We first prove that the largest subsolution $u$ of the problem exists and is star-shaped with respect to all points in $D$. This in turn implies that the free boundary $\partial \{u >0\}$ is a locally Lipschitz graph. Classical results of free boundary are used to show that the free boundary is $C^{1,\alpha}$. From here we prove uniqueness by a scaling argument , a comparison principle and using the smoothness of the free boundary. We then use quasi-concave envelope $u^*$ of $u$ which is a subsolution of the problem but larger than our solution. This contradicts the fact that our solution is the largest one and we conclude that $u^*=u$. ...

In [1], the authors considered the periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. They proved that the homogenized boundary data belongs to $W^{1,p}$ for any $1< p< \infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $\alpha$ for any $\alpha \in (0, 1)$. ...

Here we will show there exists a map $P_\Omega: W^{1,p}_0(D) \rightarrow W^{1,p}_0(\Omega)$ that satisfies the desired properties of the question and 'behaves' like a projector in the sense that $P^2_\Omega = P_\Omega$ and $P = I$ on $W^{1,p}_0(\Omega)$. Yet $P_\Omega$ is only a linear map if $p=2$.

This problem was solved in 2015 by Dipierro snd Karakhanyan. The paper is published in Advances of Mathematics, 2018, vol. 328, 40-81. The authors used a new dichotomy method, boundary Harnack inequalities, and stratification argument. Their proof applies to more general nonlinear problems. ...

For each $\varepsilon>0 $ let $ u_\varepsilon $ be the solution to the following Dirichlet problem $$ -\nabla \cdot A_\varepsilon(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x,x/ \varepsilon) \text{ on } \partial D, $$ where $D \subset \mathbb{R}^d $ $ (d\geq 2) $ is a bounded domain, $g(x, \cdot)$ is $ \mathbb{Z}^d $-periodic for any $ x\in \partial D$ and the operator is uniformly elliptic. We study the problem under strict convexity of the domain $D$ and $C^\infty$- smoothness of the data involved in the problem. ...

We consider homogenization of Dirichlet problem for divergence type elliptic operator when the operator is fixed, and the boundary data is oscillating, which is a particular case of the setting introduced in [2]. ...

In a bounded domain $D\subset \mathbb{R}^d$ ($d\geq 2$) consider homogenization of Dirichlet problem of the elliptic system \begin{equation} \begin{cases} -\nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u(x) = 0, & x \in D, \\ u(x) = g \left(x , \frac{x}{\varepsilon} \right), & x \in \partial D \end{cases} \tag{1} \end{equation} where $\varepsilon > 0$ is a small parameter and $A= A^{\alpha \beta } (x) \in M_N(\mathbb{R})$, $x\in \mathbb{R}^d$ is a family of functions indexed by $1\leq \alpha, \beta \leq d$ with values in the set of matrices $M_N( \mathbb{R})$. ...