Consider the Sobolev space $W^1$ that is the closure of $\mathcal{C}^{\infty}(\bar{D_1})$ with respect to the norm $$|\phi|_1^2=|\phi|^2_{L^2(\bar{D_1)}}+|\nabla \phi|^2_{L^2(\bar{D_1)}}.$$ I have proven that one can continuously extend the map $$e:\phi\mapsto\phi|_{\partial D_1}$$ to a map $$E:W^1\rightarrow L^2(\partial D_1).$$ My question is this: are there square integrable functions on the boundary of $D_1$ for which there is no $\phi\in W^1(D_1)$ such that $f=E(\phi)$? Thanks in advance.
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Lots of them, even some continuous ones. The trace space of $H^{1}(D)$ is $H^{1/2}(\partial D)$, a fractional Sobolev space. It has two neat characterizations: the space of Fourier series $\sum c_n \exp(int)$ such that $$\sum_{n\in\mathbb Z} |n||c_n|^2<\infty\tag1$$ and, alternatively, the space of functions $f:\partial D\to \mathbb C$ such that $$E(f):=\iint_{\partial D\times \partial D} \left|\frac{f(z)-f(w)}{z-w}\right|^2<\infty\tag2$$
In fact, (1) and (2) are the same quantity up to some constant factor. In the form (2) it is known as the
Douglas integral, named for Jesse Douglas (one of two first Fields medal recipients). The form (1) is more
convenient to see the relation with the Sobolev space on the disk. The harmonic extension of
$\sum c_n \exp(int)$ is $\phi(re^{it})=\sum c_n r^{|n|} \exp(int)$, and a short calculation shows that the
integral of $|\nabla \phi|^2$ is (1), again up to some constant factor. Recalling that harmonic functions
minimize the Dirichlet energy for given boundary values, we conclude that a function for which (1) is infinite does not have an extension of finite energy to the disk.
The form (2) is more convenient to check whether a given function is in the trace space. For example, let $f$ be $1$ on the upper half-circle and $0$ on the bottom half-circle; then (2) is infinite. Alternatively, use (1) and the fact that $|c_n|\approx 1/|n|$ for this function.