regularity theorem for the Laplace equation

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Let D be an open subset of n. Suppose that f:D is twice differentiableMathworldPlanetmathPlanetmath and satisfies Laplace’s equation. Then f has derivatives of all orders and is, in fact analytic.

Proof:   Let 𝐩 be any point of D. We shall show that f is analytic at 𝐩. Since D is an open set, there must exist a real number r>0 such that the closed ball of radius r about 𝐩 lies inside of D.

Since f satisfies Laplace’s equation, we can express the value of f inside this ball in terms of its values on the boundary of the ball by using Poisson’s formula:


Here, A(k) denotes the of the k-dimensional sphere and dΩ denotes the measure on the sphere of radius r about 𝐩.

We shall show that f is analytic by deriving a convergentMathworldPlanetmathPlanetmath power series for f. From this, it will automatically follow that f 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 f is assumed to be twice differentiable in D, it is continuousMathworldPlanetmath in D and, hence, since the sphere of radius r about s is compact, it attains a maximum on this sphere. Let us denote this maxmum by M. 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 y has been restricted by the condition |𝐲-𝐩|=r, we may rewrite this as


Assume that |𝐱-𝐩|<r/4. Then we have


Since this absolute valueMathworldPlanetmathPlanetmathPlanetmath is less than one, we may apply the binomial theoremMathworldPlanetmath to obtain the series


Note that each term in this sum is a polynomialPlanetmathPlanetmath in x-p. The powers of the various components of x-p that appear in the m-th term range between m and 2m. 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 ck1k2 depend on y trivially, but the dependence on y has been indicated for the sake of uniformity) then


Raising this to the m-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 |x-y|-n in powers of x-p converges absolutely. Furthermore, there exist constants bk1k2,km such that the term involving |(x-p)k1||(x-p)k2||(x-p)km| in the power series is bounded by bk1k2,km.

Title regularity theorem for the Laplace equation
Canonical name RegularityTheoremForTheLaplaceEquation
Date of creation 2013-03-22 14:57:29
Last modified on 2013-03-22 14:57:29
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 17
Author rspuzio (6075)
Entry type Theorem
Classification msc 26B12