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Problem set 1, PDE

Problem 1

Let u be a $latex C^1$ function defined on a neighborhood of the domain $latex \Omega$ such that $latex \Delta u = 0$ in $latex \Omega$. Then

$latex \displaystyle \int_{\partial \Omega} \partial_\nu u \; d\sigma = 0.$

Problem 2

If the power series $latex \sum a_\alpha x^\alpha$ converges at $latex x^0$, then it converges on the cube $latex |x_j|<r$ where $latex r = \min_i |x_i^0|$.

Problem 3

If u satisfies the Cauchy problem

$latex \begin{cases} \partial_t^mu = G\big(x,t,(\partial_x^\alpha\partial_t^ju)_{|\alpha|+j\le m, j<m}\big)\\ \partial_t^ju(x,0) = \phi_j(x), \quad 0\le j < m,\end{cases}$

then the functions $latex y_{\alpha j} = \partial_x^\alpha\partial_t^ju$ satisfy the system

$latex \partial_t y_{\alpha j} = y_{\alpha(j+1)} \quad |\alpha|+j < m,$                              $latex \partial_t y_{\alpha j} = \partial_{x_i}y_{(\alpha-e_i)(j+1)} \quad |\alpha|+j=m, j<m,$

$latex \displaystyle \partial_t y_{0m} = \frac{\partial G}{\partial t} + \sum_{|\alpha|+j<m} \frac{\partial G}{\partial y_{\alpha j}} y_{\alpha(j+1)} + \sum_{|\alpha|+j=m, j<m} \frac{\partial G}{\partial y_{\alpha j}} \partial_{x_i}y_{(\alpha - e_i)(j+1)},$

where $latex i = \min\{j: \alpha_j\not=0\}$, with data

$latex y_{\alpha j}(x,0) = \partial_x^\alpha \phi_j(x), \quad j<m,$                                          $latex y_{0m}(x,0) = G\big(x,0,(\partial_x^\alpha \phi_j(x))_{|\alpha|+j\le m, j<m}\big),$

Problem 4

Carry out the explicit calculations to reduce the equation $latex \Delta u = f$ to the first order system of the previous problem.