A maximum principle for exterior Dirichlet problems via rearrangements (Q1823380)

The authors study the problem \(-\Delta u=f(x)\) on an exterior domain D \(({\mathbb{R}}^ n\setminus D\) is bounded) with \(u|_{\partial \Omega}=u_ 0\geq 0\), \(u_ 0\) constant, and \(u\to 0\) at infinity. It is assumed, that \[ | f(x)| \leq \delta (| x|)| x|^{-2}\text{ for }| x| \geq R_ 0\text{ with }\int^{\infty}_{R_ 0}\delta (t)t^{-1} dt<\infty,\quad n\geq 3. \] Futher the authors use on p. 117, that \(f\in L_ 1(D)\), which does not follow from that assumption. Using rearrangement-techniques they show that \[ \max_{D}u\leq u_ 0+c_ n\int^{a}_{0}s^{-2(n- 1)/n}(\int^{A+s}_{A}\tilde f(t)dt)ds, \] where \(a=\text{meas}\{x|\) \(u(x)>u_ 0\}\), \(A=\text{meas}({\mathbb{R}}^ n\setminus D)\) and \(\tilde f(t)\) is the inverse function to \(t(\tilde f):=\text{meas}(\{x|\) \(f(x)\geq \tilde f\}\cup({\mathbb{R}}^ n\setminus D))\) and they give a condition in terms of the flux \(\int_{\partial D}(\partial u/\partial n)\) do ensuring that \(\max_{D}u=u_ 0\).