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One can build a distributional subsolution which is a Lipschitz extension of $u$ by making an extension with a positive jump in radial derivative across the boundary. Say $u$ is harmonic on $B_1$ and …

In general the answer is "no." Let $\Omega = (-\pi/2,\,\pi/2) \subset \mathbb{R}$, and for $\varphi \in C^{\infty}_0(\Omega)$ take $$u_1 = \cos(x), \quad u_2 = \cos(x) + \epsilon \varphi$$ $$a_1 = 1, …

Continuity of $d$ does not guarantee that $\nabla u$ is Holder. In fact, the regularity of $d$ is the best regularity one can expect for $\nabla u$. To see this, consider the one-dimensional divergen …

Boundedness of $f$ isn't enough to get extra regularity for $\phi_1$. Take for example $\phi = r^{\gamma}\sin(\theta)$, which solves an equation of the desired form with $f = (\gamma^2-1)\sin(\theta)$ …

The estimate $(2)$ is false even in a half-plane. Indeed, let $w$ be any solution to the heat equation on $\mathbb{R}^2 \times [0,\,\infty)$ that is even in $y$ (so $w$ solves the Neumann problem for …