K in (4.10), (4.11)and K is independent on xo, the constant d(xo) is also independent on xo, and condition (A) holds. Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 151 5.2. Proof of Proposition 5.4 (i) According to the definition of the norms (4.3), (4.4) and (4.6), the F(X) + V(X) boundedness of the operator (5.9) is equivalent to the L2 + L2 boundedness of the operators (5.13) (IDIzri + IX12)~Gij(s,D,X)(ID12rj+ IX12)kB(rJ) where (in correspondence with (4.6)) 19(rj) = 1 for rj > 0 and 19(rj) = 0 for rj = 0. Due to standard estimates for the norms of pseudodifferential operators it suffices to establish the following estimates for D,QGij : (5.14) (/ (5.15) ID,OGij(z,<,X)I 5 Cija(1 + \ (5.16) lD,"Gii(s,<,X)I 5 Ciia(I Indeed, in view of (5.15), (5.16) the left-hand side of (5.14) for 1x1 2 PO can be estimated by 5 5 The estimates (5.15), (5.16) for a = 0 with Cijo(X) independent of X for 1x1 2 po can be obtained by a repetition of the proof of Proposition 3.10 and are based on the representation (5.17) Gij(z, 5, A) = P-'(z, t,A) det ((A(z,<) - X1)jZ) . From the rule of calculation of determinants we easily derive for an arbitrary multi - index y: (5.18) 1D;det ((A(2,t)- XI)ji)l 5 CijT(l+ I Now we derive (5.20). For p = 0 the estimate follows from (5.5) and the equivalence of Zp([,A) and n(l (5.21) It follows from the definition of the Newton polygon that 1DP(z,E,41 L &3%4t,4 Combining this estimate with (5.5), we estimate the expressions in (5.21) by the right - hand side of (5.20). (ii) Direct calculations show that (A(z,D) -- AI)G(z,D, A) - I is a matrix pseudo- differential operator with the symbol 1 (5.22) T(z,E, A) = C -Ja;A(z, E)D,"G(z,E, A). l4>0 In view of the above argument our statement will be proved if we establish the esti- mates of the z-derivatives of symbols (5.22): where ~ijp(A)-+ 0 as 1x1 + 00. According to the definition (5.22) of the matrix T(z,t, A) = (Tij(z,E, A))i,j we have Using (5.15), (5.16) and the obvious estimates we obtain Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 153 Differentiating (5.24) consecutively with respect to Z, we obtain the estimate (5.26) ID:T~~(z,~,x)[ 5 const (1 + ltl)'i+mJ-l(ItP + IW-l . The last inequality permits us to estimate the left-hand side of (5.23) by -%8(rj) (5.27) const (1 + lcl)'i+mj-l (Itl'i + lAl)-$fi(ri) (ltl'j + 1x1) If ri > 0, rj > 0 the expression (5.27) is not greater than const (1 + ltl)'i+mj-l(It1 + pp)-'i(l Thus, we have proved (5.23) with ~ija(X)= ~ijgX-~,where E > 0. (iii) We give only a brief sketch of the proof. According to the commutation formula for pseudodifferential operators, G(z,D, X)(A(Z, D) - XI) - I = T")(z, D,A) + R(N), for arbitrary N = 1,2,. . . , where T(N)is a pseudodifferential operator with the symbol (5.28) The following estimates (cf. (5.15), (5.16)) are important for the proof: (5.30) la,"DtGij(~,E,A)l5 Cij,p(l+ lEl)'i+mJ-'a' (lElri + lwl(ltlrJ+ IW1 (i # d, (5.31) l~~D!Gii(Z,t,~)lI ciia0(1+ ItI)-la1(ItIri+ IXII-' > Using these estimates and repeating the argument used in the proof of (ii) we show that (cf. (5.26)) (ltl'i + IXl)~D~T~~)(Z,t,X)(Itl'J+ lXI)-% 5 COij(X), and the constant on the right-hand side tends to zero as 1x1 + 00. From the last estimate it follows that the operator (5.11) with G(A- XI) - I replaced by T(N)has norm less than 1 for large 1x1. 154 Math. Nachr. 192 (1998) As for the remainder term R"), the expression in the integrand (5.29) has a compact support with respect to y and decreases as const(1 + I< + t(y - Standard estimates of the calculus of pseudodifferential operators permit us to estimate the norm of (5.11) with G(A- XI) - I replaced by R(N). To prove (5.30), (5.31) we use (5.17). For la[ = 1, p = 0 we obtain 8rGi.j = -(8rPP-') det ((A- M>ji)+ P-'dr det ((A- AI)ja) . From this relation (5.30), (5.31) follows for IczI = 1, /3 = 0. Differentiating succes- sively the last relation, we obtain the desired estimates. 0 5.3. Proof of Proposition 5.5 We fix zo E R", n = 1,.. . ,q, and choose the numbers T and pi(.) according to (4.13), (4.14). We replace X in (5.7) by p'X and u(x)by up(z)= ((ulp(z),. . . ,uqp(z)),where 21.3P (z) = p-J(%j (p(z -- z") (j = 1, * . . , q) . As it was shown in the proof of Lemma 4.2, the limit of the left-hand side of (5.7) as p + +oo is equal to Now we shall calculate the limit of the right,-hand side. Without loss of generality we suppose that uj(z)E 23, j = 1,.. . ,q. Set 4 (5.32) fjp(z) = c p3-pb(K)Ajk(2,D)uk (p(Z - 2')) - Xp%-'J((")+' uj(p(z--')) . k=l The right-hand side of (5.7) can written as the square root of (5.33) Denk/Mennicken/Volevich, Newton Polygon and Elliptic Problems 155 According to the definition of the Fourier transform, pnI2-lj &P (PE) = (27r>-n/2 1exp( - ipt - $1 If we set z = xo + p-'y in the integrand we obtain pnI2-lJ&P(PO = (27~)-"/~exp ( - ipt. 2') s exp(-i[. y) Passing to the limit in the integral as p + +ca we obtain (5.34).As a result we prove that expression (5.33) tends to k=l Setting uK+l E =- uq E 0 we obtain inequality (4.10). Inequality (4.11)can be proved in the same way. 0 5.4. Ellipticity with parameter for systems on a compact manifold without boundary Let M be a smooth compact n -dimensional manifold without boundary. It is well - known that the Sobolev spaces Hs(M) can be defined for arbitrary real s. Thus, we can define the spaces U(M)= n HmJ(M) and F(M)= n H-lj(M). In these spaces the analogues of the norms (4.3),(4.4), (4.6)can be defined, and the corresponding spaces will be denoted by U(M,A) and F(M,A). The matrix differential operator A(x,D) defined for x E M will be considered as a continuous operator U(M,A) F(M,A) (u I+ A(x,0)~). Since Definition 5.1 concerns the principal symbol, one can trivially reformulate it for operators defined on M. The standard localization technique used in the construc- tion of parametrices of elliptic operators on manifolds permits us to prove the exact analogue of Theorem 5.3. Theorem 5.6. For the matrax operator A(x,D) with C" coeficients defined on a smooth compact manifold M the following conditions are equivalent. 156 Math. Nachr. 192 (1998) (A) The principal symbol Ao(z,<)- XI satisfies the conditions of Definition 5.1 for each x E M. (B) There ezcists a ray (1.1) and PO > 0 such that for X E C with 1x1 2 po and for arbitrary f E T(M,A) the equation A(z,D)u - XU(Z) = f (z) has a unique solution u E U(X), and the estimate holds with a constant K independent of A. (Bo) There ezists a ray (1.1) and a number po > 0 such that for uj E C"(M), j = 1,.. . , q, the a priori estimate holds with K independent of A. Acknowledgements The authors would lake to express their deep gratitude to M.S. AGRANOVICHfor fruitful discussions and very useful remarks concerning the exposition. The support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. The paper was written during the stay of the third author in Regensburg under the grant of the Deutsche Forschungsgemeinschaft. Added in proof. In [K3] some 2 x 2 block systems which are elliptic with parameter are considered on a manifold with boundary. The boundary conditions are assumed to be homogeneous and of triangular form. 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Nauk SSSR 132 (1960), No 1, 20-23; English trans]. in Soviet Math. Dokl. 1 (1960), 458-461 VOLEVICH,L. R.: On a Linear Programming Problem Arising in Partial Differential Equations, Uspekhi Mat. Nauk 18 (1963), No. 3, 155-162 CJniversitZt Regensburg Keldysh Institute NWF I - Mathematik of Applied Mathematics D - 93040 Regensburg Russian Acad. Sci. Germany Miusskaya sqr. 4 e -mail: 125047 Moscow robert.denkOmathematik.uni- regensburg.de Russia reinhard.mennickenOmathematik.uni - regensburg.de e -mail: volevichOspp.keldysh.ru