Horová, Ivana

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Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumIvana HorováOn the variational principles for partial differential equationsArchivum Mathematicum, Vol. 24 (1988), No. 1, 17--23Persistent URL: http://dml.cz/dmlcz/107305Terms of use:© Masaryk University, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVŮM MATHEMATICUM (BRNO) Vol. 24, No. 1 (1988), 17-24 ON THE VARIATIONAL PRINCIPLES FOR PARTIAL DIFFERENTIAL EQUATIONS IVANA HOROVA*) (Received April 22, 1986) Abstract. Using a modified theory of potential operators as given in ]6], we find explicit formulas for the construction of the potential for the given partial differential equation. Key words. Gateaux differential, Gateaux derivative, potential operators. MS Classification. 35 A 15. 1. PRELIMINARIES Let X be a real Banach space, X* its dual. Let F : X -> Y be a map between two Banach spaces. When Fhas aCd teaux differentialat apoint x e X in direction h e X we will denote it by DF(x> h). When the map h -» DF(x, h) from Xto Fis linear and conti­nuous, we say that Fhas a Gateaux derivative&t the point x and we denote this map by DF(x) and write DF(x).h for DF(x, h). When /is a functional (i.e. Y = R), we write <Df(x), A> = Df(x). h. Definition. Let F : X -• X* and let X0 be a closed subspace ofX. We say that F is a potential operator with respect to X0, if there exists a functional f: X -* R such that: 1. Df(x) exists for all xeX. 2. <2)f(x),/i> = <F(x),/i> for all xeX and all heX0. Such functional is called a potential for F with respect to X0. The following theorem ([6]) is an extension of the classical theorem of Vajn-berg [5]. Theorem. Let F : X -> X* have a Gateaux-derivative DF(x) at all xeX. Let the functional (DF(x). ft, fe> be continuous in Xfor allh, keX. Let X0 be a closed subspace of X. Then F is a potential operator with respect to X0, if and only if *) This article was presented in Poster session of Equadiff 6. 17 I. HOROVA <DF(x). h9 ky » <DF(x). k9 A> for all h9keX0 and all xeX. 2. FORMULATION OF THE PROBLEM AND RESULTS Let n be a positive integer, e a smooth function of real variables x9 y9 yJt,..., yh... Jr where l i f ^ w , 1 £jt ^ ... ^jr g n. Consider a partial differential equation (1) e(x'9 u(xl)9 DJtu9..., Dh .,. Dju) = 0. Necessary and sufficient conditions for equation (1) to be variational are given the following relations ([2]): (2) *-l-tf-s*- + t (Mi)"' ""W^ ° s ' s '• °yji...ji cyji..ji m=i+i \ * / °yji...jm where dtu denotes the formal (= total) derivative of a function u with respect to the coordinate xK Remark. As it is proved in [1] equations (2) can be satisfied only for r even. Let us put r == 2k. Let Q be a domain in Rn9 with the Lipschitz's boundary dQ9 Q = flu 30. Let X = C2k(Q)9X0 <=. Xresp. X! c Xbesubspaces of functions whose partial derivatives up to the (A: — l)st order resp. (2k — l)st order vanish on dQ. For the equation (1) we define an operator^ : X -+ X* by the following relation <Au9 v> = J v(x) e(xV w, ..., Dh ... Diaku) dx. a Remark. In the next we suppose e to be a sufficiently smooth function. Theorem. The operator A is potential with respect to X0 if and only if conditions (2) are satisfied. Proof. It is necessary to prove <DA(u9 v)9 w> = <DA(u9 w)9 v} for all ue X and all w9 v e X0. We have 2k de <DA(u9 v)9 w> = J w £ -5- Djr.. Djrv dx. o r = i oyJlmmmJr Now, we integrate by parts and first we apply partial integration on the last terms. We can find the following expression: <DA(u9 v)9 w> -= - J dj^w^y—jDj, ... .D^.^dx + 18 ON THE VARIATIONAL PRINCIPLES de + J w-35 *>h -Dj*-W»*S + ... da °yji..j2k - J d J * n a vd B )D1 . - Di*-its»-ids + an \ cyji..j2k / de + J w - ^ ^i,.>.,I2k-i»yI2kds ~ a.o cyji..j2k - J^2k-iU^T^ W • ^2k-a»d* + » \ cyji..j2k-i/ de + J " " ^ ^ l - ^ 2 k - 2 ^ 1 2 k - l d S + . -an °yji..j2k-i 2k - = E JC....i,A-^dx + Z--0 í ì + ï I í PÿXZÏťDsг - ö A . - ßi,^+. dS, --2 * 2 * r = l r<Sf,m:S2*Ô.O where yjk -SJ* component of the outer unit normal vector and P£;J = P((;;;̂ (x'tM» ^1iw» DjfljiU*...) are defined by the following reccurence relations: pUl 12k) |__L P(0) " -ð*Z Jik pUl 12k-2) __ ^fi A p(11,..,12k-l + l) MO) ~ -37; ai2k-i + i MO) °yjl..j2k-2 1 ^ / ^ 2 f c - 1, (X\ P^1 Jlk-r-l) __, _ pUl 12k-r) _ J pUl, ...,12k-r) \ W M12k-I I2k-r) - M/2k-!,...,12k-r+l) Q12k-1 + lM12k-1 + 1 I2k-r+ 11> 0 ^ / ̂  2k - r, 0 g r _£ 2fc. Since t> e X0> w e X0» aM boundary terms vanish. From (3) we can obtain -« . . -« - ( - ! / •# -+ z ( -D m (T)c-d,„^ - , cyji..jr m - i + i \ * / °yjt...jm From here it follows the symmetry condition is valid if and only if the conditions (2) are satisfied. .•* Lemma. Functions I^;;;mj$9 1 _£ r s_ 2k — 1, r <; /, m £2k, do no/ depend on derivatives of the order greater than 2k. Proof is an immediate consequence of reccurence relations (3). Our problem consists in constructing a potential for A with respect to X0. In the paper [6], the potential is constructed by means of a projection operator. We shall 19 I. HOROVA try to construct the potential without a definition of a projection operator. Our main result can be formulated as follows: Theorem. Let conditions (?) be satisfied and let functions P^^;;;fjl\ k ^ r ^ 2k — 1, r <5 1, m £ 2fc — 1 do not depend on the 2k-th derivatives. The functional f defined by the relation i f(u) = J ds J ue(x\ su9 sD^w, ..., sDjx ... Dj u) dx — o n 2k - j\ds jds' j Y Y I,((i::::::t)'V.s«. -0 O f t k = r .,m = r + l ... sDh ... Djk_lU9 ssfDh ... DJku9 ...9ssfDh ... DJ2k_iU) Dh ... DJruDjm ... DJruyJr+l dS. is the potential for A with respect to X0, i.e. <D/(w), v> =- <Au9 v> for all ueX and ve X0. Proof. We shall write the functional/in the form (4) / ( t / ) = / i ( w ) - / 2 ( w ) , where i fx(u) = J ds J ue(x\ su, ..., sDJt ... Dj2ku) dx, 0 ft 1 1 2 * - l 2 * - l /2(u)=fsds|ds'J £ I i f c f c ,0 * 0 0 Oft r = * l,m = r+l x(x\ su9 ..., ss'Z)^... Dj2k_iU) Dh ... DJruDJm ... DJtuyJr+ldS. 1 Now, we calculate the Gateaux derivative of/ t. We get I <Dfi (")> v> = J ds J vs(x\ su, ..., sD^ ... DJ2ku) dx o ft 1 2* fa isdsiuZ-^—Dh...DJlvdx. o ft /=o °yji...ji As we apply a partial integration for the second term and use the symmetry conditions (2) we can write i <Dfi(u)9 vy = J ds J vs{x\ su9..., sDh ... Dj2ku) dx + 0 ft 1 2k de + J sds J v £ -r D>1... Dy.w F boundary terms = o ft i--o ^y«i...ji = I "77[J re(*'' 5U' ••' sDh ••• ̂ 2*") d*] d 5 + o a 5 o 4- boundary terms = <.4w, t;> + boundary termf. 20 ON THB VARIATIONAL PRINCIPLES For boundary terms (BT) we have BT = jsdsj Y Y pgr.::/;,rv,^,.... 0 Oft r = 0 J,m=r+1 - > sDJl9 ..., Dj2k^u)Dh ... DjrvDh ... DjmuyJr+i dS. For x e X!, it is #T = 0 and (Dfx(u\ v> = <Au, v>9 ueX^veXi-Now, we can define an operator A : X -> X* as follows: <ATw, t>> = BT for all Vyue X. Then (5) <£/i("), *>> = <-4", v> - <Au, v> for all u,ve X. The operator .A is potential with respect to X0, Df(u) is potential with respect to every subspace and then A is potential with respect to X0. We shall prove that the potential for A is the functional/2(w). Let us compute <-0/2(w), v} It is 1 1 2 * - l 2 * - l <Z>/2(«), t>> = JsdsJds'J £ £ 0 0 dQ r = * I ,m=r-H • ->*-->*. - ->A«^+, + It.X ••• ->*«->*. •• VJWJ^ + + P\:;]Dh...DjrvDjm...Dj,uyJr^dS. For uGl 0 we obtain [ 1 1 ч 2 * - l 2 * - l <D/2(и),t.> = JsdsJds'J £ £ ._ 0 0 ðß r = * J,m=r+1 2* 5 P ( , ) >=* °Уji...jP + i t Х ... -V^m ... Ðлиľл+Л dS == 1 1 л Г 2 * - l 2 * - l = ísdsíf7 I I K-Чfczfe0 O 0 đS L ' = - í ,m=г+lðЯ (x', su, ..., sÐy,... Dл.1м, ss'D,,... .Dли,..., ss'ÐУl... ••• ->м-.«)Dn ••• V % • •• ->л«ľлłt dSjds'. 21 I.HOROVA From here it follows: (6) <Df2(u\v> = <Au9v> for all u e Xand veX0. Combining (4), (5), (6) we get our final result (Df(u\ v> = <.D/i(f<), v> - <Df2(u\ v> = (Au9 v> + (Au9 v> - <D/2t/, t;> and for all u e X and v e X0: Examples. We shall illustrate the preceding theory on some examples. In all cases conditions (2) are satisfied and we shall only construct the potential. '• -£( '—•i ,£ ) -£( '—' i ,£ ) - 0 ,^>s0-The potential is of the form / («)=/ . («) -f2(u), where /1W-jVd.J.{-^(|.r«i.|"*)-^.(|I«i.|'*)dxd,. i i r fiu ' fM = J s ds J ds' J MSV2 I grad u |2 -^- Vl + o o aa L °* „ 2 a(fa\3 „ 2 ,2 3u / d u \ 2 + .uA ' 2 | g r adw | 2 - ^v 2 + 2sV2w(-^--j y2 + + -v^(^>4j""»d" ,^ -s -Then for/(w) we have In the following examples the potential is constructed in a similar way. _ d (\ bu I"-2 bu\ b (\ 8u | p _ 2 bu\ . . . . _ The potential takes the following form tt . l t i\ du I*. I 5" I ' l / , , /(") = — 1 ". --— + - r - I* dx ay. JK P a II 5x | 1 ^ 1 J 3, u'u("" + uV" + i - u"2 = 0, x e <0,1>. 22 ON THE VARIATIONAL PRINCIPLES The potential is /(«) = y {2 ] "" V áx + 1 j «'« dxj. 4 g f 1 du\ d { l g " 1 0 ) e f í 5* lVl + |grad«|2 dx) ^ IVl + IgraduT^J *' * * | |z "*J  W l  |grad The potential is of the form ™-iVt+G-Ш fc* REFERENCES [1] R. W. Atheгton, G. M. Homsy, On the existence and formulation of variationalprinciples for nonlinear differential equations, Studies in Appl. Math., vol. LIV, No. 1, 1975, pp. 31 —60. [2] D. Krupka, Lepagean forms in higher order variational theořy, Proceedings of the IUTAM-ISIMM Symposium, Turin, 1982, pp. 197-238. [3] E. Tonti, Variational formulations of nonlinear differential equations, I. Acad. Roy. Belg. Bull. Cl. Sci. 55 (1969), 137-165. [4] E. Tonti, Variationalformulations of nonlinear differentialequations IL Acad. Roy. Belg. Bull. Cl.Sci. 55 (1969), 262-278. [5] M. M. Vajnberg, Variacionnyje metody issledovanija nelinejnych operatorov, Moskva, Nauka 1972. [6] A. L. Vanderbauwhede, Potential operators and variational principles: a generalization, Hadronic Journal 2, 1975, 620-641. /. Horová Department of Mathematics Faculty of Science, J. E. Purkyně University Janáčkovo nám. 2a, 662 95 Brno Czechoslovakia 23 

On the variational principles for partial differential equations

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On the variational principles for partial differential equations

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Institute of Mathematics AS CR, v. v. i.