Problem set 5, PDE

Problem 1

Classify the following second-order PDE.

  1. \displaystyle \sum_{i=1}^n \partial^2_{x_i x_i}u + \sum_{1\le i<j\le n} \partial^2_{x_i x_j}u = 0
  2. \displaystyle \sum_{1\le i<j \le n} \partial^2_{x_i x_j}u = 0

Problem 2

Use energy methods to discuss the uniqueness of the problem

\begin{cases} \Delta u - u^3 = f & \text{in }\Omega\\u = \phi & \text{on }\partial\Omega.\end{cases}

Problem 3

Let \Omega be a bounded C^1-domain in \R^n and u be a C^2-function in \bar\Omega\times[0,T] satisfying

\begin{cases} u_t - \Delta u = f & \text{in }\Omega\times(0,\infty)\\ u(\cdot, 0) = u_0 & \text{in }\Omega\\ u=0 & \text{on }\partial\Omega\times(0,\infty).\end{cases}

Then

\displaystyle \sup_{0\le t \le T} \int_\Omega |\nabla u(\cdot,t)|^2 dx + \int_0^T\int_\Omega |\partial_t u|^2 dx dt \le C\Big( \int_\Omega |\nabla u_0|^2 dx + \int_0^T\int_\Omega |f|^2 dxdt \Big),

where C is a positive constant depending only on \Omega.

Problem 4

Verify the identity

\displaystyle \frac{1}{2\pi} + \frac{1}{\pi} \sum_{k=1}^\infty r^k \cos k(\theta - \eta) = \frac{1}{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - \eta) + r^2}.

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