Can a $W^{1,2}$ map from the disk to the circle restrict to a degree one map on the boundary?

The restriction of $f$ to the boundary has degree zero. It is true also in higher dimensions. The proof presented below is based on the proof of density of $C^\infty(M,N)$ in $W^{1,p}(M,N)$, $p\geq \operatorname{dim}M$, due to Schoen and Uhlenbeck [3] (see also Theorem 2.1 in [1]).

Theorem. If $f\in W^{1,n}(B^n,S^{n-1})$ and $f|_{\partial B^n}\in C^0$, then $f|_{\partial B^n}:S^{n-1}\to S^{n-1}$ has degree zero.

Here the restriction to the boundary $f|_{\partial B^n}$ is defined as a trace of a $W^{1,n}$ function.

Proof. Let $B^n=B^n(0,1)$ and let $B^n_{1+\delta}=B^n(0,1+\delta)$ for some small $\delta>0$. First of all, we can extend the mapping $f$ to $\tilde{f}\in W^{1,n}(B^n_{1+\delta},S^{n-1})$ if $\delta>0$ is small enough. Indeed, the Sobolev extension operator $E$ is defined through averages, see [2], so extending $f$ to $B^n_{1+\delta}\setminus B^n$ gives a function $Ef\in W^{1,n}(B^n_{1+\delta},\mathbb{R}^n)$ that is continuous in the annulus $B^n_{1+\delta}\setminus B^n$. If $\delta>0$ is small enough, $|Ef|>1/2$ on $B^n_{1+\delta}\setminus B^n$ (by continuity and the fact that $|f|=1$ on $\partial B^n$) and hence $\tilde{f}=Ef/|Ef|$ in $B^n_{1+\delta}\setminus B^n$ and $\tilde{f}=f$ in $B^n$ belongs to $\tilde{f}\in W^{1,n}(B^n_{1+\delta},S^{n-1})$. If we prove that the degree of $\tilde{f}$ on the boundary of $B^n_{1+\delta}$ is zero, then also degree of $f$ on the boundary of $B^n$ is zero (by homotopy invariace of degree and continuity of $\tilde{f}$ in $B^n_{1+\delta}\setminus B^n$).

The above construction shows that we can assume that $f$ is continuous in a neighborhood of $\partial B^n$ (because $\tilde{f}$ is continuous near the boundary of the ball $B^n_{1+\delta}$ and the argument given below can be applied to $\tilde{f}$ showing that the degree of $\tilde{f}$ is zero on the boundary of the ball $B^n_{1+\delta}$).

The mapping $f$ takes values into $\mathbb{R}^{n}$ since $f:B^n\to S^{n-1}\subset\mathbb{R}^n$. Given $\epsilon>0$ define $r_{\epsilon,x}=\epsilon(1-|x|)$ and $$ f_\epsilon(x)=\frac{1}{|B(x,r_{\epsilon,x})|}\int_{B(x,r_{\epsilon,x})} f(y)\, dy. $$ That is we average $f$ over a ball of radius $\epsilon$ times the distance of $x$ to the boundary of the unit ball $B^n$. The function $f_\epsilon$ is continuous up to the boundary: as $x$ approaches $\partial B^n$, the radius of the ball over which we average tends to zero and hence $f$ is continuous up to the boundary, because $f$ is continuous in an annulus near the boundary, $f_\epsilon|_{\partial B^n}=f|_{\partial B^n}$.

According to the Poincare inequality $$ \left(\frac{1}{|B(x,r_{\epsilon,x})|}\int_{B(x,r_{\epsilon,x})}|f(y)-f_\epsilon(x)|^n\, dy\right)^{1/n} \leq C\left(\int_{B(x,r_{\epsilon,x})}|\nabla f|^n\right)^{1/n}. $$ Since $$ \operatorname{dist}(f_\epsilon(x),S^{n-1})\leq |f(y)-f_\epsilon(x)| \quad \text{for all $y$} $$ we have $$ \operatorname{dist}(f_\epsilon(x),S^{n-1})\leq C\left(\int_{B(x,r_{\epsilon,x})}|\nabla f|^n\right)^{1/n}. $$ The right hand side converges uniformly to $0$ in $x$. That is if $\epsilon$ is small enough, $f_\epsilon(x)\neq 0$ and hence $$ g(x)=\frac{f_\epsilon(x)}{|f_\epsilon(x)|} $$ is a continuous map $g:B\to S^{n-1}$, $g|_{\partial B^n}=f_\epsilon|_{\partial B^n}=f|_{\partial B^n}$. This shows that $\operatorname{deg}(f|_{\partial B^n})=0$.

[1] P. Hajłasz, Sobolev mappings between manifolds and metric spaces. In: Sobolev spaces in mathematics. I, 185–222, Int. Math. Ser. (N. Y.), 8, Springer, New York, 2009.

[2] G. Leoni, A first course in Sobolev spaces. Second edition. Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017.

[3] R. Schoen, K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), 253–268.

I believe I have an answer to my own question: the boundary map indeed must have degree zero, though I would still be curious about other proof methods.

The trick is that we can compute the degree of a map $g\colon\partial D^2\to S^1$ by computing $\int_{\partial D^2}\bar g\,dg$. This integral will be $2\pi i$ times the degree. But, moreover, this expression is well-defined for $g\in W^{1/2,2}(\partial D^2)$ because $\bar g\in W^{1/2,2}(\partial D^2)$ and $dg\in W^{-1/2,2}(\partial D^2)$. In other words, the notion of the degree of a map $\partial D^2\to S^1$ can be extended continuously from $C^\infty(\partial D^2;S^1)$ to $W^{1/2,2}(\partial D^2;S^1)$.

However, that's not the end of the story, because functions in $W^{1,2}(D^2;S^1)$ are limits of functions in $C^\infty(D^2;\mathbb C)$ that converge pointwise almost everywhere to $S^1$, and it is a nontrivial fact to show that they are actually limits of functions in $C^\infty(D^2;S^1)$, whose boundary restrictions indeed have degree zero. (Consider $e^{i\theta}$, which is in $W^{1,p}(D^2;S^1)$ for $p<2$, but to view it as a $W^{1,p}$-limit of smooth functions, we need to cut it off to zero near the origin.) Fortunately, Schoen and Uhlenbeck provide a proof in section 4 of Boundary regularity and the Dirichlet problem for harmonic maps.

In higher dimensions, it appears that this smooth approximation issue has also been resolved. See the introduction of this recent paper, for example. To generalize the above argument to higher dimensions, it remains to express the degree as an integral. Intuitively, we want to integrate the determinant of the Jacobian. For maps $g\colon\partial B^4\to S^3$, for example, the expression is \begin{equation*} \frac1{3!\cdot\operatorname{vol}(\partial B^4)}\int_{\partial B^4}g\wedge dg\wedge dg\wedge dg. \end{equation*} Explaining the notation, we view $g$ as a $\mathbb R^4$-valued zero-form on $\partial B^4$, and, correspondingly, $dg$ as a $\mathbb R^4$-valued one-form on $\partial B^4$. The $\wedge$ operation in this case means taking the usual wedge of forms on $\partial B^4$, along with taking the wedge in $\bigwedge^*\mathbb R^4$. (In particular, the operation is symmetric on $\mathbb R^4$-valued one-forms.) The resulting expression is a three-form on $\partial B^4$ with values in $\bigwedge^4\mathbb R^4\cong\mathbb R$.

Naively, because of borderline and negative regularity issues, this integral is not defined for $g\in W^{3/4,4}(\partial B^4;S^3)$. However, we can extend $g$ to a function in $W^{1,4}(B^4;\mathbb R^4)$ (not necessarily sphere-valued) so that the extension depends continuously on the boundary value. By Stokes' theorem, we see that \begin{equation*} \int_{\partial B^4}g\wedge dg\wedge dg\wedge dg=\int_{B^4}dg\wedge dg\wedge dg\wedge dg. \end{equation*} The right-hand side is indeed well-defined for $g\in W^{1,4}(B^4;\mathbb R^4)$.

See also the introduction of this Brezis and Nirenberg paper.