Solvable PDE Problems - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T16:33:57Z http://mathoverflow.net/feeds/question/65951 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65951/solvable-pde-problems Solvable PDE Problems unknown (google) 2011-05-25T11:04:52Z 2011-05-25T12:22:14Z <p>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.</p> <p>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.</p> <p>So please answer to this question if you have any suggestions.</p> <p>( Excuse me if this isn't the right place to ask this question. )</p> http://mathoverflow.net/questions/65951/solvable-pde-problems/65955#65955 Answer by Amitesh Datta for Solvable PDE Problems Amitesh Datta 2011-05-25T12:22:14Z 2011-05-25T12:22:14Z <p>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. </p> <p>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 <em>harmonic function</em> $u$ defined in the interior of $D$ (i.e., ${x\in\mathbb{R}^2:\left|x\right|&lt;1}$) 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 $\left|x\right|&lt;1$ and $F(x)=f(x)$ if $\left|x\right|=1$ is continuous. This is called the <strong>Dirichlet problem in the unit disk</strong>.</p> <p>Similarly, let $1\leq p&lt;\infty$ 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 <strong>Dirichlet problem in the upper half plane</strong>.</p> <p>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 <em>Real and Complex Analysis</em> (2nd. edition), chapter 11, and Loukas Grafakos' <em>Classical Fourier Analysis</em>, chapter 2, pages 84-87.</p> <p><strong>Solution to Dirichlet's problem in the unit disk</strong>: 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 $0\leq r &lt; 1$, where $P_r(t)= \frac{1-r^2}{1-2r\cos(t)+r^2}$ is the <em>Poisson kernel</em>. </p> <p><strong>Solution to Dirichlet's problem in the upper half plane</strong>: 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.</p> <p>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.</p>