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M 9X.,...3X N and X 0<|J3|+2J«M 1 ' * ' N Be P DEMONSTRATIO MATHEMATICA Vol. II No 1 1970 Andrzej Borzymomski SOME PARTIAL DIFFERENTIAL EQUATIONS OF EVEN ORDER INTRODUCTION Let X = JV-'V A , ^n 9x.,...3OX x O < M R n and X 0<|j3|+2j«M 1 ' * ' n be partial differential operators of elliptic, and parabolic n type respectively, where IjS = 0,; M=2p is an even i=1 J 1 posit'ive integer; X(x^,...,xn) denotes a variable point in the Euclidean space En(n»2), and t a variable in a finite interval ,(0,T). In this paper we shall consider some special cases of oper- ators X and X (see pp.5 and 18) containing, for sufficient- ly regular coefficients in-the leading parts,the p-th iterates n 2 ~ of second-order operators L = ^ a j(I) 3x 3x ' an<* L 5 7,6=1 T V d = y , a j(X,t) a__ —Jr- respectively, and possesing - 5 - 2 A.Borzymowski quasi-solutions given in a closed form. Hence,relying on some methods and results known for M=2, we can obtain fundamental solutions of the appropriate differential equations ((3) and (44)) as well as generalised potentials relative to them in a form more effective and partly under weaker assumptions than that in the theory of full equations or systems of order M. By making use of the potentials mentioned above we can subsequently solve some boundary problems for the equations (3) and (44) which, to the best of our mind, have not been considered up to now under such assumptions or with the aid of such methods. We also examine the dependence of solutions of the boundary problem related to equation (3) on the given functions appearing in this problem. The paper consists of five sections. In the first three we construct fundamental solutions for the equations (3) and (44) assuming regular continuity for their coefficients and then we develop a potential theory relative to (3) in a multiconnec- ted region. More importantly, Poisson's equality for the volume potential is proved and a limit property for transversal and oblique derivatives of the potential of a simple surface distribution is examined. To achieve this we develop Levi's parametrix method of constructing a fundamental solution as well as some methods and results of W.Pogorzelski in the theory of second order equations. In the fourth section we prove the existence of a solution of a boundary problem for equation (3), with a non-linear boundary condition containing transversal and oblique deriva- tives of the unknown function. The given functions appearing in this condition may have discontinuities in the sense of W.Pogorzelski's class . The existence of a solution of the problem is proved by applying Schauder's fixed point theorem. In the fifth section we examine the dependence of solutions of the boundary problem studied in section 4 on the functions given in that problem* i.e.the coefficients and the right-hand side function of the differential equation and the given func- tion appearing in the boundary condition. To this end we con- - 6 - Some partial differential equations of even order 3 sider two function spaces, linear and normed, and establish some relationships between the norms as well as the distances of their points. The main outcome of that section (see theorem 12) states that the solutions cf the examined boundary problem depend on the given functions in a regularly continuous way, where the continuity is understood in the sense of the distan- ces mentioned above. The sixth section is devoted to the examination of a gene- ralised Cauchy problem with p initial conditions for equation (44). This problem is effectively solved by reducing it to an integral equation. As it is not possible to apply in this case the ordinary Poisson-Weierstrass integral we make use of some other auxiliary integrals which satisfy the initial conditions but which do not satisfy the differential equation considered. Let us note that the main results in the examination of fundamental solutions and generalised potentials for the equation X [u] =0 were achieved by E. L e v i [l 5] for n=2 and b (X)eCM+1, P. J o h n [14] for arbitrary n nR '"finn and analytic coefficients, Ya.B. Lopatinskii [18J (for a system of elliptic equations with differentiable' coefficients), and in the case of M=2 by M.G e v r e y [13] and W. Pogorzelski [2l] for Holder-continuous coefficients. For the equation X [u] =0 analogical investiga- tions have been conducted for M=2 by P.D r e s s e 1 [il] , for b„ „ (X, t ) e C2, and bv W. Pogorzelski n-'V [22] and D.A r 0 n s 0 n [8J, for Holder-continuous coeffi- cients, and in the case of W>2, for parabolic systems of equations with Holder-continuous coefficients, by W.P 0 g o - r z e 1 s k i [29] and S.D. Eidelman (see [12])1. Boundary problems for the elliptic equation x[u]=0 or systems of elliptic equations have been examined by many authors by way of a priori estimates and Hilbert space methods {in which Our formula for the fundamental solution of (44) is more effective than that resulting from the theory of parabolic systems. - 7 - 4- A.Borzyraowski case very general results were obtained but,in the main,rather strong conditions concerning the coefficients and the boundary of the considered domain were assumed (comp..[6], [7], [32] , [33])], and also by means of the integral equations method by E. L e v i [lb], and S. A g m 0 n [5] for n=2, by Ya. B. Lopatinskii [17] for differentiable coefficients and arbitrary M and n, arid by W.P ogorzelski ([21] , [27]) for M=2 and the coefficients satisfying Holder's con- dition. Further, boundary value problems for elliptic equa- 0 Vi tions in class J\have been considered only in the case of the Laplace equation and spherical surfaces' (see [24], [34]). The dependence of solutions of non-linear boundary value problems on the given functions appearing in the probLems (see section 5) has not, to the best of our mind, been examined up to now. The Cauchy problem studied in section 6 is a generali- sation of the initial problems examined by M. II i c 0 1 e s c u [20] and 0. Berechet [9] for the one-dimensional ite- rated heat equation and at the same time it is some sense more general than the Cauchy problem relative to a parabolic system solved in [12] (see pp. 251-256), where the differentiability of the coefficients of the system is assumed. Also the method used for solving the problem is different from that in the papers cited above. The author wishes to express his gratitude to Professor dr Andrzej Plis for the suggestion of the subject of section 5 and for his valuable remarks concerning the whole paper. 1. CONSTRUCTION OP A FUNDAI/IENTAL SOLUTION POP. THE ELLIPTIC LIQUATION Let ^ be a given measurable bounded domain ?n E , and let |aT)i(X)| and j b^ m.. U)'] » where fttf = 1,... ,n; More general problems in class f( have been considered by H. Adamczyk and the author of this paper ifor parabolic equations of second order in [3] and [4-]. - 8 - Some partial differential equations of even order 5 fi^,... tji^ = 0,...,M-1 be finite sequences of real functions determined in the closure of a domain Si' (also bounded and measurable) containing the closure of 2 . We assume that the conditions h - . - . 1l | a s (X) - aT<f(X) X| (1) h. (klfk2>0; h1th2e(0,l]) are satisfied for X,Xg£'. We assume furthermore that a (z) The matrix [ Tj] is symmetric and the form n U(X) = 12 a^ftt)*,^(x) (2) T,(f=t V V«r (where A,...is positive definite at each point of In this section we shall study the equation >a «H-2IV.«,«) f K'°- (3) where Ao( (X)<* = ac* ft (x) acf (X) 1 * * M Eff-« 1 M oC.|...dx aj Firstly, let us assume that the coefficients a^ are constants and denote 2 - 9 - 6 A.Borzymovski Evidently, the following equality ri v ,• • M p fu] S > . Du-- L fu] (5) LJ L J is valid. This equality is of a fundamental importance for the whole present section. In the sequel we shall find the fundamental solution (f.s.) of L^ [u] = 0, preceding it by some preliminaries. Condider the function r I 1 LtKX,Y)J logi>(X,Y) for n=2m; m=1,2,...,p w(X,Y) = (6) M-n in all other cases where X(x^,...,xn) and ) are twc different points of E_E , and -iHX,Y) is their "elliptic dinstance" given by nn the formula #(X,Y) = (7) L T,i=1 T<T (a being the elements of the inverse matrix of the matrix Lemma 1. The following formula holds P 1 L ' [w(X,Y)] = J3nfp • W0(X,Y) (8) where |>(X,Y)] 2"n for n > 2 w0(X,Y) = (9) log ^(X,Y) for n = 2 We call these cases briefly the logarithmic cases. - 10 - Some partial differential equations of even order (r-1)m22p~3(p-m)!(m-2)!(p-1)! when n=2m; m=2,...,p r P~1 T [2 (p-1)ïj n = 2 (10) (-2) (p-1)!(n-4)(n-6)...(n-2p) inali other cases Proof, being similar but easier in all other cases, will be given only for n = 2m; 1 < p.
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