Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions

Question

I was looking at Theorem 12.4 of Gilbarg and Trudinger's Elliptic partial differential equations of second order (MR1814364, Zbl 1042.35002):

Theorem 12.4. Let $u$ be a bounded $C^2(\Omega)$ solution of $$ L u=a u_{x x}+2 b u_{x y}+c u_{y y}=f $$ where $L$ is uniformly elliptic, satisfying $$\lambda\left(\xi^2+\eta^2\right) \leqslant a \xi^2+2 b \xi \eta+c \eta^2 \leqslant \Lambda\left(\xi^2+\eta^2\right) \quad \quad \forall(\xi, \eta) \in \mathbb{R}^2 ;$$ where $\lambda$ and $\Lambda$ denote the eigenvalues of the coefficient matrix and $$\frac{\Lambda}{\lambda} \leqslant \gamma $$ in a domain $\Omega$ of $\mathbb{R}^2$. Then for some $\alpha=\alpha(\gamma)>0$, we have

$$ [u]_{1, \alpha}^* \leqslant C\left(|u|_0+|f/\lambda|_0^{(2)}\right), \quad C=C(\gamma).\label{1}\tag{12.22} $$

The authors remark at the end of this theorem:

The significant feature of this result is that the estimate \eqref{1} [the gradient holder bound] depends only on bounds on the coefficients and not on any regularity properties. This is in contrast with the Schauder estimates (Theorem 6.2) which depend as well on the Hölder constants of the coefficients. The Hölder estimates of Chapter 8 for divergence from equations in $n$ variables (Theorem 8.24) are also independent of the regularity properties of the coefficients, but those estimates concern the solution itself and not its derivatives. The validity of the analog of Theorem 12.4 for $n>2$ remains in doubt.

I was wondering if now there are known examples/counter-examples to this estimate in higher dimensions say $\Omega \subset \mathbb{R}^3?$

Answer 1

Yes, Safonov showed here that, for any $\alpha \in (0,\,1)$, there is a function $u$ on $\mathbb{R}^3$ that is homogeneous of degree $\alpha$ (in particular has unbounded gradient) and solves a linear uniformly elliptic PDE of the form $$a_{ij}(x)u_{ij} = 0.$$ There are simpler examples in higher dimensions, e.g. $$u(x) = \frac{1}{|x|}(x_1^2+x_2^2 - x_3^2 - x_4^2)$$ solves such a PDE in $\mathbb{R}^4$ and is Lipschitz but not $C^1$. This is in fact a component of the famous Lawson-Osserman example of a singular solution to the minimal surface system (here). In addition, there are no $1$-homogeneous solutions to such PDEs in $\mathbb{R}^3$. This is an old result of Alexandrov. A modern proof of Han-Nadirashvili-Yuan is here.