## 275- Harmonic functions and harmonic conjugates

December 23, 2008

Recall that a function $u(x,y)$ of two variables defined on an open domain $D$ is harmonic iff  $u$ is $C^2$ (i.e., all four second order derivatives $u_{xx},u_{xy},u_{yx},u_{yy}$ exist and are continuous in $D$), and $u$ satisfies Laplace equation

$u_{xx}+u_{yy}=0.$

As mentioned in problem 6 of the Fall 2008 Calculus III final exam, a function $v$ is a harmonic conjugate of $u$ iff $v$ is defined on $D$, $v_x$ and $v_y$ exist, and the Cauchy-Riemann equations hold:

$u_x=v_y$ and $u_y=-v_x$.

It follows immediately from the Cauchy-Riemann equations that if $v$ is a harmonic conjugate of a harmonic function $u$, then $v$ is also $C^2$, with $v_{xx}=-u_{yx}$, $v_{xy}=-u_{yy}$, $v_{yx}=u_{xx}$ and $v_{yy}=u_{xy}$. It is also immediate that $v$ satisfies Laplace equation because $v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0$, since continuity guarantees that the mixed partial derivatives commute. Thus $v$ is also harmonic.

In fact, modulo continuity of the second order derivatives, the harmonic functions are precisely the functions that (locally) admit harmonic conjugates.

To see this, assume first that $u$ is $C^2$ in $D$ and that it admits a harmonic conjugate $v$. Then $u_{xx}=v_{yx}$ and $u_{yy}=-v_{xy}$ so $u_{xx}+u_{yy}=0$ and $u$ was harmonic to begin with.

Conversely, assume that $u$ is harmonic in $D$. Suppose first that $D$ is (connected and) simply connected. I claim that then $u$ admits a harmonic conjugate $v$ in $D$. To see this, letting ${\mathbf F}=(-u_y,u_x)$, notice that the existence of $v$ is equivalent to the claim that ${\mathbf F}$ is a gradient vector field, since ${\mathbf F}=\nabla v$ iff $v$ is a harmonic conjugate of $u.$ But, since $D$ is simply connected, then ${\mathbf F}$ is a gradient iff it is conservative, i.e., $\displaystyle \oint_\gamma {\mathbf F}\cdot d{\mathbf r}=0$ for any simple piecewise smooth loop $\gamma$ in $D$. Fix such a $\gamma$, and let $R$ denote its interior. Then, by Green’s theorem,

$\displaystyle \oint_\gamma {\mathbf F}\cdot d{\mathbf r}=\pm\iint_R u_{xx}+u_{yy}\,dA=0,$

where the $\pm$ sign was to be chosen depending on the orientation of $\gamma$. It follows that ${\mathbf F}$ is indeed conservative and therefore a gradient, so $u$ admits a harmonic conjugate.

Finally, if $D$ is not simply connected, we cannot guarantee that such a $v$ exists in all of $D$, but the argument above shows that it does in any open (connected) simply connected subset of $D$, for example, any open ball contained in $D$.  That we cannot extend this to all of $D$ follows from considering, for example, $u(x,y)=\log(x^2+y^2)$ in $D={\mathbb R}^2\setminus\{(0,0)\}$. This is a harmonic function but it does not admit a harmonic conjugate in $D$, since there is no continuous $\arctan(y/x)$ in $D$. This example can be easily adapted (via a translation) to any non-simply connected $D$.

I close by remarking that, as mentioned in my previous post on average values of harmonic functions, one can use Green’s theorem to prove that harmonic functions $u$ satisfy the average (or mean) value property, and this property characterizes harmonicity as well, implies that $u$ is actually $C^{\infty}$ (i.e., $u$ admits partial derivatives of all orders, and they are all continuous) and has the additional advantage that it only requires that $u$ is continuous, rather than $C^2$. Similarly, one can show that the Cauchy-Riemann equations on $D$ suffice to guarantee that $u$ and $v$ are harmonic (and in particular, $C^\infty$). However, one needs to require that the equations hold everywhere on $D$. A pointwise requirement would not suffice. But I won’t address this issue here (I mention it in the notes in complex analysis that I hope to post some day).

## 175, 275 -Final exam

December 13, 2008

Just a few general remarks:

• 175: The exam is this Monday, Dec. 15, from 8:00 to 10:00 at the usual place.
• 275: The exam is this Monday, Dec. 15, from 10:30 to 12:30 at the usual place.
• It is cumulative, although emphasis is put in the material covered after the second test.
• You can use books, notes, etc as before.
• Bring your own pens, pencils, calculators, AND PAPER. I won’t have extra paper if you don’t bring enough, and the margins of the exam won’t suffice. Mark with your name every single page you turn in.

See you Monday. Good luck!

Update [Dec. 22/08]:

• Here is the exam for Calculus II – 175, and here are the solutions. (Silly typo in the solution of problem 4 corrected.)
• Here is the exam for Calculus III – 275, and here are the solutions
• I will be in my office on December 19 from about 11 until about 1, in case you want to stop by and pick up your test.
• I won’t be on campus until the Spring term. I’ll post my new office hours soon, in case you want to stop by and pick up your test once I’m back. I’ll keep the exams and homework sets I still have through the Spring term, and you can collect them at any time during office hours. Whatever remains once the term is over, I will then discard.

## BOISE EXTRAVAGANZA IN SET THEORY – Announcement 1

December 3, 2008

Friday, March 27 – Sunday, March 29, 2009

Organized by Liljana Babinkostova, Andres Caicedo, Stefan Geschke, Richard Ketchersid, and Marion Scheepers.

We are pleased to announce our eighteenth annual BEST conference.

There will be four talks by invited speakers:

Steve Jackson (University of North Texas)

Ljubisa Kocinac (University of Nis, Republic of Serbia)

Assaf Rinot (Tel Aviv University, Israel)

Grigor Sargsyan (University of California, Berkeley)

The talks will be held on Friday, Saturday and Sunday at the Department of Mathematics, Boise State University.

The conference webpage is available at URL

http://math.boisestate.edu/~best/best18

This page will be updated with information regarding lodging, abstract submission, weather, maps, schedule, etc.

Limited financial support is available to partially offset travel expenses of some participants. The criteria for granting support include whether the participant has alternative financial support for the conference, and whether the participant is presenting a talk at the conference. Preference is given to graduate students and early career researchers. The amount of support is contingent on the budget constraints. University accounting regulations require completing certain forms appropriately and submitting original receipts of expenses before issuing checks.

To apply for support, email the organizers at

best@math.boisestate.edu

Applications from graduate students must be supported by a separate email from their thesis advisor.

Anyone interested in participating should contact the organizers as soon as possible by sending an email to

best@math.boisestate.edu

The organizers will be editors for a volume in the Contemporary Mathematics series. Research papers on topics related to Set Theory and its Applications will be considered for publication in this volume. All papers will go through a thorough referee process. Former and current participants of the BEST conferences or their collaborators are especially encouraged to consider submitting a research paper. Anyone interested in submitting a paper should contact Marion Scheepers as soon as possible at marion@math.boisestate.edu with this information. More information will be posted at the conference web site.

The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Contemporary Mathematics is published by the American Mathematical Society. Support from these three organizations is gratefully acknowledged.

## 175, 275 -Homework 12

December 1, 2008

Homework 12 is due Tuesday, December 9, at the beginning of lecture. The usual considerations apply. This is the last homework set of the term. Each exercise is worth 1 point.

• 275: Turn in the problems you still have pending. Notice that the homework does not cover the last few sections of Chapter 14. However, these sections are included in the final, so make sure you try (on your own) a few exercises as practice.
• 175: Do not use the solutions manual for any of these problems. Turn in the problems you still have pending. Also: Section 8.8, exercise 28; section 8.9, exercise 40; section 8.10, exercise 19. There is no homework on the additional topics we will cover, but they will be included in the final.