Paweł Strzelecki | Some problems again

Here are some problems related to Sobolev spaces and weak solutions.

Problem 1. Consider the space $l^2$ of doubly infinite, square summable sequences $a= (a_n)$ of real numbers, with the norm

$\| a \|^2 = \sum_{n=-\infty}^{+\infty} a_n^2\, ,$

and its subspace $h^1$ of sequences such that

$\| a \|_{h^1}^2 = \sum_{n=-\infty}^{+\infty} (1+n^2) a_n^2\, < +\infty.$

Prove directly that a bounded sequence in $h^1$ has a subsequence which converges in $l^2$ . (Hint: you might use Cantor’s diagonal method.)

What does this have to do with Rellich–Kondrashov’s compactness theorem?

In Problems 2 and 3 we assume that $u\in W_0^{1,2}(\Omega)$ is an eigenfunction of the Dirichlet laplacian, i.e. a weak solution of $-\Delta u=\lambda u$ for some $\lambda >0$ .

Problem 2 (Caccioppoli inequality). Let $B_r = B(a,r)$ and $B_R=B(a,R)$ be two concentric balls contained in a domain $\Omega \subset \mathbb{R}^n$ . Prove that for some constant which depends on $R,r,n$ and $\lambda$ we have

$\int_{B_r} |\nabla u|^2 \, dx \le C_0\int_{B_R} |u|^2\, dx\, .$

Hint: Use the definition of a weak solution with a test function

$\varphi=\zeta^2 u\in W_0^{1,2}(\Omega),$

where $\zeta\in C^\infty_0$ is such that $\zeta=1$ on $B_r$ , $\zeta = 0$ on $\mathbb{R}^n\setminus B_R$ and $\zeta\ge 0$ .

Problem 3 (a bit more involved). Consider $u_\varepsilon=u\ast\varphi_\varepsilon$ , where

$\varphi_\varepsilon = \varepsilon^{-n} \varphi (x/\varepsilon)$

is a standard mollifier. Prove that for each compact set $K\Subset\Omega$ , each $m\in \mathbb{N}$ and each multiindex $\alpha$ with $\vert\alpha\vert=m$ there is a number $N>0$ such that

$\bigl(D^\alpha u_{1/j}\bigr), \qquad j=N, N+1,N+2,\ldots$

is a Cauchy sequence in $L^2(K)$ . Use this fact, the completeness of $W^{m,2}$ , and the Sobolev imbedding, to conclude that all eigefunctions of the Laplacian in a bounded domain $\Omega$ are smooth in the classical sense in the interior of $\Omega$ .

Hint: What equation and in which domain does $u_\varepsilon$ satisfy? Do its derivatives satisfy the same equation? In which sense (weak or classical)? Once you answer these questions, try to use the result of Problem 2.