Internat. J. Math. & Math. Sci. VOL. 12 NO 2 (I989) 385-390 385 KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS ABDULLAH ALTIN Faculty of Science University of Ankara Besevler, -Ankara, Turkey EUTIQUIO C. YOUNG Department of Mathematics Florida State University Tallahassee, Florida 32306 (Received September 7, 1988) ABSTRACT. The classical Kelvin principle concerns invariance of solutions of the By employing a hyperbolic- Laplace equation with respect to inversion in a sphere. polar coordinate system, the principle is extended to cover a class of singular equations, which include the ultrahyperbolic equation. Kelvin principle, Laplace equation, ultrahyperbolic equation, KEY WORDS AND PHRASES. Lorentzian distance. 1980 AMS SUBJECT CLASSIFICATION CODES. i. 35LI0, 35C99. INTRODUCTION. As is well known the classical Kelvin principle introduced in 1847 (Thomson [i]) concerns solutions of the Laplace equation. For solutions of some class of elliptic differential equations and their iterated forths in n independent variables, n z 2 the extension of Kelvin principle is usually proved using rectangular coordinates [2], Germain and Bader [3], Huber [4], Weinstein [5] ). (Diaz and Martin In a generalization of Kelvin principle was established by Weinstein [5] for equation 2 n r. i=l using polar coordinates. k u + x.1 i x. 1 u k.l 0, x.1 Following Weinstein method we shall give in this Kelvin principle for solutions of 1960 the the class of < paper a new formulation singular partial of differential equations n L(u) where . (i E i=l (2,,v_ + ui u) 2 x.1 i S n) and 1 x.l B. m Z x.1 j=l 2 8j 8u ( +---- +yj yj yj r2 s m) (i u i are real P(u) 0 parameters, r is the (i.]) ],orentzia metric defined by 2 r n F. i=l 2 x. 1 m I j=l 2 y_ (1.2) A. ALTIN AND E.C. YOUNG 386 and P is a general linear operator of arbitrary vanishing for u Zp L The domain of the operator of class C2(D)nCq(), n, denote points in R u 2. in R n+m order q in the variables Zl, z 2, 0. where x m R and R is the set of all real valued functions u(x,y,z) (Xl,. p, x n ), ,ym p D x R is a regularity domain of (Yl respectively, and (Zl,...,z) and z p x R HYPERBOLIC-POLAR COORDINATE SYSTEM FOR EQUATION (i. i). First let us consider the n+m-dimensional Laplace operator 2 n Au 2 3u 7. + x. i=l 2 m u 7 j=l 1 (2.1) 2 Xn+j and introduce the polar coordinates x x x x x x r I r cos8 cos8 ...cos8 cos8 cos8 ...cos8 sin0 1 2 m+l m-i m m+n-2 m+n-i 2 m 3 cosOlcos82 .cOSSm_ 1 cos0 m cos8 m+l ...sin8 m+n-2 r cos0 cos0 ...cos0 cose sin0 1 2 m-i m m+l n r cosO n+l e. i cose 2 ...cos0 r sine m sin01 n for N m-i (2.2) r cos8 sine 2 i n+m-i Xn+m where 0 cosolcoso2...cOSSm_iCOSSmCOSOm+ 1...cosSm+n_2cosom+n_ 1 l, r n+m-2, 0 N 8 [x 2 I n+m-i + x2 n+m + 2 N and )% (2.3) Under this change of variables, the polar form of the Laplace operator is given by n+m Au 2 2 u ---Z ---Z D u 1 where the operator I u + n+m-I + Dr r Dr i=l Dx. 2 r depends only on the variables 1 (u) (2.4) @I, n+m-i is the and r Euclidean distance given by (2.3). Now in (2.1) and (2.2) let i /-i and cos(iCj) 8m+.] @.] for Cj and sin(iOj i let ch iy4 Xn+ i sh ., and n-l. e icj Since for (D/DXn+ j) i 2 m with 2 u (D/Dyj) u, the operator (2.1) reduces to the ultrahy- perbolic operator 2 [h= 7. i:l u ---T x. 2 Z Du2 j:l On the other hand, the polar coordinate system (2.2) takes the form (2.5) KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS x x x x I 2 3 n Yl Y2 r chich2"’’ChCm-lChCmCSl’’’cSn-2CS@n-i r chich2...Chm_iChmCOSl...cOSn_2Sin@n_l r chich2...Ch#m_iChmCOS@l...sinn_ 2 r chich 2.. "ChCm-lChmsinl r chich2...Chm_iShm chich#2...ShCm_ 1 r r Ym_l (2.6) chish 2 r she Ym 38F I where r is the Lorentzian distance given by (1.2). We refer this to as "polar- hyperbolic transformation". In the polar-hyperbolic coordinate system, the operator (2.5) assumes the form n Z []u i=l where 2 2 2 m Du ---,/- Dx.I 2 Du E j=l Du + [ 2 r Dr yj u + n+m-i depends only on the variables Dr 1 + -u r u 2 + Au= u rr + r is given in (1.2). 2, 3 are given by 1 --_Uo0 r u r r (2.7) ’n-i and i’’’" ’m’l For example, the polar forms of Au for n Au 1 --z2(u) r (u + r OO 0 + cos sin O 1 Uo sin2 0 u The corresponding forms for the hyperbolic equations are given by [ rl [h where u xx + u yy r --Ur r 2 U r 2 + zz 1 --U r Urr u 1 + u 1 r r x 2 + Y 2 Uee --z + 2 sh0 1 --Ucho O ch2o and r x u ch0cos, y r chOsin z r sh0 3. A FUNDAMENTAL THEOREM AND MAIN RESULT. In [5] using his main three recursion formulas, Weinstein gave the followir, theorem which will be used to establish our main result. THEOREM I. Let v v(r,l,...,n_l) 2 D v ---2 D r + k Dv 1 r Dr r where k is a real or complex number and for v satisfy the differential equation 2 (v) (3.1) is a linear, differential operator vanishing l-k Then w satisfies the same 0 which is independent of the variable r. A. ALTIN AND E.C. YOUNG 388 equation (3.1) in the variables w(p, v(i/P,l n_l 1 P’ where i’’’’’ n-i i/r p and n_l Using Theorem 1 we can now establish an extension of Kelvin principle to ultrahyperbolic equations THEOREM 2. i. 1 ). u(x l,...,xn,yl,...,ym Let u , Then w r -I Xl ur x Yl r r ’zl’’’’’zp) Ym -, r z be a solution of the equation ,z I (3.2) P is also a solution of the same equation (i.i), where n I n + m 2 + m Z u. + Z 8. i i=l j=l (3.3) and r is the Lorentzian distance defined by (1.2). Let us consider the polar-hyperbolic transformation (2.6) which can be written in the following form PROOF. x r i functions of i l,...,n (3.#) gj(,), fi(,), gj(,) or without ’m’l"" "’n-i r yj where the notations i 1 m subscripts f(,), g(,) denote We note that from (3.#) we have x. r x. Yk Yk Dr 3yj where 6jk + Dr 7. m + yj is the Kronecker delta. __r xj x.] Dr x i=l (i < 7. i=l + 7. i=l Yk 6jk x] 6jk i YJ From (1.2) we have _< n) and YJ __Dr y. and we may express the partial derivatives (i _< j <_ m) r 8i/8xj, 8i/Sx4j and 8i/8yj the denominator of which is the Jacobian of the transformation (2.6). as a quotient It can shown that (x I,.. 3(r, 1 ,Xn,Y 1 m,i Ym n+m- I r h ,) Cn_l The numerator of this quotient contains obviously only a factor r n+m-2 hence be KELVIN PRINCIPLE FOR A CLASS OF SINGULAR EQUATIONS On the other hand, since x. Du Dx. m Du Dr r Du Yj Du Dyj r Dr m 1 1 Du 1 Du xj Dxj r z + Dr yj Dyj and 2 l(U), r 1 Du + r Dr 1 Du i Fji( n-i Du i=l D. # 1 < j < n 2(u)’ r 1 Since + n n + m r l-(n+m-i + w 1 + m r _k x the substitution P 2 xi/r (i) + Du 7. j=l 8j) x i "’m’@l "’’’n-i (i.i), then our equation (i.i) 1 + (u) (3.6) 0 r by Theorem 1 k ,..,- ,-Yl ,..,- I u(r Y r r r ,Zl,..,zP Here we note that, since n 2 and I Xi=le i + 7.j= i 8j satisfies the equation (i.i). by m X e. i:l n -< m depend only on the ariables becomes 2 (3.5) -< If we substitute the expressions (2.7) and (3.5) into D u D r 3x r i=l D# ix i we see that where the operators _Du r i=l + H..(,) (,), =--G --Fji(,), 389 (xi/r 2) i=l Z 2 m 2 1 (yj/r2) :j=l 7. r i/r in the solution u means replacing the variables x_. and y_. yj/r 2, ] respectively. We note that, since the r defined by (1.2) is not real for the solutions of (i.i) is valid only in the domain D x n D x D n m D {(x,y) n+m. x e D n y D m 7. i=l 2 x. 1 , 7.n.1=i X.l2 where m > < Z i-=- y 2 J is a hyperconoidal domain in R Here D and D are the spherical domains centered m n n p is the regularity domain of R at the origins of R and R respectively, and m, u with respect to the variable z. A. ALTIN AND E.C. YOUNG 390 (ii) In equation (i.i) if we have addition instead of subtraction of the two summations, then Theorem 2 remains valid, where r n 2 m 2 2 Z x. + 7. 1 i=l j=l yj This includes Weinstein’s [5] and Altin’s [2] results. 7u where Y= const, the formula (3.2) gives the In the special case Pu result obtained in [2] (iii) (iv) If we multiply both sides of the equation (i.i) by -I, we get the equation 2 m Z j:l -n(u) where r ( u + 2 yj zmj=l Yj2 Enj=l x.]2 -r 2 2 n j u yj yj j u u Z (-----2 + x. x. j:l This shows that if Ym Xl -r -r -r -r 0 r U(Xl,..,Xn,Yl,..,Ym,Z l,..,zp) is a solution of the equation (i.i), then by Theorem 2 yi x n -X r U( 1 Wl (u) + x. Zp) zl where X is given by (3.3) It is clear is also a solution of the same equation (i.i), 2 2 that this solution is valid in a different domain where r I > 0 that is, r < 0 (v) It is clear that by a simple linear transformation, Theorem 2 also holds for a more general equation of the form 2 n a2( i Z where a n o i bi,e i o o "(nl’’’’’m w 8 e. u i 1 + m u Z ? 8 are real parameters (a are fixed points in r -X 2 b.32( u( alr where 2 anr nz (i-i). 2 o r 2 i=l a. n 2 ’ 0 u) + q 0, b. 0), 1 Dn and Dm, n-n i- 8. u nm-nm2 blr m Z j=l 2 bmr jb. 0 r o (,..., on respectively nl-n (u) + and Here ’Zl’’’’z P o J)2 ] REFERENCES I. THOMSON, W. Extraits de deus Lettres Addressees a M. Liouville, J. Math. Pures AppI. 12(1847), 256. 2. ALTIN, A. 3. GERMAIN, P. and BADER, R. Some Expansion Formulas for a Class of Singular Partial Differential Equations, Proc. Amer. Math. Soc. 85(1982), 42-46. Sur le Problems de Tricomi, Rend. Circ. Mat. Palermo 2(1953), 53-70. 4. HUBER, A. 5. WEINSTEIN, A. Some Results on Generalized Axially Symmetric Potential Theory, Proc. Conf. Partial Diff. Eqs., University of Maryland, 1956, 147-155. 359-365. 6. On a Singular Differential Operator, Ann. Mat. Pura Appl. 49(1960), DIAZ, J. B. and MARTIN, M. H. Riemann’s Method and the Problem of Cauchy, II. The Wave Equation in n Dimension, Proc. Amer. Math. Soc. 3(1952), 476-483.