JOHN'S DECOMPOSITION THEOREM FOR GENERALIZED. METAHARMONIC FUNCTIONS. DAVID COLTON. 1. Introduction. A result of basic importance in ...
JOHN'S DECOMPOSITION THEOREM FOR GENERALIZED METAHARMONIC FUNCTIONS DAVID COLTON 1. Introduction A result of basic importance in the theory of metaharmonic functions [14] is John's decomposition theorem [12, 11], which states that any metaharmonic function regular in an exterior domain can be uniquely decomposed into the sum of a solution regular in the entire space and one which satisfies the Sommerfeld radiation condition
-^-wA =0
(1)
uniformly in all directions. Here r is the modulus of the position vector and n is the dimension of the space. This theorem for example plays a central role in the derivation of asymptotic expansions and uniqueness theorems [4, 11]. In this paper we consider regular solutions of the singular partial differential equation " d2u
d2u
2v du
-TT + T T + - T 2
+M==0
'
(2)
dx dyr y dy where v is a fixed real parameter and which for v equal to a positive integer can be interpreted as being a metaharmonic function in n = 2v + 2 variables depending only on the two variables x = xi} y = (x22 + ... +xn2)*. Solutions regular on some portion of the axis y = 0 are known as generalized metaharmonic functions [10]. The axis y = 0 is a singular curve of the regular type [8,9] with exponents 0 and 1—2v. Consequently there always exist solutions of (2) which are regular on some portion of the axis, and for 2v ^ 0, — 1, —2, ... each such regular solution can be continued across the axis as an even function of y. Therefore for 2v # 0, — 1, — 2, ... every solution regular on some portion of the axis is a real analytic function of x and y2 in some domain D that is symmetric with respect to the singular line y = 0. Hence a generalized metaharmonic function can be expressed as u(x, y) = u(r, £) where x = r cos 9, y = r sin 6, £ = cos 6. For v > 0, we have at our disposal Weinacht's (renormalized) fundamental solution [15] K
n(xy n(x,y,
xOOty) yo) = =
[ r ( v ) ] 2 22i-2v
| / r ' t f ^ O O s i n 2 " - 1 tdt,
where R = [(x-x0)2+y2+yo2-2yyocost]i and H v (1) denotes a Hankel of the first kind of order v (Weinacht's original formula was expressed in Neumann's function rather than Hankel's function). For future reference that Cl(x, y; x0, y0) is a n analytic function of x, y and v for v > 0, (x, y) ^ By using Green's second formula for equation (2) [9]
J
( ^ f r )
H y2v[uLv(w)-WLv(u)]dxdy,
8DnR+
Received 9 December, 1968. [J. LONDON MATH. SOC. (2), 1 (1969), 737-742] JOUR. 4
(3)
function terms of we note (x0, y0).
(4)
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DAVID COLTON
where^R+ = {(x, y) \ y > 0}, Lv = -5-3 + j~i + — ~r and n is the outward normal to D, the following theorem for equation (2) can be proved in the same manner as in John's work, viz an application of formula (4) with u a regular solution of equation (2) and w set equal to the fundamental solution Q (cf. [5; p. 315] or [11]): THEOREM 1. Assume v > 0. Let u be a regular solution of equation (2) in the exterior of a bounded domain D. Then u can be uniquely decomposed as
u = U+V,
(5)
where U and V are regular solutions of (2), U is regular in the entire plane, and V satisfies the radiation condition
>-*)- °
(6)
uniformly for —1 ^ ^ < 1. For v < 0 this standard method of analysis is no longer applicable due to the fact that the integrals defined in formulae (3) and (4) do not exist for v < 0 and v < —\ respectively, and it is to this problem we now address ourselves. The approach adopted is to analytically continue the function defined by (3) into the range v < 0 and then to apply a relationship motivated by formula (4) where the path of integration is chosen to lie on the Riemann surface of the integrand 2v
/ _dw_ _ du_
The resulting decomposition theorem is of particular interest in that it now turns out that the radiation condition (6) must hold uniformly for £ contained in a region lying in the complex £ plane, and not simply for — 1 ^ £ < 1 as in the case of Theorem 1. An example will be given showing that this reult is best possible. This seems to indicate that analytic function theory not only provides a powerful method for studying generalized metaharmonic functions, but, as in the case with singular ordinary differential equations, is in fact the correct and natural avenue of approach. 2. Analytic continuation of the fundamental solution Setting x = r cos 8, y = r sin0, xQ = p cos cp, y0 = p sin q>, we can express R as R = [r2 + p 2 — 2rp(cos 8 cos cp + sin 8 sin
(22) (23)
2i — the uniform convergence of the series (17) and (20), and the orthogonality property (16), we have V(r, O =
_A1
d[a-'Jv+H{a))
—
(25)
JOHN'S DECOMPOSITION THEOREM FOR GENERALIZED METAHARMONIC FUNCTIONS
741
where the series (25) is uniformly convergent for a < Ro < r ^ Ri < oo, t, contained in some ellipse T in the complex £ plane inclosing [—1, +1]. From Theorem 4.3 of [1] it is seen that (25) is a regular solution of equation (2) for r > a satisfying lim r v+ *
(f-H
uniformly for £ e T. From equation (18) we therefore can write u(r, Q = l/(r, f) + F(r, 0 ; where
r > a,
«, U(r, 0 = r" v £ an Jv+n(r) Cnv(£)
(27) (28)
is uniformly convergent for Rt ^ r ^ i? 0 > a. In view of equation (13), this implies that U(r, £) is a real analytic function of r and ^ in the entire (x, y) plane and this fact along with formula (27) shows that U(r, £) is an everywhere regular solution of equation (2). The decomposition (27) is unique since if U(r, £) is a generalised metaharmonic function which is regular in the entire plane and also satisfies the radiation condition (26), then the orthogonality property (16) and the series representation (18) shows that u(r, £) must be identically zero. We have thus proved the following theorem: THEOREM 2. Assume v < 0, 2v # — 1, —2, —3, .... Let u be a regular solution of equation (2) in the exterior of a bounded domain D. Then u can be uniquely decomposed as u = U+V, (29)
where U and V are solutions of (2), U is regular in the entire plane, and V satisfies the radiation condition
4^-'^
=0
(30)
Or J uniformly for t, contained in some ellipse T in the complex ^ plane inclosing [—1, + 1 ] in its interior. Example. For v < —£ the ellipse T cannot be replaced by the line segment [—1,4-1]. For in this case U(r, £) = el>5 is a solution of equation (2) regular in the entire plane which also satisfies the radiation condition, i.e. the decomposition is no longer unique. Remark. For v > — \ the condition that (30) holds uniformly in T is implied by the weaker requirement that (30) holds only for £ e [— 1, +1] [1]. References 1. D. Colton, " A contribution to the analytic theory of partial differential equations", / . Differential Equations, 5 (1969). 2. , " Jacobi polynomials of negative index and a nonexistence theorem for the generalized axially symmetric potential equation ", SIAM J. App. Math., 16 (1968). 3. , " On the analytic theory of a class of singular partial differential equations ", to appear in the Proceedings of the Symposium on Analytic Methods in Mathematical Physics (Gordon and Breach, 1969).
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JOHN'S DECOMPOSITION THEOREM FOR GENERALIZED METAHARMONIC FUNCTIONS
4. D. Colton, " A priori estimates for solutions of the Helmholtz equation in exterior domains and their applications ", to appear in / . Math. Anal. Appl. 5. R. Courant and D. Hilbert, Methods of mathematical physics, Vol. //(Wiley, 1962). 6. A. Erd&yi, Higher transcendental functions, Vol. I (McGraw-Hill, 1953). 7. , Higher transcendental functions, Vol. II (McGraw-Hill, 1953). 8. , " The analytic theory of systems of partial differential equations ", Bull. Amer. Math. Soc, 57 (1951). 9. , " Singularities of generalized axially symmetric potentials", Comm. Pure Appl. Math., 9 (1956). 10. R. Gilbert, Function theoretic methods in partial differential equations (Academic Press, 1969). 11. P. Hartman'and^C. Wilcox, " On solutions of the Helmholtz equation in exterior domains ", Math. Z., 75 (1961). 12. F. John, Recent developments in the theory of wave propagation (N.Y.U. Lecture Notes, 1955). 13. G. Szego, Orthogonal polynomials (Amer. Math. Soc. Colloquium Publications, 1959). 14. I. Vekua, New methods for solving elliptic equations (John Wiley, 1968). 15. R. Weinacht, " Fundamental solutions for a class of singular equations ", Cont. Diff. Eqns., 3 (1964).
Indiana University, Indiana, U.S.A.
