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# regularity theorem for the Laplace equation

Warning: This entry is still in the process of being written, hence is not yet complete.

Let $D$ be an open subset of $\mathbb{R}^{n}$. Suppose that $f\colon D\to\mathbb{R}$ is twice differentiable and satisfies Laplace’s equation. Then $f$ has derivatives of all orders and is, in fact analytic.

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

$f({\bf x})={1\over r^{{n-1}}A(n-1)}\int_{{|{\bf y}-p|=r}}f({\bf y}){r^{2}-|{% \bf x}-{\bf p}|^{2}\over|{\bf x}-{\bf y}|^{n}}\,d\Omega({\bf y})$ |

Here, $A(k)$ denotes the area of the $k$-dimensional sphere and $d\Omega$ denotes the measure on the sphere of radius $r$ about ${\bf p}$.

We shall show that $f$ is analytic by deriving a convergent 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 continuous 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

${1\over|{\bf x}-{\bf y}|^{n}}$ |

which appears in the integral. We may write this quantity more explicitly as

$\left(|{\bf y}-{\bf p}|^{2}-2({\bf x}-{\bf p})\cdot({\bf y}-{\bf p})+|{\bf x}-% {\bf p}|^{2}\right)^{{-{n\over 2}}}.$ |

Since the values of the variable $y$ has been restricted by the condition $|{\bf y}-{\bf p}|=r$, we may rewrite this as

${1\over r^{n}}\left(1+{-2({\bf x}-{\bf p})\cdot({\bf y}-{\bf p})+|{\bf x}-{\bf p% }|^{2}\over r^{2}}\right)^{{-{n\over 2}}}.$ |

Assume that $|{\bf x}-{\bf p}|<r/4$. Then we have

$\left|{-2({\bf x}-{\bf p})\cdot({\bf y}-{\bf p})+|{\bf x}-{\bf p}|^{2}\over r^% {2}}\right|\leq{2|({\bf x}-{\bf p})\cdot({\bf y}-{\bf p})|\over r^{2}}+{|{\bf x% }-{\bf p}|^{2}\over r^{2}}\leq$ |

${2|{\bf x}-{\bf p}|\>|{\bf y}-{\bf p}|\over r^{2}}+\left({|{\bf x}-{\bf p}|% \over r}\right)^{2}\leq 2\cdot{1\over 4}+\left({1\over 4}\right)^{2}={9\over 1% 6}<1.$ |

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

${1\over|{\bf x}-{\bf y}|^{n}}={1\over r^{n}}\left(1+{-2({\bf x}-{\bf p})\cdot(% {\bf y}-{\bf p})+|{\bf x}-{\bf p}|^{2}\over r^{2}}\right)^{{n\over 2}}=$ |

$\sum_{{m=0}}^{\infty}{\left({n\over 2}\right)^{{\underline{m}}}\over m!}\left(% {-2({\bf x}-{\bf p})\cdot({\bf y}-{\bf p})+|{\bf x}-{\bf p}|^{2}\over r^{2}}% \right)^{m}$ |

Note that each term in this sum is a polynomial 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

${-2(x-p)\cdot(y-p)+|x-p|^{2}\over r^{2}}=\sum_{{k=0}}^{n}c_{k}(y)\>(x-p)_{k}+% \sum_{{k_{1},k_{2}=0}}^{n}c_{{k_{1}k_{2}}}(y)\>(x-p)_{{k_{1}}}(x-p)_{{k_{2}}},$ |

(actually, the coefficients $c_{{k_{1}k_{2}}}$ depend on $y$ trivially, but the dependence on $y$ has been indicated for the sake of uniformity) then

$\sum_{{k=0}}^{n}|c_{k}(y)|\>|(x-p)_{k}|+\sum_{{k_{1},k_{2}=0}}^{n}|c_{{k_{1}k_% {2}}}(y)|\>|(x-p)_{{k_{1}}}|\>|(x-p)_{{k_{2}}}|\leq{9\over 16}.$ |

Raising this to the $m$-th power, we see that, if we define

$\left({-2(x-p)\cdot(y-p)+|x-p|^{2}\over r^{2}}\right)^{m}=\sum_{{k_{1},k_{2},% \ldots,k_{m}=0}}^{n}c_{{k_{1}k_{2},\cdots k_{m}}}(y)(x-p)_{{k_{1}}}(x-p)_{{k_{% 2}}}\cdots(x-p)_{{k_{m}}},$ |

then we have

$\sum_{{k_{1},k_{2},\ldots,k_{m}=0}}^{n}|c_{{k_{1}k_{2},\cdots k_{m}}}(y)|\>|(x% -p)_{{k_{1}}}|\>|(x-p)_{{k_{2}}}|\cdots|(x-p)_{{k_{m}}}|\leq\left({9\over 16}% \right)^{m}$ |

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 $b_{{k_{1}k_{2},\cdots k_{m}}}$ such that the term involving $|(x-p)_{{k_{1}}}|\>|(x-p)_{{k_{2}}}|\cdots|(x-p)_{{k_{m}}}|$ in the power series is bounded by $b_{{k_{1}k_{2},\cdots k_{m}}}$.

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## Comments

## (hopefully not infinite) descent

A good part of the reason why this entry is taking so long to complete is that I keep finding myself backtracking recursively to add background material. Since it makes an interesting and amusing story and comments on the state of the website, I think I'll relate this tale of recursive descent. (It's nothing too original since I'm sure most of you could tell similar stories.) To begin with, I wrote the current entry while revising my proof of the Riemann mapping theorem when I realized that the proof used the regularity theorem, which had had not yet been entered in the encyclopedia. So I went off to write this entry. In the course of writing it, I used the Poisson formula, which is also not defined. Oh, but that uses the measure on the sphere, which isn't documented either. Hence, I find myself about to add an entry on something as basic as solid angle and measure on the sphere!

Don't get me wrong. This isn't a complaint. I'm not at all upset, just a bit amused. As I see it, this is an opportunity to fill in some basic gaps. If it means taking a week to finish the original proof so be it, because when I'm done not only will I have a really nice proof, but will have added a bunch of related material which was somehow overlooked.

I guess the moral of this story is something like this. A few days ago, Aaron congratulated us on 4000 entries. That certainly is an achievement to be proud of. Only let's be careful not to rest on our laurels! It'll probably take another 4000 entries to fill all the little gaps in the knowledge base so that one could type in a topic at, say an advanced graduate level, and be 99% sure that all the more basic terms, definitions, and theorems it refers to have already been entered. To the best of my knowledge, Planet Math is already the most comprehensive collection of math on the web (and, on top of that, it's GNU free, so you don't have to worry about a publishing company suing your pants off) and, when we get to the point of 99% completeness for standard topics, I can easily see it becoming a standard reference like Math reviews --- if you need to track down a reference, you go to Math Reviews, whilst if its a definition or a theorem that you need, you go to Planet Math.

## Re: (hopefully not infinite) descent

Nicely put - I do wonder how many more entries we'd need for

"completeness", and whether we can come up with a reliable way

to estimate that number. I've had my own entertaining paper

chases when trying to understand a definition or theorem

from basic principles. If we had a good way to estimate how

complete things really are, I'd almost suggest we do a freeze

on new topics that aren't being added for completeness's sake

until the KB is sufficiently complete. OTOH, if participants

are willing to go to the lengths you're doing here, we could

probably acheive a decent level of completeness IRT. I think it

would be nice if we could adjust the incentive stucture to favor

filling gaps in the KB.

## Re: (hopefully not infinite) descent

Here's an idea for estimating completeness and filling holes in our knowledge base. Granted it's crude and has its problems, but it's the sort of practical thing that could be done right away if enough of us find it worth doing.

Let's start with the typical math curriculum. It runs something like follows:

Elementary / High school

Arithmetic

Algebra

Euclidean Geometry

Trigonometry

Analytic Geometry

Calculus

University (Bachelor & Master's degrees)

Real Analysis

Complex Analysis

Linear and Multilinear Algebra

Abstract Algebra (Groups, Rings, Fields)

Set Theory

Logic

Probability & Statistics

Topology (Point Set and Algebraic)

Differential Geometry

Algebraic Geometry

Number Theory

Discrete Math (Combinatorics, Game Theory)

Functional Analysis

Each of these subjects is more or less standardized. For each of these subjects, there are several textbooks which describe more or less the same theorems and definitions. The differences beteween them are more a matter of style and approach than of material. I mean, for example, that most people will agree that a course of complex analysis ought to include a discussion of Cauchy's integral theorem and residue calculus although they might disagree on whether it is better to introduce holomorphic functions using Taylor series or the Cauchy-Riemann equations. (although, again, both topics would eventually be covered in most any course of complex analysis)

My proposal is as follows. Take any one of these topics and pick up a typical text on the subject. Thumb through the book. Every time you encounter a theorem or a definition, check to see whether it appears in the encyclopedia. This is particularly easy, if one combines it with the creation of a link page or a topic entry since then noosphere automatically looks for those topics which are already in the encyclopaedia. For example, last night I added the entry on theorems in complex analyis, then skimmed through Knopp's textbook and made sure all the theorems in the book appeared in the list. When time allows, I plan to skim trough other complex analyisis texts and systematically add any theorems which are not already in the list. Also, in the meanwhile, at least one more member has added theorems to the list. If this goes on for a while, I think we can be reasonable assured of having a complete (for all practical purposes) list of theorems in complex analysis and once the theorems in the list which do not yet have entries are added, the complex analysis department of Planet Math should be reasonably complete.

As I said, this method may not be perfect, but at least it's practical. I also think that we would have reached quite a milestone if our knowledge base were complete enough that all the material to be found in textbooks for standard mathematical subjects were included.

From my recent experience with complex analysis, I would say that this goal is within reach. Here's some data. It took me between a half-hour to an hour to skim Knopp and come up with the list of theorems. The list contains 26 theorems; of these, only 3 are not yet entered. Extrapolating from this, I would say that I plan to look through 4 to 12 more books on complex analysis (and specialized monographs on subtopics like conformal mapping) for theorems. Skimming through the books should take something like 10 hrs total. It's reasonable to estimate that, when done, there will be somewhere between 50 and 100 theorems on the list. At the current rate, that would suggest something like a dozen theorems which are not yet in the database. Afterwards, it might also be nice to look through the enecyclopedia (here, subject classification search will be most helpful) and make sure that all the theorems in complex analysis are indexed in the list. All in all, this seems to involve something like 15 hrs of work plus adding 12 theorems to the encyclopedia. It's work, but not prohibitively much. If I dropped everything else I was doing with respect to Planet Math to concentrate on this project, I could pull it off in a month. Maybe, that's what I'll do for February.

Now, from the curriculum list I drew up, there are around 20 areas that need to be covered. Assuming that for each of these areas, one would want to make a reasonably complete list of theorems and a list of concepts and write a topic entry and a bibliography, that's something like 40 or 50 projects each of which would involve the mythical man-month of labor and would entail skimming through about 100 math books. That's a bit much for a single person, but if a dozen members were serious enough about it to spend half their time on the project, it should be feasible in a year or two.

Now that Planet Math has reached 4000 entries and we have so much material entered, maybe it would be a good time to pull back a bit from adding new material to organize the existing material and fill in gaps systematically. Even if Joe's suggestion of a temporary moratorium on new stuff is a bit extreme, maybe it wouldn't be such a bad idea if the regulars around this joint would take some time out to fill in the gaps before moving on.

Ray

## Re: (hopefully not infinite) descent

Maybe we should make an entry based on your list. (Draft below, but its your

list of topics, so perhaps you should make the entry.) I like your strategy for

building a list of basic topics to cover. If I know which pages are being used

to keep track of what's been covered, I can probably write some quick routines

that will give a numerical estimate of PM's completeness WRT these lists.

====================================================================

The purpose of this entry is to list the basis courses in the typical

mathematics curriculum. Each of the items listed will eventually link to a page

that gives a more detailed overview of that subject. PlanetMath will eventually

have fairly complete coverage of all of these topics.

\begin{description}

\item[School]

\begin{itemize}

\item Arithmetic

\item Algebra

\item Euclidean Geometry

\item Trigonometry

\item Analytic Geometry

\item Calculus

\end{itemize}

\item[University]

\begin{itemize}

\item Real Analysis

\item Complex Analysis

\item Linear and Multilinear Algebra

\item Abstract Algebra (Groups, Rings, Fields)

\item Set Theory

\item Logic

\item Probability

\item Statistics

\item Topology (Point Set and Algebraic)

\item Differential Geometry

\item Algebraic Geometry

\item Number Theory

\item Discrete Math (Combinatorics, Game Theory)

\item Functional Analysis

\end{itemize}

\end{description}

## Re: (hopefully not infinite) descent

I like the idea. In addition to helping make sure that the data base is reasonably complete, it could serve another useful purpose. After all, many users of Planet Math are students (and teachers) who might appreciate an entry which helps relate the contents of the encyclopaedia to what they're studying and suggests what entries they should be looking at based on where they are in their studies. Another point that might be worth keeping in mind when putting together such pages is the fact that Planet Math entries are written at varying levels of sophistication and this might be a way to make sure that there is somewhere an exposition of an elementary topic at an elementary level. For example, if an article on factorization of polynomials begins with "Let R be a ring and R[x] the ring of polynomial,...", then it's probably going to go over the head of a 7-th grader studying how to factor polynomials and it might be time to add a more elementary article on the same topic (which could point to the more advanced article as a suggestion for advanced reading down the road.)

To some extent, this would bump into the "overview of Planet Math" and the pages related to that entry, but I don't think that's a real problem. First, as our illustrious founder said a few days ago in reference to a graphics guide, some duplication can be good. Second, the curriculum guide and the overview approach the same goal from different vantage points. The curriculum guide is written for the student seeing the material for the first time whilst the overview is written more from the vantage point of the research mathematician who is considerable more mature mathematically. As I see it, having several different portals into and different organizational systems for the same body of knowledge is a good thing; having an alphabetical index, a subject index, a search engine, an overview, and a curriculum guide ought to make Planet Math more useful an accessible as well as help us make sure that it is reasonably complete.

In a few days, I'll add this curriculum-based entry and start on some of the related pages. Right now, I'll rest.

## question for the regularity theorem

When I first saw the formulation of the theorem I was quite suprised. Ussually one considers Laplace equation together with boundary conditions. Then in this situation the following factors influence the regularity of solution, namely:

- regularity of the boundary

- regularity of the boundary data

and all of this are missing here. So, am I understanding right, that INDEPENDANTLY of the boudary conditions, and INDEPENDANTLY of the boundary (it could be even not Lipshitz continuous), the function sattisfing Laplace equation inside domain is ANALYTIC? Is it just a feature Laplace equation, or the same is true for some other PDE (partial diff.eq.), like Helmholz equation?

Thanks in advance.

## Re: question for the regularity theorem

That's absolutely correct and, as you say, it is is an amazing fact. For most differential equations regularity of the solution depends on regularity of the boundary data, but for the Laplace equation, you could have boundary data which are not differentiable, maybe even discontinuous or even distributional, yet the unique solution of the Laplace equation will still be analyic on the interior of the domain. (I once amazed a class in electromagnetic theory I taught by exhibiting a solution of the Laplace equation inside a cube which was perfectly analytic on the interior, but which was a Dirac delta-distribution on the boundary of the cube.) I know that this property is shared by certain other elliptic equations. (many of which which may be thought of as generalizations of the Laplace equation with variable coefficients). I believe that the Helmoltz equation also has this property that all solutions are analytic, but don't quote me on that just yet, since I haven't thought it through and checked it 100%. If so, it might be a good thing to add another entry on (or perhaps a topic entry on PDE's all of whose solutions are analytic).

The fact there are so many properties of solutions of the Laplace equation which do not at all depend on the boundary conditions or the nature of the boundary is why there is a sub-branch of analysis that studies the properties of such functions. For more information, see the topic entry on potential theory and the references contained there.

## Re: question for the regularity theorem

Thanks a lot for the answer. When I saw word 'regularity' I was immediately thinking about 'Sobolev regularity', i.e. belonging of the solution to some Sobolev space. And I think that such regularity DO depend on the boundary shape and boundary data. In particular in your example:

> solution of the Laplace equation

> inside a cube which [is] perfectly analytic on the interior,

> but which [is] a Dirac delta-distribution on the boundary of the cube.

that solution probably doesn't belong even to L^2.

Anyway, thanks once again for quite complete explanation.

Regards

Serg.