Homework 10

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1. Warming up

a.- Solve

in a rectangle of sides $(L_1,L_2)$, with the following boundary conditions and sources:

1. Homogeneous Dirichlet boundary conditions and $f$ arbitrary in $L^2$.

2. Homogeneous Neumann boundary conditions and $f$ arbitrary in $L^2$. What further condition is need to assert a solution exists?

3. Mixed boundary conditions, look at what means a Robin boundary condition.

b.- From the structure of the solutions check that if $f \in H^s$ then $u \in H^{s+2}$.

c.- Solve for inhomogeneous Dirichlet boundary conditions. Assume you can decompose the boundary data into the boundaries values for the eigenfunctions of the Laplacian at the rectangle. What can you conclude about the smoothness of the solution in the interior?

2. Clasification

Exercise 1.1 Classify the following PDEs as elliptic, parabolic, or hyperbolic:

1. $\Delta u + u = 0$ (Helmholtz equation)

2. $u_{xx} + 2u_{xy} + u_{yy} = 0$

3. $u_{xx} - u_{yy} + 2u_x = 0$

4. $\text{div}(a(x)\nabla u) = f(x)$, where $a(x) > 0$.

Exercise 1.2 Show that the PDE $u_{xx} + \cos(x) u_{xy} + \sin(x) u_{yy} = 0$ is elliptic in a region of $\mathbb{R}^2$. Find this region.

3. Maximum Principles

Exercise 3.1 Prove the weak maximum principle for $\Delta u = 0$ in a bounded domain $\Omega \subset \mathbb{R}^n$: If $u \in C^2(\Omega) \cap C(\overline{\Omega})$, then

Exercise 3.2 Let $u$ satisfy $\Delta u \geq 0$ in $\Omega$. Show that if $u$ attains a maximum at an interior point, then $u$ is constant.

Exercise 3.3 For $\Delta u + c(x)u = 0$ with $c(x) \leq 0$, prove that if $u$ attains a non-negative maximum inside $\Omega$, then $u$ is constant.

4. Existence and Uniqueness

Exercise 4.1 Use Lax-Milgram to show that the Dirichlet problem

has a unique weak solution in $H^1_0(\Omega)$ for $f \in L^2(\Omega)$.

Exercise 4.2 For the Neumann problem $-\Delta u = f$ with $\frac{\partial u}{\partial n} = g$: (a) Find a necessary condition on $f,g$ for solvability. (b) Show solutions are unique up to additive constants.

5. Variational Methods

Exercise 5.1 Show that minimizers of

over $H^1_0(\Omega)$ solve $-\Delta u = f$.

Exercise 5.2 Prove the first Dirichlet eigenvalue $\lambda_1$ of $-\Delta$ satisfies:

7. Nonlinear Elliptic Equations

Exercise 7.1 Show that a weak solution exists for

when $f \in L^2(\Omega)$.

Exercise 7.2 Prove that for $-\Delta u = e^u$ in $\Omega \subset \mathbb{R}^n$ bounded,