Harmonic functions, Poisson kernels

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1 June 17, 16 Harmonic functions, Poisson kernels Paul Garrett garrett/ [This document is garrett/m/complex/notes 14-15/8c harmonic.pdf] 1. Mean-value propert. Poisson kernel for disk 3. Dirichlet problem for disk 4. Poisson kernel for upper half-plane The two-dimensional Euclidean Laplacian is x + Man of these ideas are not specific to two dimensions: the Laplacian on R n is [1] x x n A twice-continuousl-differentiable complex-valued function u on a non-empt open set U C R satisfing Laplace s equation u is also called harmonic. In some contexts, a harmonic function is understood to be real-valued. With the notation we have z 1 x i z z z 1 x + i z z 1 4 A holomorphic function u satisfies the Cauch-Riemann equation u/z, so ever holomorphic function is harmonic. Similarl, ever ever conjugate-holomorphic function is harmonic. for holomorphic f, the real and imaginar parts Refz 1 are harmonic, and real-valued. fz + fz Imfz 1 i fz fz Harmonic functions have a mean-value propert similar to holomorphic functions. This ields the Poisson formula, recovering interior values from boundar values, much as Cauch s formula does for holomorphic functions. The solution of the Dirichlet problem is a converse: ever function on the boundar of a disk arises as the boundar values of a harmonic function on the disk. We prove this for relativel nice functions on the boundar, but with adequate set-up the same can be proven for an distribution generalized function on the boundar. [1] Up to scalar multiples, the Laplacian is the unique second-order differential operator on R n that is translationinvariant, rotation-invariant, and annihilates constants. In fact, ever translation-invariant and rotation-invariant differential operator on R n is a polnomial in the Laplacian. In some natural non-euclidean spaces there are motioninvariant differential operators not expressible in terms of the corresponding Laplacian. 1

2 Paul Garrett: Harmonic functions, Poisson kernels June 17, Mean-value propert among other features, in two dimensions harmonic functions form a useful, strictl larger class of functions including holomorphic functions. For example, harmonic functions still enjo a mean-value propert, as holomorphic functions do: [1..1] Theorem: Mean-value propert For harmonic u on a neighborhood of the closed unit disk, u 1 ue iθ dθ Proof: Consider the rotation-averaged function vz 1 ue iθ z dθ for z 1 Since the Laplacian is rotation-invariant, v is a rotation-invariant harmonic function. In polar coordinates, for rotation-invariant functions vz f z, the Laplacian is v x + f x + x x z f z + z f z 1 z f x z 3 f + x z f + 1 z f z 3 f + z f f + 1 z f The ordinar differential equation f + f /r on an interval, R is an equation of Euler tpe, meaning expressible in the form r f + Brf + Cf with constants B, C. In general, such equations are solved b letting fr r λ, substituting, dividing through b r λ, and solving the resulting indicial equation for λ: λλ 1 + Aλ + B Distinct roots λ 1, λ of the indicial equation produce linearl independent solutions r λ1 and r λ. However, as in the case at hand, a repeated root λ produces a second solution r λ log r. Here, the indicial equation is λ, so the general solution is a + b log r. When b, the solution a + b log r blows up as r +. Since f v u is finite, it must be that b. a rotation-invariant harmonic function on the disk is constant. its average over a circle is its central value, proving the mean-value propert for harmonic functions. /// [1..] Remark: One might worr about commutation of the Laplacian with the integration above. In the first place, it is clear that we must have this commutativit. Second, the best and most final argument for such is in terms of Gelfand-Pettis also called weak integrals of function-valued functions, rather than temporar elementar arguments. [1..3] Remark: The solutions a + b log r do indeed exhaust the possible solutions: given f + f /r on, R, we see r f is constant because r f r f + f r f /r + f r The class of harmonic functions includes useful non-holomorphic real-valued functions. For example, realvalued logarithms of absolute values of non-vanishing holomorphic functions are harmonic: log fz 1 log f + log f 1 holomorphic + anti-holomorphic

3 so is annihilated b 4 z z. Paul Garrett: Harmonic functions, Poisson kernels June 17, 16. Poisson s formula and kernel for the disk The mean-value propert will ield [..1] Corollar: Poisson s formula For u harmonic on a neighborhood of the closed unit disk z 1, u is expressible in terms of its boundar values on z 1 b uz 1 Up to scalars, the Poisson kernel function is ue iθ 1 z z e iθ dθ for z < 1 P e iθ, z 1 z z e iθ Proof: Pre-composition h f of a harmonic function h with a holomorphic function f ields a harmonic 1 z function. With ϕ z the linear fractional transformation given b matrix ϕ z, the mean-value z 1 propert for u ϕ z gives uz u ϕ z 1 u ϕ z e iθ dθ Linear fractional transformations stabilizing the unit disk map the unit circle to itself. Replace e iθ b e iθ ϕ 1 z e iθ uz u ϕ z 1 ue iθ dθ Computing the change of measure will ield the Poisson formula. This is computed b ie iθ θ θ θ eiθ ie iθ 1 ze iθ + izeiθ e iθ z 1 ze iθ ieiθ ize iθ + ize iθ ie iθ z 1 ze iθ ieiθ 1 z 1 ze iθ giving the asserted integral. θ θ 1 e iθ ie iθ 1 z 1 ze iθ 1 zeiθ e iθ z eiθ 1 z 1 ze iθ 1 z z e iθ /// 3. Dirichlet problem on the disk As a sort of converse to the existence of the Poisson formula and Poisson kernel, the Dirichlet problem on the closed unit disk in C R is posed b specifing of a continuous function f on the circle S 1 {z : z 1}, and asking for harmonic u on z < 1, extending to a continuous function on z 1, such that u S 1 f. That is, the following asserts that the collection of harmonic functions is a correct collection of functions in which to pose the problem of reconstituting the interior values of a function from its boundar values, if we are to have existence and uniqueness. In contrast, the collection of holomorphic functions on a disk is too small, since the boundar values have Fourier series with no negative terms see below. On another hand, an class of functions strictl larger than harmonic functions will include non-zero functions with boundar values identicall. 3

4 Paul Garrett: Harmonic functions, Poisson kernels June 17, 16 [3..1] Corollar: Given a continuous function f on the circle S 1 {z : z 1}, there is a unique harmonic function u on the open unit disk extending to a continuous function on the closed unit disk and u S 1 f. In particular, uz 1 fe iθ For real-valued f, the function u is also real-valued. Proof: This proof re-discovers the Poisson kernel. 1 z z e iθ dθ for z < 1 To skirt some secondar analtical complications, suppose that f is at least C 1, so that the partial sums of the Fourier series of f converge uniforml and absolutel pointwise to f, so where fe it n Z fn 1 Note that on z 1 with z e it we can write In these terms, e int fn e int lim N e inθ fe iθ dθ z n for n and z e it fn e int z n for n < and z e it fn e int 1 e inθ fe iθ dθ e int 1 fe iθ e inθ e int dθ 1 1 ze fe iθ iθ N+1 1 ze iθ + zeiθ ze iθ N+1 1 ze iθ dθ After moving z to z < 1, we can take the limit N, and fn z n + n< n fn z n 1 fe iθ 1 zeiθ + 1 ze iθ 1 ze iθ dθ The left-hand side is the sum of an anti-holomorphic and a holomorphic function, so is harmonic. Simplifing slightl, 1 zeiθ + 1 ze iθ 1 ze iθ 1 zeiθ + ze iθ z 1 ze iθ fn z n + n< n fn z n 1 1 z 1 ze iθ 1 z e iθ z 1 z z e iθ fe iθ 1 z z e iθ dθ To see that we recover the original boundar values, let z re it with r >, so lim r 1 1 fe iθ 1 z z e iθ dθ lim fn r n e int + fn r n e int r 1 n n< n fne int fe it using the good convergence of the Fourier series for f on the circle. /// 4

5 Paul Garrett: Harmonic functions, Poisson kernels June 17, Poisson kernel for upper half-plane Again using the fact that h f is harmonic for h harmonic and f holomorphic, we can transport the Poisson kernel P e iθ, z for the disk to a Poisson kernel for the upper half-plane H via the Cale map C : z z + i/iz + 1. The Cale map gives a holomorphic isomorphism of the disk to the upper half-plane, and of the circle with i removed to the real line. In the relation uz 1 fe iθ P e iθ, z dθ replace z b C 1 z with z H, and replace e iθ b C 1 z / with t R: using dθ de iθ /ie iθ, u C 1 z 1 d f C 1 t P C 1 t, C 1 z R The change of measure can be determined b the natural computation t + δ i it + δ + 1 t + δ i it + 1 iδ t i it δ 1 + iδ1 it 1 + Oδ it + 1 d t + δ i it + 11 iδ1 it 1 t + δ i it iδ1 it 1 1 t i it δ 1 it t i i it + 1 it + 1 t i it δ it Oδ dt dt 1 + it1 it dt 1 + t With z x + i in H, the Poisson kernel itself becomes P H t, x + i P disk C 1 t, C 1 z 1 C 1 z C 1 z C 1 t 1 z i iz+1 z i iz+1 1 it 1 iz 1 + iz z i1 it 1 izt i 1 it x x + 1 z i itz t t itz i z with all these coordinate changes, u C 1 z 1 f C 1 t R 1 it 4 4 z t 1 it x t + x t + dt 1 f C 1 t π R x t + dt + Oδ That is, replacing f C 1 b f and u C 1 b u, for suitable function f on the real line, a harmonic function u on the upper half-plane with boundar value f on R is given b ux + i 1 ft π R x t dt Poisson formula on H + 5