Solvable PDE Problems

2011-05-25T11:04:52Z

I'm currently in a PDE course where one of the requirements is to find a common PDE problem and explain how to solve it.

The problems found easily on google won't help me, since every student has to find a different PDE problem and they are all have been chosen by other students.

So please answer to this question if you have any suggestions.

( Excuse me if this isn't the right place to ask this question. )

Answer by Amitesh Datta for Solvable PDE Problems

2011-05-25T12:22:14Z

The PDE that I shall suggest is quite common and therefore it is likely that it has already been selected by another student. However, the analysis of this PDE is vast and very interesting.

The motivation is as follows: let $D$ be the unit disk in the plane (i.e., ${x\in \mathbb{R}^2: \left|x\right|\leq 1}$) and let $f$ be a continuous function defined on the boundary of $D$. We wish to find a harmonic function $u$ defined in the interior of $D$ (i.e., ${x\in\mathbb{R}^2:\left|x\right|<1}$) whose boundary values are $f$; i.e., $u$ is a continuous function required to satisfy the Laplace equation $u_{xx}+u_{yy}=0$ and the function $F$ defined on $D$ by the rule $F(x)=u(x)$ if $\left|x\right|<1$ and $F(x)=f(x)$ if $\left|x\right|=1$ is continuous. This is called the Dirichlet problem in the unit disk.

Similarly, let $1\leq p<\infty$ and let $f\in L^p(\mathbb{R})$. We wish to find a harmonic function $u$ defined in the upper half plane such that $u(x,0)=f(x)$ almost everywhere on the real line. This is called the Dirichlet problem in the upper half plane.

There exist approaches to both problems that use general measure theory in a particularly enlightening manner. I will briefly sketch the solutions; if you wish to see a more comprehensive treatment, you may look at Walter Rudin's Real and Complex Analysis (2nd. edition), chapter 11, and Loukas Grafakos' Classical Fourier Analysis, chapter 2, pages 84-87.

Solution to Dirichlet's problem in the unit disk: the general approach is to define $u$ as the Poisson integral of $f$. More precisely, we define $u(re^{i\theta})=\frac{1}{2\pi} \int_{-\pi}^{\pi} P_r(\theta - t)f(t) dt$ for $0\leq r < 1$, where $P_r(t)= \frac{1-r^2}{1-2r\cos(t)+r^2}$ is the Poisson kernel.

Solution to Dirichlet's problem in the upper half plane: the general approach is to first define the Poisson kernel $P_t(x)=c\frac{t}{x^2+t^2}$ (for $t>0$, $x\in\mathbb{R}$, and $c=\frac{1}{\pi}$) and then define $u(x,t)=(P_t * f)(x)$; the convolution of $P_t$ and $f$ on the real line. Since ${P_t}_{t>0}$ is an approximate identity on $\mathbb{R}$, it follows that $u(x,t)$ converges to $f(x)$ in $L^p$ as $t\to 0$. In fact, this convergence is a.e. (the proof is non-trivial; one approach is to use maximal functions) and this implies that we have solved the Dirichlet problem in the upper half plane.

I hope that I have helped and I apologize for the somewhat sketchy proofs! I have certainly noted some non-trivial facts and I recommend you to look at Rudin and Grafakos for the details. Of course, I should add that the solutions that I have presented will be much more meaningful if you are familiar with measure theory and elementary complex analysis.

Solvable PDE Problems - MathOverflow [closed]

Solvable PDE Problems

Answer by Amitesh Datta for Solvable PDE Problems