# Is the Dirichlet boundary condition continuous?

First, let's fix some notation:

Let $$\mathbb D^n \subseteq \mathbb{R}^n$$ be the closed $$n$$-dimensional unit ball. Given a real-valued function $$f \in C^{\infty}(\mathbb D^n)$$, we denote by $$\omega_f$$ the unique harmonic function satisfying $$\omega_f|_{\partial \mathbb D^n}=f|_{\partial \mathbb D^n}$$.

Now, suppose that $$f_k \in C^{\infty}(\mathbb D^n)$$ converge in $$W^{1,2}(\mathbb D^n)$$ to $$f$$, and let $$\omega_{f_k}$$ be the associated sequence of "harmonic projections". Is it true that $$d\omega_{f_k} \to d\omega_f$$ in $$L^2$$ or $$L^1$$?

The trace theorem implies that if $$f_k \to f$$ in $$W^{1,2}(\mathbb D^n)$$, then $$f_k|_{\partial \mathbb D^n}\to f|_{\partial \mathbb D^n}$$ in $$L^2(\partial \mathbb D^n)$$, so $$\omega_k|_{\partial \mathbb D^n}\to \omega|_{\partial \mathbb D^n}$$ in $$L^2(\partial \mathbb D^n)$$.

• what does $d\omega_{f_k}$ mean? (and $L^2$ of what?) – mathworker21 Sep 12 '19 at 13:43
• This is the differential (or the gradient if you prefer) of the function $\omega_{f_k}$. Perhaps I should write $d(\omega_{f_k})$ to make it clear. If you think in terms of gradients, then $d(\omega_{f_k}) \in L^2(\mathbb D^n,\mathbb R^n)$. Equivalently, you can think of each component of the gradient as an element in $L^2(\mathbb D^n)$. – Asaf Shachar Sep 12 '19 at 13:45

If I'm not mistaken, the answer is yes: $$d\omega_{f_k}$$ converges to $$d\omega_f$$ in $$L^2$$.
Proof: W.l.o.g. we can assume that $$f=0$$ (everything here is linear). $$f_k \to 0$$ in $$W^{1,2}(\Omega)$$ implies that $$f_k|_{\partial \Omega} \to 0$$ in $$W^{1/2,2}(\partial \Omega)$$ (See here for example https://en.wikipedia.org/wiki/Sobolev_space#Traces, or the theorem I quote below). Moreover, as a map $$W^{1,2}(\Omega) \to W^{1/2,2}(\partial \Omega)$$, the trace operator has a continuous inverse, that is, for every $$v\in W^{1/2,2}(\partial \Omega)$$ there exists $$u\in W^{1,2}(\Omega)$$ with $$u|_{\partial \Omega} = v$$ such that $$\|u\|_{W^{1,2}(\Omega)} \lesssim \|v\|_{W^{1/2,2}(\partial \Omega)}$$. See Leoni's "A first course on Sobolev spaces", second edition, Theorem 18.40.
Back to our case: the above theorem implies the existence of $$g_k \in W^{1,2}$$ with $$g_k|_{\partial \Omega} = f_k$$ such that $$\|g_k\|_{W^{1,2}(\Omega)} \lesssim \|f_k\|_{W^{1/2,2}}(\partial\Omega)$$. In particular, $$\|dg_k\|_{L^2(\Omega)} \lesssim \|f_k\|_{W^{1/2,2}}(\partial\Omega)$$. But $$\|d\omega_{f_k}\|_{L^2(\Omega)}\le \|dg_k\|_{L^2(\Omega)}$$ since the harmonic map is the minimizer of the Dirichlet energy with these boundary conditions. Therefore $$f_k|_{\partial \Omega} \to 0$$ in $$W^{1/2,2}(\partial \Omega)$$ implies $$d\omega_{f_k} \to 0$$ in $$L^2$$.