Partial differential equations: modern theory

Let $\varphi$ be a test function with $\varphi(x) = 2 + 3x + 5x^2$ for $|x| < 1$. Compute $\langle \delta', \varphi \rangle$, where $\delta'$ is the distributional derivative of the Dirac delta.

On $\Omega = (-1, 1)$, which functions belong to $H^1(\Omega)$? Select all that apply.

On $V = H^1_0(0, 1)$ (norm $\|u\|_{H^1}^2 = \int u^2 + u'^2$), which bilinear forms are coercive? Select all that apply.

Hadamard's example $u_n(x, y) = \sin(nx)\sinh(ny)/n^2$ shows that the Cauchy problem $\Delta u = 0$, $u(x, 0) = g$, $\partial_y u(x, 0) = h$ is ill-posed. Which of Hadamard's conditions fails?

In the proof of Lax–Milgram, $A$ is defined by $B(u, v) = \langle Au, v \rangle$. Which argument shows that the range of $A$ is closed?

Suppose hcoer : ∀ u, C * ‖u‖ * ‖u‖ ≤ B u u and hu : ∀ v, B u v = ℓ v. Which conclusion is the stability estimate of Lax–Milgram (and is provable from these hypotheses)?

For a smooth solution of $u_t = u_{xx}$ on $(0, 1)$ with $u(0, t) = u(1, t) = 0$, let $E(t) = \int_0^1 u^2\,dx$. Using the sharp Poincaré inequality $\int_0^1 u_x^2 \ge \pi^2\int_0^1 u^2$, which bound holds?

Solve $-u'' = f$ on $(0, 1)$, $u(0) = u(1) = 0$, by finite differences: with $h = 1/N$ and $x_i = ih$, the unknowns $u_1, \dots, u_{N-1}$ satisfy $\frac{-u_{i-1} + 2u_i - u_{i+1}}{h^2} = f(x_i)$ with $u_0 = u_N = 0$. Write solve_poisson(f, n) returning the list $[u_0, u_1, \dots, u_N]$, solving the tridiagonal system by forward elimination and back substitution (the Thomas algorithm).