regularity theorem for the Laplace equation
Warning: This entry is still in the process of being written, hence is not yet .
Since satisfies Laplace’s equation, we can express the value of inside this ball in terms of its values on the boundary of the ball by using Poisson’s formula:
Here, denotes the http://planetmath.org/node/4495area of the -dimensional sphere and denotes the measure on the sphere of radius about .
We shall show that is analytic by deriving a convergent power series for . From this, it will automatically follow that has derivatives of all orders, so a separate proof of this fact will not be necessary.
Since this involves manipulating power series in several variables, we shall make use of multi-index notation to keep the equations from becoming unnecessarily complicated and drowning in a plethora of indices.
First, note that since is assumed to be twice differentiable in , it is continuous in and, hence, since the sphere of radius about is compact, it attains a maximum on this sphere. Let us denote this maxmum by . Next, let us consider the quantity
which appears in the integral. We may write this quantity more explicitly as
Since the values of the variable has been restricted by the condition , we may rewrite this as
Assume that . Then we have
Note that each term in this sum is a polynomial in . The powers of the various components of that appear in the -th term range between and . Moreover, let us note that we can strengthen the assertion used to show that the binomial series converged by inserting absolute value bars. If we write
(actually, the coefficients depend on trivially, but the dependence on has been indicated for the sake of uniformity) then
Raising this to the -th power, we see that, if we define
then we have
Because of the fact that one may freely rearrange and regroup the terms in an absolutely convegent series, we may conclude that the expansion of in powers of converges absolutely. Furthermore, there exist constants such that the term involving in the power series is bounded by .
|Title||regularity theorem for the Laplace equation|
|Date of creation||2013-03-22 14:57:29|
|Last modified on||2013-03-22 14:57:29|
|Last modified by||rspuzio (6075)|