As Terence Tao has demonstrated brilliantly
through his live journal, writing about one’s work in math online can stimulate
wild discussions and promote apparent productivity that ultimately leads to
real ones. I, on the other foot, am not as brilliant so must resort to writing
about others’ work, in fact, not even original work, but text book content
often time, while chewing on rum icecream.
Professor Ryzhik has decreed that I read the small book on Elliptic PDE
by Prof Qing Han and Fanghua Lin. With all due respect, the book is the most
compact course on the subject, with a density comparable to a neutron star.
Nonetheless working through some of the missing links didn’t prove as helpful
as both of us originally wished, since I am no great computation guy, and like
to criticize possible typos or mistakes under a magnifying eyeglass. So here we
go; I will pinpoint some tough bones in the first one and half chapter that I
actually swept through in the past week or so. To make the best out of my
soliloquated rambling here, one should ideally have a copy of <<Elliptic
Partial Differential Equation>> by Han and Lin in hand. At least one
should take the pain of wiki-ing the undefined concepts and terminologies.
Notice that most of acronyms are formed based on words appearing one or two
lines before, hence should be self contained.
begins with a full discussion of the connection between mean value property,
harmonic functions, analyticity, and the maximum principle. MVP(M),
harmonicity(H), and analyticity(A) are fully equivalent except A -> H/M:
proof of H -> M and M-> H direction consists of basically Gauss-Green’s
theorem applied to the unit ball, and its boundary, the unit sphere of one
dimension lower. The integral of Laplacian of u over a ball can be expressed as
integral of radial derivative of u over the sphere. Harmonic functions would
then give zero derivative, but when radius is 0, the integral evaluates to 1
since the spherical measure has to be normalized. This establishes the MVP.
Conversely, if the Laplacian of u integrates to 0 in any small ball, then by
continuity it must be 0.
A bit trickier is the proof of H/M -> smoothness,
and basically one has to convolve the function u satisfying MVP with a smooth
radial bump function (doesn’t have to be constant near 0) psi on the unit ball
that interates to 1. One shows the convolution with psi acts like the identity
operator on harmonic functions. And convolution with a smooth function is
smooth, because of differentiation under integral sign property.
Analyticity is proved by estimating the
alpha-th derivative of a harmonic u, which can be shown to be sub-factorial.
theorem with remainder term easily gives harmonic u is also analytic. Analytic
functions are certainly not harmonic as one dimensional cubic functions
The rest can be properly called
applications. First we see that Maximum principle (MP) is a consequence of the
fact that at a maximum x of u, the Laplacian must be nonpositive much like
second derivative test of one dimensional functions.
Another analogue with 1-d situation is the
gradient estimate, which bounds the norm of derivative of a harmonic u in terms
of its maximal absolute value in a ball. The analogy is that harmonic functions
can be thought of as higher dimensional analogues of linear functions, since
for linear functions second derivative must vanish, but here the trace of the
Hessian must vanish. And for linear functions the derivative or slope is
bounded from above by the difference of the lowest value and highest value divided
by the length of the spanning interval. Curiously, Riemann surfaces are complex
analogues of 1-d real manifolds, but there is an infinite family of them: in
fact even for those diffeomorphic to the torus C/L, where L is a lattice, there
are infinite many holomorphic classes, labeled by the lattice isomorphism
Harnack inequalities have mystified me for
a long time, since they appear in various complicated forms with not a small
amount of conditions. But in one sentence they state that the values of a
harmonic u do not differ by a lot in a ball, in terms of ratios. Again 1-d
linear functions are the right analogues. Imagine you have a linear function u positive
on the interval (a,b), then a compact set in (a,b) is contained in [c,d] with
a<c<d<b. the biggest ratio of u at two different points in [c,d] is
u[d]/u[c] or u[c]/u[d], Harnack basically says that this ratio is bounded in
terms of [c,d] and [a,b] only, independent of u. The reason is that if this
ratio gets too large, then u will go below zero before getting out of (a,b)