Partial Differential Equations

Assignment 3

1. [20 points] Determine the types of the following equations, and reduce them to canonical form.

   1. $u_{xx} + 2e^{x+y}u_{xy} + e^{2y}u_{yy} = 0.$

   2. $u_{xx} + 2u_{xy} + 4u_{xz} + 5u_{zz} + u_x + 2u_y = 0 \; (\text{in } \mathbb{R}^3).$

   3. $u_{xx} - 2 \sin x u_{xy} - \cos^2 x u_{yy} - \cos x u_y = 0.$

   4. $\sin^4(2x) u_{xx} + 4\sin^4(2x)u = \frac{\partial^2 u}{\partial t^2}.$

2. [10 points] Use Gauss-Divergence Theorem to drive the following integration by parts formula

   1. $$\int_\Omega v\frac{\partial u}{\partial x_i}=-\int_\Omega u\frac{\partial v}{\partial x_i}+\int_{\partial\Omega}uv\nu^i$$
   2. $$\int_\Omega v\Delta u=-\int_\Omega\nabla u\nabla v+\int_{\partial\Omega}\frac{\partial u}{\partial \nu}v$$

3. [20 points] Let$\Omega$be a open bounded subset of$\mathbb{R}^n$with$\nu$the outward unit normal vector at$\partial\Omega$,$A$be an$n\times n$matrix and$L=\text{div}(A\nabla u)$be an elliptic operator on$\Omega$. Discuss the uniqueness of solution (if it exists) for the following PDEs:

   1. $$Lu=f \text{ in } \Omega,\quad u=g \text{ on } \partial\Omega$$
   2. $$Lu=f \text{ in } \Omega,\quad A\nabla u \cdot \nu = g \text{ on } \partial\Omega$$
   3. $$Lu-u=f \text{ in } \Omega,\quad A\nabla u \cdot \nu = g \text{ on } \partial\Omega$$