Some boundary Harnack principles with uniform constants (Q2172612)

The authors investigate certain uniform boundary Harnack principles. The first result for dimension \(2\) arises as follows. Let \(K\subset B(0,1)\cap \overline{\mathbb{H}}_-\), where \(B(0,1)\) is the unit Euclidean ball with center at \(0\) and \(\overline{\mathbb{H}}_-=\{(x_1,x_2)\in \mathbb{R}^2:x_1\leq 0\}\). Set \(D=B(0,1)\setminus K\). Let \(U_0\) be the connected component of \(B(0,1/16)\cap D\) which contains \((1/32, 0)\). Then the authors obtain the uniform boundary Harnack principle for this case: There is a positive constant \(C_0\), independent of \(K\), such that if \(u\) and \(v\) are positive harmonic functions on \(D\), continuous on \(\overline{D}\), and vanish on \(K\), then \[ \frac{u(x)/v(x)}{u(y)/v(y)}<C_0,\quad x,y\in U_0. \] This estimate relies on the fact that \(K\) is connected, and does not generalize for \(d\geq 3\). It cannot be extended to \(x,y\in B(0,1/16)\cap D\). The paper under review also studies another uniform boundary Harnack principle in higher dimensions.