Complex Anal. Oper. Theory DOI 10.1007/s11785-017-0671-7
Complex Analysis and Operator Theory
Harmonic Families of Hypersurfaces Eleutherius Symeonidis1
Received: 10 October 2016 / Accepted: 30 March 2017 © Springer International Publishing 2017
Abstract We establish a general concept of families of hypersurfaces over which the integral of a harmonic function remains invariant. Passing to the limit these hypersurfaces often degenerate, a fact that renders the majority of the classical mean value properties, and also helps to find new ones. Keywords Harmonic function · Mean value property · Quadrature identity Mathematics Subject Classification 31B05 · 31B35
1 Introduction Let be an open subset of Rn , n ≥ 3, h : → R a harmonic function. For reasons of convenience in the subsequent formulas we assume that 0 = (0, 0, . . . , 0) ∈ . It is well known that for a ball with center 0 ∈ Rn and radius r > 0, which is contained in , we have the mean value property 1 ωn
S n−1
h(r x) dσ (x) = h(0) ,
Communicated by Lucian Beznea.
B 1
Eleutherius Symeonidis
[email protected] Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85071 Eichstätt, Germany
E. Symeonidis
where σ is the Euclidean measure on the sphere S n−1 = {x ∈ Rn : |x| = 1} and ωn its area. In spherical coordinates, this formula takes the form 1 ωn
π
π
...
0 n−2
· sin
0
s1 sin
2π
0 n−3
h (r cos s1 , r sin s1 cos s2 , . . . , r sin s1 sin s2 . . . sin sn−1 )
s2 . . . sin sn−2 dsn−1 dsn−2 . . . d s1 = h(0).
The spheres are not the only hypersurfaces, over which mean value properties are known. Let us recall such a property over ellipsoids of revolution. We assume that the prolate ball x12 cosh2 r
+
x22 + · · · + xn2 sinh2 r
≤1
(r > 0) with foci at (−1, 0, . . . , 0) ∈ Rn and (1, 0, . . . , 0) ∈ Rn is contained in . Then, it holds (see [6]): 1 ωn
π
...
0 n−2
0
π 2π
h(cosh r cos s1 , sinh r sin s1 cos s2 , . . . , sinh r sin s1 . . . sin sn−1 )
0 n−3
s1 sin s2 . . . sin sn−2 dsn−1 dsn−2 . . . ds1 1 ( n2 ) n−3 = h(x, 0, . . . , 0)(1 − x 2 ) 2 dx . n−1 √ ( 2 ) π −1 · sin
These two mean value properties are proven in a different way. Nevertheless, our experience in the two-dimensional situation (see [3,5]) has given us reasons to believe that there should exist a common treatment of these and many other higher dimensional mean value properties. It is the purpose of this article to present a general concept, which can be applied to establish these two mean value properties and many others. The basic idea of this concept is to look at the aforementioned mean value properties in a slightly different way. Not the equality with the right side is of prior importance, but the fact that the left side does not depend on the parameter r . In the first property, the mean value of a harmonic function over concentric spheres does not change. In the second one, the concentric spheres are replaced by confocal ellipsoids of revolution. The equalities with the right sides follow from these invariance properties, because concentric spheres can shrink to the center, confocal ellipsoids of revolution to the interfocal segment. It is an easy exercise to obtain the right sides as limits of the left sides for r → 0. If we primarily understand a mean value property as an invariance property, we are naturally led to the concept of a “harmonic” family of hypersurfaces.
2 Harmonic Families of Ellipsoids In this section we develop the concept of a harmonic family of hypersurfaces, in order to derive a mean value property with respect to arbitrary ellipsoids in Rn , which
Harmonic Families of Hypersurfaces
generalizes that one in the introduction. In fact, this surface mean value property can be derived from the volume mean value property over ellipsoidal bodies in [1]. Yet, our primary purpose in this article is to convey the concept of a harmonic family of hypersurfaces, and the case of ellipsoids is its very first application. A family of ellipsoids centered at 0 ∈ Rn and having their axes on the coordinate axes is given by the parametrization F(r, s1 , . . . , sn−1 ) := (a1 (r ) cos s1 , a2 (r ) sin s1 cos s2 , . . . , an (r ) sin s1 . . . sin sn−1 ) , (2.1) where s1 , . . . , sn−2 ∈ [0, π ], sn−1 ∈ [0, 2π ] serve as parameters of each ellipsoid, and r ∈ [0, R] (R ∈ (0, ∞)) is the family parameter, which determines the lengths a1 (r ), . . . , an (r ) of the semiaxes. Now let h be an arbitrary harmonic function on a region that contains the ellipsoids of the family together with their interiors. We shall call the family harmonic, if the integral
π 0
... 0
π
2π
h (F(. . .)) sinn−2 s1 sinn−3 s2 . . . sin sn−2 dsn−1 dsn−2 . . . ds1
0
(2.2) is invariant with respect to r . In our approach, this concept of invariance will be attributed to the divergence theorem. In this context we require the partial derivative of the parametrization F with respect to r to be orthogonal to all the other partial derivatives. In fact, this is the condition in the following theorem. Theorem 1 If a1 (r )a1 (r ) = a2 (r )a2 (r ) = · · · = an (r )an (r ) for all r , then (2.2) is invariant with respect to r . For the proof we shall need the higher-dimensional vector product, which we recall in a definition and three lemmas. Definition 1 Let e1 = (1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1) ∈ Rn . For v1 = (v11 , . . . , v1n ), . . . , vn−1 = (vn−1,1 , . . . , vn−1,n ) ∈ Rn we define their vector product by
v1 × · · · × vn−1
v11 .. := . vn−1,1 e1
. . . v1n .. .. . . ∈ Rn , . . . vn−1,n . . . en
where e1 , . . . , en are treated as mere symbols in the computation of the determinant. Lemma 1 The vector product v1 × · · · × vn−1 has the following properties, by which it is uniquely determined. 1. ∀ j ∈ {1, . . . , n − 1} v1 × · · · × vn−1 ⊥ v j .
E. Symeonidis
v1 .. . 2. = |v1 × · · · × vn−1 |2 ≥ 0, where v1 , . . . , vn−1 and their vector v n−1 v1 × · · · × vn−1 product are treated as row vectors. 3. |v1 × · · · × vn−1 | is equal to the volume of the parallelotope spanned by v1 , . . . , vn−1 . Proof (1) and (2) follow straightforwardly from the definition. For (3) we remark that the volume in question is equal to the volume of the orthotope (rectangular polytope) vn−1 w j spanned by w1 := v1 , w2 := v2 − wv21ww11 w1 , . . . , wn−1 := vn−1 − n−2 j=1 w j w j w j as defined by the Gram–Schmidt orthogonalization process. (For simplicity we denote by just vw the inner product of vectors v and w.) The rest is a consequence of the next lemma and the relation |v1 × · · · × vn−1 | = |w1 × · · · × wn−1 |, which follows inductively. Lemma 2 For pairwise orthogonal vectors v1 , . . . , vn−1 ∈ Rn it holds: |v1 × · · · × vn−1 | = |v1 | · . . . · |vn−1 | . Proof The statement follows from Definition 1 and the properties of the determinant. Lemma 3 The volume of the parallelotope spanned by vectors v1 , . . . , vn−1 ∈ Rn is equal to |v1 |2 v1 v2 v2 v1 |v2 |2 .. .. . . vn−1 v1 vn−1 v2
1 . . . v1 vn−1 2 . . . v2 vn−1 .. . .. . . . . . |vn−1 |2
Proof The determinant in the statement is the so-called Gram determinant of the vectors v1 , . . . , vn−1 . If we introduce orthonormal coordinates in the span of v1 , . . . , vn−1 , we easily see that this determinant is the square of the determinant that has the coordinates of these vectors as its columns (or rows). So, the statement follows from the well known fact that a determinant represents the volume of the parallelotope spanned by its columns (up to the sign). Now we are ready to prove the theorem. Proof of the theorem By (2.1) we have: ∂ F(r, s1 , . . . , sn−1 ) = a1 (r ) cos s1 , a2 (r ) sin s1 cos s2 , . . . , an (r ) sin s1 . . . sin sn−1 , ∂r
Harmonic Families of Hypersurfaces
∂ F(r, s1 , . . . , sn−1 ) ∂ F(r, s1 , . . . , sn−1 ) × ··· × = sinn−2 s1 sinn−3 s2 . . . sin sn−2 ∂s1 ∂sn−1 −a1 sin s1 a2 cos s1 cos s2 . . . an cos s1 sin s2 . . . sin sn−1 0 −a2 sin s2 . . . an cos s2 sin s3 . . . sin sn−1 · , .. .. .. .. . . . . e2 ... en e1
(2.3) where we suppressed the argument r in the determinant. Here we observe that the cofactor of each e j contains the product a1 ·. . . · aˆj ·. . . ·an as a factor. After extracting these factors, the further computation is reduced to the case a1 = · · · = an = 1 of the unit sphere. In this case, it is easily seen ∂F F , 1 ∂ F , . . . , sin s1 ...1sin sn−2 ∂s∂n−1 are pairwise orthogonal unit vectors, and that ∂s 1 sin s1 ∂s2 all of them are orthogonal to F, which (still in the case a1 = · · · = an = 1) is a unit vector too. From the above lemmas it follows that the vector product of ∂F 1 ∂F ∂s1 , . . . , sin s1 ... sin sn−2 ∂sn−1 is equal to n F, where n ∈ {1, −1}. Testing this for s1 , . . . , sn−1 → 0 we see that n = (−1)n+1 . This reasoning implies that the determinant in (2.3) is equal to (−1)n+1 ·(a2 . . . an cos s1 , a1 a3 . . . an sin s1 cos s2 , . . . , a1 . . . an−1 sin s1 . . . sin sn−1). We now observe that due to the assumption in the theorem the quantities a j a1 · . . . · aˆj · . . . · an
=
a j a j a1 · . . . · an
are equal for all j ∈ {1, . . . , n}. Denoting their value by α(r ), the previous computations show that ∂F ∂F ∂F · (−1)n+1 sinn−2 s1 . . . sin sn−2 = α(r ) · × ··· × ∂r ∂s1 ∂sn−1 (we have suppressed the arguments of the partial derivatives). We finally conclude: d dr
π
0
=
π
... 0
π
...
0
= (−1)
0 n+1
2π
h (F(r, s1 , . . . , sn−1 )) sinn−2 s1 . . . sin sn−2 dsn−1 dsn−2 . . . ds1
0
π 2π 0
α(r ) 0
∇h (F(. . .))· π
... 0
π
0
∂ F(. . .) n−2 sin s1 . . . sin sn−2 dsn−1 dsn−2 . . . ds1 ∂r
2π
∇h (F(. . .))
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∂ F(. . .) ∂ F(. . .) × ··· × dsn−1 dsn−2 . . . ds1 ∂s1 ∂sn−1 = α(r )
h(x1 , . . . , xn ) dx1 . . . dxn = 0 , ·
E(r )
x2
where E(r ) denotes the n-dimensional ellipsoidal ball a (r1 )2 + · · · + 1 according to Gauss’ divergence theorem in n dimensions.
xn2 an (r )2
≤ 1,
The following mean value property with respect to ellipsoids is a consequence of this theorem. Corollary 1 If h is a harmonic function on the ellipsoidal ball
x12 x22 x2 + +· · ·+ An2 2 A1 A22 n
≤ 1,
where A1 ≥ A2 ≥ · · · ≥ An−1 > An > 0, then the following mean value property holds: π 2π π ... h (A1 cos s1 , A2 sin s1 cos s2 , . . . , An sin s1 . . . sin sn−1 ) 0
0
0
· sinn−2 s1 . . . sin sn−2 dsn−1 dsn−2 . . . ds1 h(x1 , x2 , . . . , xn−1 , 0) 2 = dx1 . . . dxn−1 , 2 2 (A21 − A2n ) . . . (A2n−1 − A2n ) E 1− x1 − · · · − xn−1 A2−A2 A2 −A2 1
where E ⊂ Rn−1 denotes the ellipsoidal ball
n
x12 A21 −A2n
n−1
+ ··· +
n
2 xn−1 A2n−1 −A2n
≤ 1.
Proof Let an (r ) := An − r for r ∈ [0, An ]. If we define a j (r ) :=
an (r )2 + A2j − A2n for 1 ≤ j ≤ n − 1,
the conditions of the theorem are satisfied. Therefore, the values of (2.2) for r = 0 and r = An are the same:
π
...
0
0 n−2
π
2π
h (A1 cos s1 , A2 sin s1 cos s2 , . . . , An sin s1 . . . sin sn−1 )
0
s1 . . . sin sn−2 dsn−1 dsn−2 . . . ds1 π 2π ... h A21 − A2n cos s1 , A22 − A2n sin s1 cos s2 , . . . , 0 0 0
2 An−1 − A2n sin s1 . . . sin sn−2 cos sn−1 , 0 sinn−2 s1 . . .
· sin =
π
sin sn−2 dsn−1 dsn−2 . . . ds1 π π π =2 ... h A21 − A2n cos s1 , A22 − A2n sin s1 cos s2 , . . . , 0 0 0
2 An−1 − A2n sin s1 . . . sin sn−2 cos sn−1 , 0
Harmonic Families of Hypersurfaces
sinn−2 s1 . . . sin sn−2 dsn−1 dsn−2 . . . ds1 .
(2.4)
The statement follows by applying the change of variables formula. Remark 1 The ellipsoid (x1 , . . . , xn−1 , 0) :
x12 A21 − A2n
is called focal ellipsoid of the ellipsoid
x12 A21
+ ··· + + ··· +
2 xn−1
A2n−1 − A2n xn2 A2n
=1
= 1.
Remark 2 In the case when An−1 = An , the innermost integral (and possibly some of the following ones) in (2.4) can be computed, thus leading to a simpler formula. This corresponds to a degeneration of the focal ellipsoid. In the case of a prolate ball (A2 = · · · = An ) see, for instance, [6]. Remark 3 For the sake of completeness we finish by stating the two-dimensional analogue of this mean value property: If h is a harmonic function on (an open neighborhood 2 2 of) the elliptic disk ax 2 + by2 ≤ 1, where a > b > 0, it holds that
√ a 2 −b2 √ − a 2 −b2
h(x, 0) 1 dx = √ 2 a 2 − b2 − x 2
π −π
h(a cos s, b sin s) ds .
For a proof see [2].
3 Harmonic Families of Hyperboloids The mean value property that we have just derived has an analogue in the case of hyperboloids. The author thanks professor Masaharu Nishio for the suggestion to study this case. To determine this analogue, we follow the path of the previous section and make the necessary adjustments. We consider the family of hyperboloids in Rn with equations x12 x22 xn2 − − ··· − = 1 , where x1 > 0 . 2 2 a1 (r ) a2 (r ) an (r )2 This family is described by the parametrization H (r, s, s1 , . . . , sn−2 ) := (a1 (r ) cosh s, a2 (r ) sinh s cos s1 , a3 (r ) sinh s sin s1 cos s2 , . . . , an (r ) sinh s sin s1 . . . sin sn−2 ) , where s ∈ [0, ∞), s1 , . . . , sn−3 ∈ [0, π ], sn−2 ∈ [0, 2π ] serve as parameters of each hyperboloid, and r ∈ [0, R] (R ∈ (0, ∞)) is the family parameter. The functions a1 (r ), . . . , an (r ) are assumed positive.
E. Symeonidis
Now let h be a harmonic function on a region that contains all hyperboloids of the family together with their interiors, that is, the sets x12 x22 xn2 − − · · · − ≥ 1 , x1 > 0 . a1 (r )2 a2 (r )2 an (r )2 We call the family harmonic, if the integral ∞ π
0
0
π
... 0
2π
h(H (. . .)) sinhn−2 s sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds
0
(3.1)
is invariant with respect to r . Theorem 2 If a1 (r )a1 (r ) = −a2 (r )a2 (r ) = · · · = −an (r )an (r ) for all r , then (3.1) is invariant with respect to r , under the condition that lim
x1 →∞ E (x ) r 1
∂h (x1 , x2 , . . . , xn ) dx2 . . . dxn = 0 , ∂ x1
where Er (x1 ) = (x2 , . . . , xn ) :
x22 a2 (r )2
+ ··· +
xn2 an (r )2
≤
x12 a1 (r )2
−1 .
Proof We begin by computing partial derivatives and a vector product. ∂ H (r, s, s1 , . . . , sn−2 ) = a1 (r ) cosh s, . . . , an (r ) sinh s sin s1 . . . sin sn−2 , ∂r ∂ H (. . .) ∂ H (. . .) ∂ H (. . .) × × ··· × = sinhn−2 s sinn−3 s1 . . . sin sn−3 ∂s ∂s1 ∂sn−2 a1 sinh s a2 cosh s cos s1 . . . an cosh s sin s1 . . . sin sn−2 0 −a2 sin s1 . . . an cos s1 sin s2 . . . sin sn−2 · .. .. .. .. . . . . e1 e2 ... en (here and in what follows we often suppress the argument r ). Recalling the identities sin(is) = i sinh s and cos(is) = cosh s we observe that this determinant resembles the one in (2.3) if we replace there a1 by ia1 , s1 by is, and s j by s j−1 for 2 ≤ j ≤ n − 1. Its value is therefore (−1)n+1 (a2 . . . an cosh s, −a1 a3 . . . an sinh s cos s1 , . . . , −a1 . . . an−1 sinh s sin s1 . . . sin sn−2 ) . Now, the assumption in the theorem allows us to define α :=
a2 a1 an =− = ··· = − a2 . . . an a1 a3 . . . an a1 . . . an−1
Harmonic Families of Hypersurfaces
as a function of r , with the help of which we can write ∂H ∂H ∂H ∂H × ··· × · (−1)n+1 sinhn−2 s sinn−3 s1 . . . sin sn−3 = α(r ) · × ∂r ∂s ∂s1 ∂sn−2 (we have suppressed the arguments of the partial derivatives). Therefore, the differentiation of (3.1) with respect to r gives: d dr
∞ π
0
...
0
0
0
π
...
0
2π
h (H (. . .)) sinhn−2 s . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds
0
∞ π
=
π
0
2π
∇h (H (. . .)) ·
0
∂ H (. . .) ∂r
· sinhn−2 s sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds π 2π ∞ π ... ∇h (H (. . .)) = −α(r ) · 0
0
0
0
∂ H (. . .) ∂ H (. . .) ∂ H (. . .) ·(−1) · × × ··· × dsn−2 dsn−3 . . . ds1 ds ∂s ∂s1 ∂sn−2 = −α(r ) ·
h(x1 , . . . , xn ) dx1 . . . dxn = 0 n
Y (r )
according to Gauss’ divergence theorem, where Y (r ) denotes the interior of the correx2
x2
x2
sponding hyperboloid of the family, that is, the set a (r1 )2 − a (r2 )2 − · · · − a (rn )2 > 1, n 1 2 x1 > 0. (Due to the condition in the theorem, the flux of ∇h through {x1 } × Er (x1 ) vanishes as x1 → ∞.) The following mean value property with respect to hyperboloids is a consequence of this theorem. Corollary 2 If h is a harmonic function on the hyperboloidal body xn2 A2n
x12 A21
−
x22 A22
−···−
≥ 1, x1 > 0, where A2 ≥ A3 ≥ · · · ≥ An−1 > An > 0, such that lim
x1 →∞ E(x ) 1
∂h(x1 , x2 , . . . , xn ) dx2 . . . dxn = 0 , ∂ x1
where E(x1 ) := (x2 , . . . , xn ) :
∞ π 0
π
...
0 n−2
0
2π
x22 A22
+ ··· +
xn2 A2n
≤
x12 A21
− 1 , then
h(A1 cosh s, A2 sinh s cos s1 , . . . , An sinh s sin s1 . . . sin sn−2 )
0 n−3
s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds h(x1 , x2 , . . . , xn−1 , 0) dx1 . . . dxn−1 2 , = · 2 2 2 2 x n−1 x 12 x 22 (A1 + A2n )(A2 − A2n ) . . . (An−1 − A2n ) Y − − ··· − −1 · sinh
s sin
A21 +A2n
A22 −A2n
A2n−1 −A2n
E. Symeonidis
where Y ⊂ Rn−1 denotes the hyperboloidal body x1 > 0.
2 xn−1 x12 x22 − A2 −A 2 −· · ·− A2 −A2 A21 +A2n n n 2 n−1
≥ 1,
Proof Let an (r ) := An −r for r ∈ [0, An ]. If we define a j (r ) := an (r )2 + A2j − A2n for 2 ≤ j ≤ n − 1, a1 (r ) := A21 + A2n − an (r )2 , the conditions of the theorem are satisfied. Therefore, the values of (3.1) for r = 0 and r = An are the same:
∞ π
0
0
π
... 0
2π
h(A1 cosh s, . . . , An sinh s sin s1 . . . sin sn−2 )
0 n−3
s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds · sinhn−2 s sin ∞ π π π =2 ... h A21 + A2n cosh s, A22 − A2n sinh s cos s1 , . . . , 0 0 0 0
2 An−1 − A2n sinh s sin s1 . . . sin sn−3 cos sn−2 , 0 · sinhn−2 s sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds . The statement follows by applying the change of variables formula.
(3.2)
Remark 4 As in the case of ellipsoids, if An−1 = An , at least one integral in (3.2) can be computed, thus leading to a simpler formula. Remark 5 For the sake of completeness we finish by stating the two-dimensional analogue of this mean value property: If h is a harmonic function on the “hyperbolic 2 2 disk” ax 2 − by2 ≥ 1, a, b > 0, x > 0, which decays sufficiently fast at infinity, it holds that ∞ h(x, 0) 1 ∞ dx = h(a cosh r, b sinh r ) dr . √ √ 2 −∞ x 2 − a 2 − b2 a 2 +b2 For a proof see [4].
4 Harmonic Families of Hypersurfaces of Revolution In this section we consider families of hypersurfaces of revolution and parametrize them by S(r, s, s1 , . . . , sn−2 ) := (a(r, s), d(r, s) cos s1 , d(r, s) sin s1 cos s2 , . . . , d(r, s) sin s1 . . . sin sn−2 ) , d(r, s) ≥ 0, (4.1) where we regard s ∈ I (I a closed interval in R), s1 , . . . , sn−3 ∈ [0, π ], sn−2 ∈ [0, 2π ] as the coordinates of each hypersurface, and r ∈ [0, R) (R ∈ (0, ∞]) as the family parameter. Furthermore, we assume that all the hypersurfaces are closed, provided that they are bounded. More precisely, we assume that d(r, s) = 0 for all r ∈ [0, R) and s ∈ ∂ I .
Harmonic Families of Hypersurfaces
If we proceed as in the previous sections, we arrive at the following theorem for harmonic functions h on a domain that contains all hypersurfaces of the family together with their interiors, that is, the sets {(a(r, s), x2 , . . . , xn ) : x22 + · · · + xn2 ≤ d(r, s)2 , s ∈ I, r ∈ [0, R)}. Theorem 3 If there exist smooth functions r → b(r ) and s → j (s) such that ∂ S(. . .) ∂ S(. . .) ∂ S(. . .) ∂ S(. . .) × ··· × · b(r ) , · j (s) sinn−3 s1 . . . sin sn−3 = × ∂r ∂s ∂s1 ∂sn−2 then the integral I
π
π
...
0
0
2π
h (S(. . .)) j (s) sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds
0
(4.2)
is invariant with respect to r , provided that the integral
d(r,s) π 0
· j (s)ρ
0 n−2
... sin
π
0 n−3
2π
∂1 h (a(r, s), ρ cos s1 , . . . , ρ sin s1 . . . sin sn−2 )
0
s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 dρ
vanishes for s → ∂∞ I , where ∂∞ I denotes the boundary of I in R ∪ {∞, −∞} and ∂1 h the partial derivative of h with respect to its first variable. Remark 6 If I is compact, the last condition is trivially satisfied due to the assumption in the beginning. Proof It holds: d dr
I
π
0 n−3
π
... 0
2π
h (S(r, s, s1 , . . . , sn−2 )) j (s)
0
· sin s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds π π 2π ∂ S(. . .) = j (s) ... ∇h (S(. . .)) · ∂r I 0 0 0
π
· sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds = b(r ) I
·
0
...
∂ S(. . .) ∂ S(. . .) ∂ S(. . .) × ··· × dsn−2 dsn−3 . . . ds1 ds. × ∂s ∂s1 ∂sn−2
0
π
2π
∇h(S(. . .))
0
Now, if I is compact, this integral covers the whole flux of ∇h through a closed hypersurface, due to the aforementioned assumption. If I is unbounded, the last condition in the theorem shows that the flux of ∇h through the “top” and the “bottom” of the enclosed solid vanishes. Therefore, in both cases, by Gauss’ divergence theorem the last term is equal to
E. Symeonidis
± b(r )
D(r )
h(x1 , . . . , xn ) dx1 . . . dxn = 0 ,
where D(r ) denotes the enclosed solid.
This theorem gives a sufficient condition for a family of hypersurfaces of revolution to be harmonic. In the sequel we examine its consequences on the appearance of the family, that is on the functions a(r, s) and d(r, s). By changing the variables r and s if necessary, we may assume without restriction that b(r ) ≡ 1 ≡ j (s). Then (denoting partial derivatives by subscripts), ∂ S(r, s, s1 , . . . , sn−2 ) = (ar (r, s), dr (r, s) cos s1 , . . . , dr (r, s) sin s1 . . . sin sn−2 ) , ∂r ∂ S(. . .) ∂ S(. . .) ∂ S(. . .) ∂ S(. . .) × × × ··· × ∂s ∂s1 ∂s2 ∂sn−2 √ n−2 n−3 n−4 = 2ds d sin s1 sin s2 . . . sin sn−3 as 1 √ √ cos s1 √1 sin s1 cos s2 . . . √1 sin s1 . . . sin sn−2 ds 2 2 2 2 0 − sin s cos s1 cos s2 . . . cos s1 sin s2 . . . sin sn−2 1 0 − sin s2 . . . cos s2 sin s3 . . . sin sn−2 . · 0 . .. .. .. .. .. . . . . e e2 e3 ... en 1 We observe that this determinant is of the form (2.3) if we take there a2 = · · · = an = 1, s1 = π4 , and replace si by si−1 for 2 ≤ i ≤ n − 1 and a1 by − adss . Thus, this determinant is equal to
1 as as as · √ ,− √ cos s1 , − √ sin s1 cos s2 , . . . , − √ sin s1 . . . sin sn−2 . 2 ds 2 ds 2 ds 2
(−1)
n+1
If we insert what we have computed into the hypothesis of the theorem, we arrive at the system ar = (−1)n+1 d n−2 ds =
∂ (−1)n+1 d n−1 , ∂s n−1
dr = (−1)n d n−2 as .
(4.3)
This nonlinear system of the first order can be reduced to a single (nonlinear) equation of the second order in the following way. Let the function (r, s) → v(r, s) be such that a = vs . Then, (vr )s = (vs )r = ar = n+1 n−1 ∂ (−1)n+1 d n−1 , so vr = (−1)n−1d + w(r ) with an appropriate function w(r ). If we ∂s n−1 n+1 n−1 replace v by v − w, we obtain a = vs and vr = (−1)n−1d . Furthermore, if we take into account the second equation of (4.3), we have vrr = (−1)n+1 d n−2 dr = −d 2n−4 as = −d 2n−4 vss .
Harmonic Families of Hypersurfaces
Since d n−1 = (−1)n+1 (n − 1)vr , we finally obtain the desired equation, namely 2n−4 vrr = − (−1)n (1 − n)vr n−1 · vss .
(4.4)
5 Applications I. The infinite cylinder x22 + · · · + xn2 = d 2 , d > 0 This cylinder becomes part of the family (4.1) (for r = 0), if we put the initial conditions a(0, s) = s, d(0, s) = d for all s ∈ R. For the function v at the end of the previous section, this amounts to vs (0, s) = s and vr (0, s) =
(−1)n+1 d n−1 n−1
(5.1)
for all s ∈ R. Guided by the first initial condition, we seek for a solution of (4.4) under the functions of the form v(r, s) =
s2 + β(r ) . 2
—Let n be odd. Then, (4.4) and (5.1) imply 2n−4
2n−4
β = −(n − 1) n−1 · (β ) n−1 ,
β (0) =
d n−1 . n−1
The differential equation is equivalent to 4−2n
2n−4
β (β ) n−1 = −(n − 1) n−1 . For n = 3, together with the initial condition it gives 3−n 2n−4 n−3 n−1 n − 1 3−n · (β ) n−1 = −(n − 1) n−1 · r + ·d (n − 1) n−1 , 3−n 3−n n−1 1 3−n (n − 3)r + d 3−n , β = n−1
and finally β(r ) =
2 1 3−n d 3−n + (n − 3)r , 2(1 − n)
so v(r, s) =
2 1 s2 3−n − d 3−n + (n − 3)r . 2 2(n − 1)
(5.2)
E. Symeonidis
It follows that a(r, s) = vs (r, s) = s and d(r, s) =
n−1
1 3−n (n − 1)vr (r, s) = d 3−n + (n − 3)r .
(5.3)
For n = 3, (5.2) and the initial condition for β (0) yield d2 ln β (r ) = −2r + ln 2 and eventually β(r ) = −
d 2 −2r e 4
up to additive constants. Thus, we can take v(r, s) =
d2 s2 − e−2r , 2 4
which implies a(r, s) = s and d(r, s) =
2vr (r, s) = de−r .
(5.4)
—Let n be even. In this case, (4.4) and (5.1) imply 2n−4 β = − (1 − n)β n−1 ,
β (0) = −
d n−1 . n−1
Up to an additive constant, the solution is β(r ) =
2 1 3−n d 3−n − (n − 3)r , 2(1 − n)
so v(r, s) =
2 1 s2 3−n − d 3−n − (n − 3)r . 2 2(n − 1)
It follows that a(r, s) = vs (r, s) = s and d(r, s) =
(1 − n)vr (r, s) =
n−1
n−1
1 3−n (1 − n)β (r ) = d 3−n − (n − 3)r . (5.5)
Harmonic Families of Hypersurfaces
Now, if a harmonic function h vanishes sufficiently fast at infinity, so that the last condition in the theorem is satisfied, the integral
∞
π
−∞ 0 n−3
· sin
π
... 0
2π
h(s, d(r, s) cos s1 , . . . , d(r, s) sin s1 . . . sin sn−2 )
0
s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds
is invariant with respect to r , whence, by letting r → ∞ in the odd case, r → −∞ in the even one and taking (5.3), (5.4), (5.5) into account, this integral is equal to ωn−1 where ωn−1 =
n−1
2π 2 , ( n−1 2 )
∞ −∞
h(s, 0, . . . , 0) ds ,
the area of the unit sphere in Rn−1 .
II. The infinite cone x22 + · · · + xn2 = α 2 x12 , x1 ≥ 0, α > 0 This cone becomes part of the family (4.1) if we put the initial condition d(0, s) = αa(0, s) ,
a(0, s) ≥ 0 , (5.6) for all s ∈ I . For the function v this means that n−1 (−1)n+1 (n − 1)vr (0, s) = αvs (0, s) for all s ∈ I . To solve (4.4) in this situation, we try with a function of the form v(r, s) = γ (r )δ(s) , where γ (r ) = 0, δ(s) > 0 for all r ∈ [0, R), s ∈ I . Then, (4.4) becomes 2n−4 γ (r )δ(s) = − (−1)n (1 − n)γ (r )δ(s) n−1 γ (r )δ (s) .
(5.7)
Since δ(s) is positive, so has to be (−1)n+1 γ (r ), unless n = 3. It follows that γ
2n−4
2n−4 (−1)n+1 γ n−1 · γ
=−
(n − 1) n−1 · δ δ
2n−4 n−1
δ
=: λ ∈ R .
(5.8)
The differential equation for δ(s) gives by integration n−3 2 2 (n − 1) n−1 δ = −λδ n−1 + c
with c ∈ R. It is not our purpose to find every solution of (5.7) here, but one that satisfies our initial condition. We assume c = 0 (and λ < 0) and continue. Another integration leads to √ 5−3n n−2 δ(s) n−1 = ± −λ(n − 2)(n − 1) 2(n−1) s + c˜
E. Symeonidis
with c˜ ∈ R. Once more we restrict ourselves to the special solution n−1
δ(s) = (μs) n−2 ,
(5.9)
5−3n √ √ where either μ = −λ(n − 2)(n − 1) 2(n−1) and s ∈ I = [0, ∞), or μ = − −λ(n − 5−3n 2)(n − 1) 2(n−1) and s ∈ I = (−∞, 0]. To solve the other part of (5.8), we have to consider two cases separately. —Let n be odd. 2n−4 The differential equation γ = λγ · γ n−1 gives by integration
γ (r ) = c +
λ γ (r )2 n−1
n−1 2
(5.10)
with c ∈ R. Since λ < 0 and γ (r ) has to be positive (unless n = 3, see the remark below), we infer c > 0. It now follows: 1 n−1 μγ (r )(μs) n−2 , n−2 n−1 n+1 d(r, s) = (−1) (n − 1)vr (r, s) = n−1 (−1)n+1 (n − 1)γ (r )δ(s) √ 1 λ n−1 γ (r )2 (μs) n−2 . n−1 c+ = n−1
a(r, s) = vs (r, s) = γ (r )δ (s) =
We observe that all hypersurfaces of the family remain cones with the same (positive x1 -) axis and the origin as their vertex. The initial condition (5.6) now reads √
n−1
n−1 c+
λ n−1 γ (0)2 = α μγ (0) n−1 n−2
(5.11)
5−3n √ 2(n−1) > 0 (whence I = [0, ∞)) (and μγ (0) > 0). We choose μ = −λ(n −2)(n −1) > 0. and solve to get γ (0) = (α 2(n−1)c +1)(−λ) From (5.10) it follows that γ (r ) is an increasing function on [0, R), we can take
R = ∞, and limr →∞ γ (r ) =
(n−1)c −λ .
Therefore,
1 (n − 1)c n−1 μ (μs) n−2 = lim a(r, s) = r →∞ n−2 −λ
√
n−1
√ 1 n − 1 c(μs) n−2 ,
(5.12)
lim d(r, s) = 0
(5.13)
r →∞
(the cones shrink to their common axis). Remark 7 In the case n = 3, if γ (r ) is assumed negative, we take γ (0) < 0 and μ < 0 for (5.11) and proceed as in the following case of even n.
Harmonic Families of Hypersurfaces
—Let n be even. 2n−4 Here, the differential equation is γ = λγ · −γ n−1 and gives by integration γ (r ) = − c +
λ γ (r )2 n−1
n−1 2
(5.14)
with c > 0. The expressions for a(r, s), d(r, s) are as in the odd case, as well as the initial condition (5.11). From (5.14) it follows that γ (r ) is a decreasing function on [0, R), so we take (n−1)c (α 2 +1)(−λ)
γ (0) = − − (n−1)c −λ .
< 0, whence we can have R = ∞, and limr →∞ γ (r ) =
5−3n √ Since a(r, s) ≥ 0, we have to choose μ = − −λ(n − 2)(n − 1) 2(n−1) < 0, whence I = (−∞, 0]. Thus, the limits of a(r, s) and d(r, s) for r → ∞ remain as in (5.12) and (5.13). In both cases now, if a harmonic function h vanishes sufficiently fast at infinity, so that the last condition in the theorem is satisfied, the integral
I
π 0
... 0
π
2π
h(a(r, s), d(r, s) cos s1 , . . . , d(r, s) sin s1 . . . sin sn−2 )
0
· sinn−3 s1 . . . sin sn−3 dsn−2 dsn−3 . . . ds1 ds is invariant with respect to r , whence, by letting r → ∞ and taking (5.12) and (5.13) into account, this integral is equal to ωn−1
h
√ √ 1 n−1 n − 1 c(μs) n−2 , 0, . . . , 0 ds .
I
References 1. Khavinson, D.: Holomorphic partial differential equations and classical potential theory. Departamento de Análisis Matemático de la Universidad de la Laguna, 38271 La Laguna (1996) 2. Royster, W.C.: A Poisson integral formula for the ellipse and some applications. Proc. Am. Math. Soc. 15, 661–670 (1964) 3. Symeonidis, E.: Harmonic families of planar curves. RIMS (Research Institute for Mathematical Sciences, Kyoto University) Kôkyûroku Bessatsu B43, 171–181 (2013) 4. Symeonidis, E.: A mean value property of harmonic functions on the interior of a hyperbola. Acta Math. Univ. Comen. LXXXI(2), 143–149 (2012) 5. Symeonidis, E.: Harmonic deformation of planar curves. Int. J. Math. Math. Sci. (2011). doi:10.1155/ 2011/141209 6. Symeonidis, E.: A mean value property of harmonic functions on prolate ellipsoids of revolution. Acta Math. Univ. Comen. LXXVII(1), 55–61 (2008)
