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I have been reading An Introduction to PDE written by Pinchover and Rubinstein, but there is a proof of a corollary that is not clear for me.

Once they proof weak maximum principle for heat equation and use it to prove continuous dependence of solutions with parameters and uniqueness a corollary says the following

Let \begin{align*} u(t,x)=\sum_{n=1}^{\infty}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t} \end{align*} be a formal solution of the heat problem \begin{align*} \begin{cases} u_{t}-ku_{xx}=0 &x\in(0,L),\,t>0\\ \begin{cases} u(t,0)=0 &t\geq0\\ u(t,L)=0 &\\ u(0,x)=f(x) &x\in[0,L]\\ \end{cases} \end{cases} \end{align*}

If series \begin{align*} f(x)=\sum_{n=1}^{\infty}B_{n}\sin\frac{n\pi x}{L} \end{align*} converges uniformly in $[0,L]$, then the series for $u$ converges uniformly in $[0,L]\times[0,T]$, and $u$ is classical.

For the proof uses the Cauchy criterion for uniform convergence for the series of $f(x)$, that is, for $\epsilon>0$ exists $N_{\epsilon}$ such that for $l\geq k\geq N_{\epsilon}$ is true that \begin{align*} \left\vert S_{l}-S_{k}\right\vert<\epsilon,\,\forall x\in[0,L] \end{align*} where I mean for $S_{k}$ \begin{align*} S_{k}=\sum_{n=1}^{k}B_{n}\sin\frac{n\pi x}{L} \end{align*} that is \begin{align*} \left\vert \sum_{n=1}^{l}B_{n}\sin\frac{n\pi x}{L}-\sum_{n=1}^{k}B_{n}\sin\frac{n\pi x}{L}\right\vert< &\epsilon,\,\forall x\in[0,L]\\ \left\vert \sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}\right\vert<& \end{align*}

Obviously, for each $n$ we have that $u_{n}(t,x)=B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t}$ is a classical solution of the heat equation and, by superposition principle, a finite sum of them are again classical solutions, that is \begin{align*} v(t,x)=\sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t} \end{align*} is classical. Here, the autors say that, by the weak maximum principle, we have \begin{align*} \left\vert\sum_{n=k}^{l}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t}\right\vert<\epsilon,\,\forall x\in[0,L] \end{align*} Then they complete the proof saying that by Cauchy criterion the series solution converges uniformly on the square to a continuous function $u$ that satisfies boundary and initial conditions, therefore $u$ is classical.

WHAT I DON'T UNDERSTAND IS THE JUMP USING THE MAXIMUM PRINCIPLE. CAN ANYONE EXPLAIN IT TO ME IN A CLEAR AN EXPLICIT WAY?

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2 Answers 2

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Well, I have finally figured out this one. Simply they are not using directly maximum principle but continuity with initial and boundary conditions (which is proved using maximum principle) applied to sums of $u_{n}$ appropriate and the fact that Fourier series for $f$ converge uniformly. Then you can prove that formal $u$ converges uniformly in the rectangle to a continuous function that satisfies boundary and initial conditions and therefore solution is classical.

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Actually, uniform boundedness of
\begin{align*} \bigg|\sum_{n=k}^{\ell}B_{n}\sin\frac{n\pi x}{L}\mathrm{e}^{-k\left(\frac{n\pi}{L}\right)^{2}t}\bigg|<\varepsilon,\,\text{for all}\, x\in[0,L], \end{align*} does follow from the weak maximum principle, as $$ u_{k,\ell}(x,t)=\sum_{n=k}^{\ell}B_{n}\sin\frac{n\pi x}{L}\mathrm{e}^{-k\left(\frac{n\pi}{L}\right)^{2}t}, $$ satisfies heat equation as well, and hence it satisfies the weak maximum principle. Thus its maximum and minimum is achieved on the boundary, and as it vanishes for $x=0$ and $x=L$, then $$ \max_{(x,t)\in [0,L]\times[0,T]}\bigg|\sum_{n=k}^{\ell}B_{n}\sin\frac{n\pi x}{L}e^{-k\left(\frac{n\pi}{L}\right)^{2}t}\bigg|=\max_{x\in[0,L]}\bigg|\sum_{n=k}^{\ell} B_{n}\sin\frac{n\pi x}{L} \bigg| < \varepsilon. $$

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