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regularity theorem for the Laplace equation
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(Theorem)
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Warning: This entry is still in the process of being written, hence is not yet complete.
Let be an open subset of
. Suppose that
is twice differentiable and satisfies Laplace's equation. Then has derivatives of all orders and is, in fact analytic.
Proof: Let be any point of . We shall show that is analytic at . Since is an open set, there must exist a real number such that the closed ball of radius about lies inside of .
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 area 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
Since this absolute value is less than one, we may apply the binomial theorem to obtain the series
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
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"regularity theorem for the Laplace equation" is owned by rspuzio. [ full author list (2) ]
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(view preamble)
Cross-references: bounded, converges absolutely, uniformity, coefficients, binomial, range, components, powers, polynomial, sum, series, binomial theorem, absolute value, restricted, integral, compact, continuous, indices, equations, multi-index notation, variables, necessary, power series, convergent, sphere, measure, Poisson's formula, boundary, terms, radius, closed ball, real number, point, proof, analytic, orders, derivatives, Laplace's equation, differentiable, open subset
There is 1 reference to this entry.
This is version 14 of regularity theorem for the Laplace equation, born on 2005-01-21, modified 2007-06-02.
Object id is 6655, canonical name is RegularityTheoremForTheLaplaceEquation.
Accessed 2546 times total.
Classification:
| AMS MSC: | 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions) |
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Pending Errata and Addenda
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