Suppose that $u\in H^1(\Omega)$ where $\Omega$ is a smooth bounded domain, and $Lu=0$, here $L$ is a uniformly elliptic operator. For example, $$Lu=\sum a_{ij} D_{ij} u$$ and there exists $\lambda>0$ such that $$\lambda^{-1}\lvert \xi\rvert^2 \leq a_{ij}\xi_i\xi_j\leq \lambda\lvert \xi\rvert^2\qquad \forall \xi=(\xi_1,\cdots,\xi_n)\in\mathbb{R}^n.$$
Let $S\subset\partial \Omega$ be smooth (or good enough), then there exists a solution of $w$ such that $$ \begin{cases} Lw=0& \mathrm{in}~\Omega ,\\ w=u& \mathrm{on}~S,\quad w=0~\mathrm{on}~\partial\Omega \backslash S\\ \end{cases}. $$
Question When reading papers about PDE, I often come across that the authors using the above results without citation, but I don't know how to prove it. At least, we may need to show the following admissible set is not empty: $$ \mathcal{A} =\left\{ v\in H^1\left( \Omega \right) :v=u~on~S;~v=0~on~\partial \Omega \backslash S \right\} . $$ Could you please tell me how to prove it, or any references would be very welcome.