Weak convergence in $L^p$ implies strong convergence in $H^{-1}$

Question

Let $B(0,1) \subset \mathbb{R}^d$ and $p > d$. Suppose $f_n$ weakly converges to $f$ in $L^p(B(0,1))$. Prove that $f_n$ strongly converges to $f$ in $H^{-1}(B(0,1))$. Here $H^{-1}(B(0,1))$ is the dual space of $H^1_0(B(0,1))$ with respect to $L^2-$topology.

I checked many books such as Evans' PDE, Brezis' FA or Mazya's Sobolev Spaces but didn't found this result.

Any ideas to prove it? (References are also welcome). Thank you.

Answer 1

I know how to prove this using "Fourier series." I wasn't able to prove it otherwise when I tried way back when. (But see daw's comment above...)

$B(0,1)$ is nice enough that we know there is an orthonormal system $\{e_{n}\}_{n \in \mathbb{N}} \subseteq H^{1}_{0}(B(0,1))$ of $L^{2}(B(0,1))$ consisting of eigenfunctions of the Laplacian. That is, for each $n \in \mathbb{N}$, \begin{equation*} -\Delta e_{n} = \lambda_{n} e_{n} \end{equation*} where $\{\lambda_{n}\}_{n \in \mathbb{N}} \subseteq (0,\infty)$ is increasing and unbounded. Also, if we equip $H^{1}_{0}(B(0,1))$ with the inner product \begin{equation*} (u,v)_{H^{1}_{0}(B(0,1))} = \int_{B(0,1))} Du \cdot Dv \, dx, \end{equation*} then $(\lambda_{n}^{-\frac{1}{2}} e_{n})_{n \in \mathbb{N}}$ is an orthonormal basis for this Hilbert space.

It's useful to note that we can use these to build Fourier series that are similar to those in the torus. Most useful is the following fact: \begin{equation*} \|u\|_{H^{1}_{0}(B(0,1))}^{2} = \int_{B(0,1)} \|Du\|^{2} \, dx = \sum_{n = 1}^{\infty} \lambda_{n} \langle u, e_{n} \rangle^{2}, \\ \|f\|_{H^{-1}_{0}(B(0,1))}^{2} = \sum_{n = 1}^{\infty} \lambda_{n}^{-1} [f,e_{n}]^{2}. \end{equation*} Above $\langle \cdot, \cdot \rangle$ is the (standard) $L^{2}(B(0,1))$ inner product and $[\cdot,\cdot]$ is the dual pairing between $H^{1}_{0}(B(0,1))$ and $H^{-1}(B(0,1))$.

The identities above follow from the definition of $H^{1}_{0}(B(0,1))$ and $H^{-1}(B(0,1))$ and the properties of $(e_{n})_{n \in \mathbb{N}}$ mentioned above. Notice that the first identity says that $u$ is in $H^{1}_{0}(B(0,1))$ if and only if its "Fourier coefficients" decay fast enough; $f$ is in $H^{-1}(B(0,1))$ if and only if its "Fourier coefficients" don't grow too fast. The latter will be used in what follows.

Claim 1: If $(u_{n})_{n \in \mathbb{N}}$ converges weakly in $L^{2}(B(0,1))$ to $u$, then $u_{n} \to u$ strongly in $H^{-1}(B(0,1))$.

Proof: Since $u_{n} \rightharpoonup u$, we know that $\|u_{n}\|_{L^{2}(B(0,1))}$ is bounded in $n$ and $\langle u_{n},e_{m} \rangle \to \langle u, e_{m} \rangle$ for each fixed $m \in \mathbb{N}$. Thus, \begin{align*} \limsup_{n \to \infty} \|u_{n} - u\|_{H^{-1}(B(0,1))} &\leq \lim_{n \to \infty} \sum_{j = 1}^{N} \lambda_{j}^{-1} \langle u_{n} - u, e_{j} \rangle + \limsup_{n \to \infty} \sum_{j = N + 1}^{\infty} \lambda_{j}^{-1} \langle u_{n} - u, e_{j} \rangle \\ &\leq \lim_{N \to \infty} \lambda_{N + 1}^{-1} \sup \left\{ 2 \|u_{n}\|^{2}_{L^{2}(B(0,1))} \, \mid \, n \in \mathbb{N} \right\} \\ &= 0. \end{align*}
This proves strong convergence.

Claim 2: Weak convergence in $L^{p}(B(0,1))$ for some $p \in [2,\infty)$ (or weak-$*$ convergence when $p = \infty$) implies strong convergence in $H^{-1}(B(0,1))$.

Proof: Recall that the hypothesis implies weak convergence in $L^{2}(B(0,1))$.