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.

It

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.

Hence Taylor’s

theorem with remainder term easily gives harmonic u is also analytic. Analytic

functions are certainly not harmonic as one dimensional cubic functions

testify.

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

classes.

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)

contradicting positivity.