Consider the elliptic eigenvalue problem $$ \begin{cases} \int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\ \qquad \qquad \qquad \quad u&= \ \ \ 0\quad\text{on} \quad \partial B(r), \end{cases} $$ where $B(r)$ is a ball of radius $r$ and $\phi \in W_0^{1,2}(B(r))$ is a standard test function. By the classical theory it is known that the principal eigenfunction $u$ (solving the above equation) is non-negative and minimizes the Rayleigh quotient. Instead of merely non-negativity, I am interested in finding a quantitative lower bound in $\overline B(r)$.

I was able to find out that if the coefficient matrix $A$ is assumed to be uniformly elliptic, $C^1$-function of $x$, the principal eigenfunction behaves like the distant function to the boundary: \begin{equation}\tag 1 u \ge c(1-\sigma)r \quad \text{in} \quad \overline B(\sigma r), \quad \text{for} \quad 0<\sigma \le 1. \end{equation} Obviously, due to the scale invariance of the equation, the constant $c$ must depend on $u$. Now my question is threefold:

Does equation (1) hold if we merely assume the coefficient matrix to be "bounded and measurable"? With this I mean that $A$ is measurable function of $x$ and the following hold:

- $A(x) p \cdot p \ge \lambda|p|^2, \quad x, p \in \mathbb{R}^n, \quad\lambda \in \mathbb{R}_+,$
- $|A(x)p| \le \Lambda |p|, \quad \Lambda \in \mathbb{R}_+$.

If yes, what would be a good reference for the result in this generality? If not, what are the minimal assumptions which are needed for $A$ to satisfy in order to equation (1) to hold?

What is the exact behavior of the constant $c$ in (1) and what would be a good reference for this? In particular is it possible to prove for sufficiently smooth $A$ that $$\tag 2 \frac{1}{|B(r)|}\int_{B(r)} u \, dx \le \frac{C}{(1-\sigma)^\gamma}\inf_{B(\sigma r)} u $$ for some constants $C$ and $\gamma$ which only depend on the a priori information and, in particular, do not depend on $u$ or $r$.