December 3, 2009

Weyl’s lemma and approximations to the identity

Posted by Akhil Mathew under analysis | Tags: , , , |
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Theorem 1 (Weyl) Let {f \in L^2(U)}, where {U} is the unit disk with Lebesgue measure. If  \displaystyle \int_U f \Delta \phi = 0   for all {\phi \in C^{\infty}(U)} with compact support, then {f} is harmonic (in particular smooth). 

I dropped out of the groove for a couple of days due to other activities; I’m back today to talk about Weyl’s lemma (for the Laplacian—it generalizes to elliptic operators), a tool we will need for the special case of the Hodge decomposition theorem on Riemann surfaces.   The result states that a “weak” solution to the Laplace equation is actually a strong one. (more…)

 

November 30, 2009

A primer on harmonic functions

Posted by Akhil Mathew under analysis | Tags: , , , |
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The topic for the next few weeks will be Riemann surfaces.  First, however, I need to briefly review harmonic functions because I will be talking about harmonic forms.  I will have more to say about them later, and I actually won’t use most of today’s post even until then.  But it’s fun.

Some of this material has also been covered by hilbertthm90 at A Mind for Madness.

Definition

A {C^2} function {f} on an open subset of {\mathbb{R}^n}, {n >1}, is called harmonic if it satisfies the Laplace equation\displaystyle \Delta f = \sum \frac{\partial^2f}{\partial x_i^2} = 0. For now, we are primarily interested in the case {n=2}, and we will identify {\mathbb{R}^2} with {\mathbb{C}}.  In this case, as is well-known, harmonic functions are locally the real parts of holomorphic functions.

The Poisson Integral

The following fact is well-known: given a continuous function {f} on the circle {C_1(0)}, there is a unique continuous function on the closed unit disk {\overline{U}} which is harmonic in the interior and coincides with {f} on the boundary.The idea of the proof is that {f} can be represented as a Fourier series,

\displaystyle f(e^{it}) = \sum_{n \in \mathbb{Z}} c_n e^{int}

 where the {c_n} are obtained through the orthogonality relations

\displaystyle c_n = ( f, e^{-int} )

 where the inner product is the {L^2} product taken with respect to the Haar measure on the circle group. This convergence holds in {L^2}, because the exponentials form an orthonormal basis for that space. Indeed, orthonormality can be checked by integration, and the Stone-Weierstrass theorem implies their linear combinations are dense in the space of continuous functions on the circle. It is even the case that convergence holds uniformly if {f} is well-behaved (say, {C^2}). But this is only for motivational purposes, and I refer anyone interested to, say, Zygmund’s book on trigonometric series for a whole lot fo such results.

Now, it is clear that the functions\displaystyle z \rightarrow r^n e^{int}, \ z \rightarrow r^n e^{-int}

 are harmonic (where {t = Arg(z), r = |z|}) as the real parts of {z^n, \bar{z}^n}.

It thus makes sense to define the extended function {\tilde{f}} as\displaystyle \tilde{f}(re^{it}) = \sum_n c_n r^{|n|} e^{int}. (more…)