Quasi-linear PDEs and low-dimensional sets
In this interesting paper, the authors establish refined boundary Harnack estimates for non-negative solutions to operators of \(p\)-Laplacian type \(\text{div}\,A(x,\nabla u)\) with \(A: \mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n\) satisfying \[ \begin{aligned} &\alpha^{-1}|\eta|^{p-2}|\xi|^2\leq \sum_{i,j=1}^n \dfrac{\partial A_i}{\partial x_j}(x,\eta)\xi_i\xi_j\quad \text{and}\quad\Big(\sum_{i,j=1}^n \Big|\dfrac{\partial A_i}{\partial x_j}(x,\eta)\Big|^2\Big |^{1/2}\leq \alpha |\eta|^{p-2}, \\& \big|A(x,\eta)-A(y,\eta)\big|\leq \beta |x-y|^\gamma|\eta|^{p-1} ,\\ &A(x,\eta)=|\eta|^{p-1}a(x,\eta/|\eta|),\end{aligned} \] with suitable constants \(\alpha,\beta, \gamma\). In difference to the traditional setting of boundary value problems in a bounded domain of \(\mathbb{R}^n\) having a boundary with Hausdorff dimension in the range \([n-1,n),\) the key novelty of the paper is that the solutions considered vanish only on a low-dimensional set \(\Sigma\subset\mathbb{R}^n,\) and the quantitative and scale-invariant estimates are derived in the context of low-dimensional Reifenberg flat sets. Results regarding degenerate elliptic operators and the Martin boundary problem in these settings are also obtained.
scientific article; zbMATH DE number 6892858