Why does Boundary $H^2$ regularity fail for trace non-zero functions?

Question

In Evans, Section 6.3, Theorem 4; we know that if $\Omega$ is a bounded region in $\mathbb{R}^n$ with smooth boundary (say), and $u\in H_0^1(\Omega)\cap H^2(\Omega)$ then we have the bound $\|u\|_{H^2}\le C(\|u\|_{L^2}+\|\Delta u\|_{L^2})$.

I've been agonizing over constructing a counterexample in $\Omega\subseteq \mathbb{R}^2$ to illustrate why no bound can hold when $u\in H^2(\Omega)$ alone. My first thought was to consider a sequence of harmonic functions whose gradients diverge in $L^2(\Omega)$, but I couldn't figure it out.

Answer 1

Your idea is good. Consider the sequence $u_n(x,y) = e^{nx}\sin ny$, in any fixed bounded domain $\Omega$. The functions are harmonic. Also, $$|\nabla u_n|^2 = n^2 e^{2nx}\sin^2 ny + n^2 e^{2nx}\cos^2 ny = n^2 e^{2nx}$$ which is substantially larger than $u_n^2 \le e^{2nx}$.

Answer 2

You can deduce this from the general theory, without explicitly constructing counterexamples. Assume that you had an estimate of the form $||u||_{H^2} \leq C(||u||_{L^2} + ||\Delta u||_{L^2})$ for $u \in H^2(\Omega)$. Consider the vector space $V = \{u \in H^2(\Omega) \, | \, \Delta u = 0 \}$ consisting of harmonic functions on $\Omega$ as a normed vector space with the $||\cdot||_{H^2}$ norm. Look at the closed unit ball $B$ of $(V, ||\cdot||_{H^2})$. The embedding $H^2(\Omega) \hookrightarrow L^2(\Omega)$ is compact so any sequence $u_n \in B$ has a subsequence $u_{n_k}$ that is Cauchy in $L^2(\Omega)$. The estimate $||u||_{H^2} \leq C ||u||_{L^2}$ shows that $u_{n_k}$ is also Cauchy in $H^2(\Omega)$ which implies that $B$ is compact. Thus $V$ must be a finite dimensional vector space. However, this is absurd as without imposing any boundary conditions there are infinitely many linearly independent harmonic functions in $H^2(\Omega)$ on any open subset where $n \geq 2$ (at least when $\Omega$ is bounded).

This argument generalizes in the sense that any time you have an estimate of the form $||x||_{X} \leq C(||Dx||_{Y} + ||Kx||_{Z})$ where $X,Y,Z$ are Banach spaces, $D \colon X \rightarrow Y$ is a bounded operator and $K \colon X \rightarrow Z$ is a compact operator it implies that $D$ is a semi-Fredholm operator, meaning it has a closed image and a finite dimensional kernel. In our example, $D = \Delta$ and $K$ is the compact inclusion $H^2(\Omega) \hookrightarrow L^2(\Omega)$. Since such estimates imply that the kernel of $D$ is finite dimensional, the spaces involved should already encode appropriate boundary value conditions for $D$. In our example, $X = H^2(\Omega) \cap H^1_0(\Omega)$ and the $H^1_0(\Omega)$ part handles the boundary value conditions.