Problem Set 5

1. Classification of PDE

Show that the equation \[ y u_{xx} + (x + y) u_{xy} + x u_{yy} = 0, \]

is hyperbolic everywhere except along $y = x$. Put the PDE into canonical form and then find the general solution.

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2. Classification of PDE II

Study the PDE \[ yu_{xx} - u_{yy} = 0. \]

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3. Continuous dependence on data in the wave equation

By using d'Alembert's formula applied to the wave equation \begin{gather} c^2 u_{xx} = u_{tt} \label{wave1} \\ u(x,0) = f(x), u_t(x,0) = g(x), \quad -\infty < x < \infty, \label{wave2} \end{gather}

show that the solution must depend continuously on the data, $f$ and $g$ on any strip $\{(x,t) | -\infty < x < \infty, 0 \leq t \leq T\}$.

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5. Solution of Spherical Laplace

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A neat ill-posed problem

Here is an example that shows that elliptic problems are (can be?) ill-posed if we give them Cauchy data. Suppose we wish to solve \begin{gather*} \nabla^2 u = u_{xx} + u_{yy} = 0 \quad \text{in $y > 0$}
u(x,0) = 0
u_y(x,0) = 0. \end{gather*}

Of course, $u = 0$ is one solution to the problem. Now suppose were to perturb the boundary very slightly: \begin{gather*} \nabla^2 v = v_{xx} + v_{yy} = 0 \quad \text{in $y > 0$}
v(x,0) = 0
v_y(x,0) = e^{-\sqrt{n}} \sin nx = g(x). \end{gather*}

Notice that the Neumann condition is very small, with $|g(x)| \to 0$ as $n \to \infty$. However, we can verify that one solution to this problem is \[ v(x,y) = \frac{e^{-\sqrt{n}}}{n} \sin(nx)\sinh(ny). \]

Regardless of how large $n$ is, following $\sinh(ny) \sim e^{ny}/2$, we have \[ |v(x,y)| \sim \frac{|\sin (nx)|}{2n} e^{-\sqrt{n}+ny} \to \infty, \]

for any fixed $y$ and for any $x \neq \pi\mathbb{Z}$. So even though the boundary condition is such that \[ \max_{x\in \mathbb{R}} |v_y(x,0)| \to 0 \]

we are not approaching the solution $u = 0$ but are instead having \[ \max_{(x,y)\in \mathbb{R}^2} |v(x,t)| \to \infty \]

and so the problem is ill-posed and does not vary continuously with the initial data.