Harmonic extension through conical surfaces

This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.


Introduction
The Schwarz reflection principle gives a simple formula for extending a harmonic function h on a domain ω ⊂ R N through a relatively open subset E of ∂ω on which h vanishes, provided E lies in a hyperplane. A corresponding reflection formula holds when E lies in a sphere. When N ≥ 3 and N is odd, Ebenfelt and Khavinson [6] (cf. Chapter 12 of [16]) have shown that a point to point reflection law can only hold when the containing real analytic surface is either a hyperplane or a sphere. Thus more sophisticated methods are needed for extending a harmonic function which vanishes on any other type of set E. This is the background to the following problem, which was posed by Dima Khavinson at various international conferences: if h is harmonic on an infinite cylinder and vanishes on the boundary, does it extend harmonically to all of R N ? Of course, in the planar case, where h is harmonic on an infinite strip, the answer is readily seen to be positive by repeated application of the Schwarz reflection principle. In higher dimensions the problem was eventually also shown to have an affirmative answer [7] by analysis of the Green function of the cylinder. Subsequently, the authors investigated extension properties of harmonic functions on an annular cylinder {x ∈ R N −1 : a < x < b} × R that vanish on either one or both of the cylindrical boundary components (see [8,10,11]). The domain reflection results that emerged were noteworthy, given that reflection formulae for the harmonic functions themselves fail to exist. This raises the following general question.
Problem 1 For a domain ω in R N and a subset E of ∂ω identify a larger domain ω E such that each harmonic function on ω which vanishes continuously on E has a harmonic extension to ω E .
Naturally we should assume that E is contained in a real-analytic surface, but the question is interesting even in the particular case where E is contained in the zero set of a polynomial. The cylindrical case corresponds to the polynomial (x , x N ) → x 2 − 1. The next most natural case to consider is a cone. The analogue of Khavinson's question above would then be: if h is harmonic on an infinite cone and vanishes on the boundary, does h extend harmonically to all of R N , except for the negative axis of the cone? Again, in the planar case, such extension follows by repeated application of the Schwarz reflection principle. A typical point of R N (N ≥ 3) will be denoted by x = (x , x N ), where x ∈ R N −1 and x N ∈ R, and we will write θ x = cos −1 (x N / x ) when x = 0. Let 0 < θ * < π. We will show that harmonic functions h on the infinite cone = (θ * ) = {x ∈ R N \{0} : θ x < θ * } that vanish on ∂ have an extension to the set (π) = {x ∈ R N \{0} : θ x < π} = R N \({0 } × (−∞, 0]).
The proof of Theorem 1 is technically more challenging than the corresponding result for the cylinder. However, it also yields tools applicable to reflection results for functions that are harmonic on a domain of the form (θ 0 , θ * ) = {x ∈ R N \{0} : θ 0 < θ x < θ * } and vanish on ∂ (θ * ). Strikingly, a dichotomy emerges between the cases where θ * ≤ π/2 and θ * > π/2 , as we will now see.
We now have a reasonably complete set of harmonic extension results for conical surfaces to complement those known for cylinders. Our hope is that these will suggest further steps towards addressing the broader question in Problem 1.
The extension of harmonic functions through conical surfaces is obviously related to extension properties of the Green function for a cone, and harmonic functions on conical domains are naturally related to Legendre functions. The plan of the paper is thus as follows. In Sect. 3 we assemble and develop some relevant material concerning Legendre functions. This is subsequently used, in conjunction with contour integration, to establish an expansion of the fundamental function that is adapted to the geometry of cones, and then two different expansions for the Green function of the cone (θ * ). The first of these latter expansions is used to establish the second and also has later application. The second yields extension properties of the Green function that are used in proving Theorem 1. Theorems 2 and 3 rely on both Theorem 1 and further extension properties of the Green function. These latter properties are established using bounds for ratios of conical functions that may be of independent interest.

Sharpness of results
The domain of extension in Theorem 2 is formed by angular reflection. This is natural, since in the planar analogue of the result the function h is harmonic in an angle and would extend to an angle of twice the aperture by Schwarz reflection. The sharpness of this result in higher dimensions is demonstrated by the following example.
If x , y ∈ R N −1 , then we define φ x ,y ∈ [0, π] by the equation whenever the denominator is non-zero. We also recall that cos θ x = x N / x . The following result shows how some of the above functions relate to harmonicity.
is harmonic on (π) when suitably interpreted on the positive x N -axis.
Proof We will give the details when N ≥ 4 and leave the adjustments required when Since it is enough to show that Thus it remains to check that Next, by (6), and (11) follows. The above calculation is not valid when θ x = 0, or when φ x ,y ∈ {0, π}. In the latter case we can use the continuity of C is a removable singularity for the harmonic function h, by Corollary 5.2.3 of [1]. A similar argument, combined with Lemma 5(vii), shows that the positive x N -axis is also removable for h.
Then any function of the form where A, B ∈ R, is harmonic on (π) when suitably interpreted on the positive x N -axis.
in the proposition and use the fact that P .
is harmonic on (π) when suitably interpreted on the positive x N -axis.
Proof We put w = iλ in the proposition, take real and imaginary parts of h, and expand cos(λ log x + c) using the addition formula.
Functions of the form P −μ − 1 2 +iλ are known as conical (or Mehler) functions. We record below some of their further properties for future reference.
Proof (i) This is clear from the Mehler-Dirichlet formula (9).
(iii) By definition, Since the coefficients in the expansion are all positive, we now see that the function θ → P −μ − 1 2 +iλ (cos θ) is the product of two positive increasing functions on (0, π).
In the next three sections we will adapt an argument outlined on pp.69-72 of Dougall [3] for R 3 to establish expansions of the Green function for (θ * ) in all dimensions.

An expansion of the fundamental function
If x, y ∈ R N , then we define γ x,y ∈ [0, π] by the equation It will be convenient to define We recall from p. 1938 of [5] (cf. equation (80) in [13]) an addition formula for P −μ ν , namely when θ x +θ y < π. (The restriction in [5] that φ x ,y < π may be removed by dominated convergence, in the light of (5) and the asymptotic behaviour of P −μ ν for large μ, as described in (14.15.1) of [19].) Since cos γ −x,y = − cos γ x,y , sin γ −x,y = sin γ x,y , and analogous formulae hold for θ −x and φ −x ,y , we can replace x by −x in (13), and use (4) and (12) to obtain when θ −x + θ y < π, that is, when θ y < θ x . When μ = 0 the appropriate analogue of (13) may be found by combining equations (14.18.1) and (14.9.3) of [19]. This leads to the formula Equations (14) and (15) are valid when γ x,y , θ x , θ y ∈ (0, π) and θ y < θ x . Let and suppose that y < x and 0 < γ x,y < π. Then (2) and parts (i), (ii) of Lemma 5 show that where For any κ ∈ N let c(κ) denote the contour around the boundary of the rectangle oriented anticlockwise. The function z → P 3−N 2 and §15.2(ii) of [19]). Thus the residue theorem yields since the singularities of the integrand in Z ∩ 2−N 2 , 0 are removable. By Lemma 5(vii) the above integrand is bounded in modulus by on the top and bottom sides of the contour, and by x on the right hand side. Since we can parametrize the reverse path −c(κ) on the left hand side of the rectangle as (The convergence of this integral will become clear below.) Since we see that (− cos γ x,y ) is real and symmetric in λ, by Lemma 9(i). Combining this with (16), we see that Noting that by (5.11.9) in [19], we see from Lemma 5(vii) that the integral in (21) converges absolutely even when y = x . It follows from dominated convergence and symmetry that (21) is valid for any non-zero choices of y and x , provided γ x,y ∈ (0, π). Since by (12) and (20), we see from (21) that We now make the additional assumption that 0 < θ y < θ x < π, and deal first with the case where N ≥ 4. We can combine (24) with (14) to see that In view of the positivity of P .
In addition, Thus the integral in (25) still converges when we replace the summand by its absolute value. In particular, we can thus allow γ x,y to range over (0, π], by dominated convergence. When N = 3 we instead combine (15) with (24) to see that The analogue of (26) again holds, so the expansion in (27) has the same absolute convergence property. We have established (25) and (27) for any x, y ∈ R N \{0} satisfying 0 < θ y < θ x < π. The integrals and summations are interchangeable, by Fubini's theorem.

An expansion for the Green function
We assume in this section that x, y ∈ R N \{0} and θ x , θ y ∈ (0, π).
When N ≥ 4, y ∈ and x ∈ we define Since the function θ → P 3−N 2 −k − 1 2 +iλ (cos θ) is positive and increasing on (0, π), by Lemma 9, we see that when θ x ≤ θ * . It now follows from (26), with θ x = θ * , and (5), that the right hand side of (28) is absolutely convergent, and from dominated convergence that h y is continuous on , when suitably interpreted at points where θ x = 0. Further, by Fubini's theorem and Corollary 8, the function h y satisfies the volume mean value property in , and so is harmonic there. It tends to 0 at infinity, by (26) again with θ x = θ * . Since h y (x) = x − y 2−N on ∂ , by (25), it follows from the minimum principle that h y is the greatest harmonic minorant of · − y 2−N on . Hence, when 0 < θ y < θ x < θ * , it follows from (25) and (28) that the Green function of is given by where The integration and summation can be interchanged in (29), by the absolute convergence of the expansions in (25) and (28). If θ x < θ y , then we replace g k (λ, θ x , θ y ) by g k (λ, θ y , θ x ) in (29), by the symmetry of the Green function. When N = 3 analogous reasoning shows that when θ y < θ x . This was asserted long ago in p.71(1) of [3], though full details of the proof were not provided. (That paper used P μ ν to denote what today is called P −μ ν , as can seen from the definition on p.48.)

A second expansion for the Green function
We denote by n k,m the mth positive zero of the entire function ν → P and note from Lemma 5(vi) that Suppose that y < x and θ x , θ y ∈ (0, π), and let We recall that (z) is holomorphic except for simple poles at the nonpositive integers, and that Hence, by (12), the singularities of the function lie at the integers j satisfying j ≥ k, and the residue at j is then The singularities of f at such points are thus removable, in view of Lemma 5(ii).
The remaining singularities of f in {Rez > 2 − N } are simple poles at the points (n k,m ) m≥1 .
When N ≥ 4 we then see from (29), an interchange of summation and integration, and (32), that and when N = 3 we use (30) in place of (29) to see that We temporarily assumed above that θ y < θ x . If θ x < θ y , then we define x * = ( x / y )y and y * = ( y / x )x. We then observe that G (x * , y * ) = G (y, x) = G (x, y), by (29) (or (30)) and the symmetry of the Green function, to arrive at (33) (or (34)) again. Our earlier assumption that θ x , θ y are non-zero can be dropped provided the formulae are suitably interpreted. Thus these formulae hold when θ x = θ y and y < x . The corresponding formulae when x < y are obtained by interchanging x and y in (33) and (34).

Extending the Green function of the cone
In preparation for the main result of this section we note the following lemma.
Proof It follows from parts (x), (iv) and then (viii) of Lemma 5 that 2 ν + 1 2 Theorem 11 Let y ∈ and a > 1, and define Then the formulae in (33) and (34) converge absolutely and uniformly to a harmonic function on ω (1) y,a , and when x and y are interchanged they converge absolutely and uniformly to a harmonic function on ω (2) y,a . In particular, G (·, y) has a harmonic extension G (·, y) to the set ( \{y}) ∪ ω (1) y,a ∪ ω (2) y,a . Further, Proof Suppose first that N ≥ 4 and x > a y . We assume, without loss of generality, that 1 < a ≤ 2, and define By (31) we see that which will allow us to apply Lemma 10 and some results from Lemma 5. By Lemma 5(viii), and, by Lemma 10, where Using the Legendre duplication formula, we see that Thus, by (36) and (37), When θ * /2 < θ y < θ * we combine Lemma 5(ix) with the mean value theorem and use the concavity of sin θ on (0, π) to see that Using (38) again we see that in view of (37) and (5). We next consider the case where 0 ≤ θ y ≤ θ * /2. Let If h is a harmonic function on , then h 2 is subharmonic there, and so we can use the volume mean value inequality to see that By Corollary 7 we can apply this inequality to the harmonic function given by (interpreted, as usual, in the limiting sense on {0} N −1 × (0, ∞)) to see that Since the sets are bounded above by a constant C(a, N ), we can use (39)-(41), (5) and (31), to see that It follows that the expression for G (x, y) in (33) converges absolutely to a harmonic function in ω (1) y,a and satisfies the estimate (35) there. For the set ω (2) y,a we interchange x and y in (33) and argue similarly. Analogous reasoning applies when N = 3.

Proof of Theorem 1
We will adapt the approach taken in Theorem 19 of [10]. Theorem 1 follows from the result below on letting c → ∞. We define
(More precisely, the Riesz decomposition theorem shows that h * i − G i is harmonic on , and the representation then follows from the fact that h * i and G i both vanish at the boundary.) Let a > 1. It follows from Theorem 11 that the formula defines a harmonic extension of h from ∩ A(c ) to the intersection of the sets and Since c may be arbitrarily close to c, and a may be arbitrarily close to 1, the result follows.

Bounds for ratios of conical functions
Several authors have considered bounds on ratios of modified Bessel functions: see, for example, [20] and the references provided there. In this section we establish corresponding bounds on ratios of conical functions in preparation for the proofs of Theorems 2 and 3. We begin with two elementary lemmas concerning Riccati equations.
Then t 0 > a, by hypothesis. If t 0 < b, then h (t 0 ) = 0 and so This yields a contradiction, since h > h (t 0 ) on (a, t 0 ). Thus t 0 = b as claimed.

Lemma 14 Suppose that
where h, A, B and C are all positive.

Proof
Since h + C > 0 and A > 0, we see that h and h − B have opposite signs.

Proposition 15
Let 0 < θ 1 < θ 2 < π and μ, λ ∈ R. Then Proof We note from (14.10.2) in [19] that and combine this with Lemma 5(iii) to see that We also know from (14.10.1) in [19] that and combine this with (46) to see that and so Equation (45) follows on substituting t = cos θ .
It follows from Lemma 14 that h μ (θ ) is bounded above by the positive root of the equation namely, which equals f 2 (θ ). Further, from (47),
Proof We modify the previous proof. Again we will assume, for simplicity, that N ≥ 4. This time we note that when (x, y) ∈ ω 2,δ .

Proof of Theorem 2
Let θ 0 < θ − < θ + < θ * and 1 < c < c . As in the proof of Theorem 12, we can represent h in (θ * )\ (θ + ) ∩ A(c ) as the potential G of a signed measure on the union of the sets G (x, y)d (y).
It follows from Theorem 12 that h a has a harmonic extension to the intersection of the sets (42) and (43), and from Proposition 17 that h b has a harmonic extension to the set (2θ * − θ + )\ (θ + ). The result now follows on letting c → ∞ and θ + → θ 0 +.

Proof of Theorem 3
We follow the above argument except that we use Proposition 18 to see that h b has a harmonic extension to the set x ∈ R N \{0} : θ + < θ x and tan θ x 2 tan θ + 2 < tan θ * 2 2 .
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