On a K-homogeneous metric (Q6615661)

Let \(\gamma=(\gamma_i)_{1\le i\le n}\in\mathbb R^n\) and \(R^n_+=\{x=(x_i)_{1\le i\le n}\in \mathbb R^n \text{ when } \gamma_i=0 \text{ and }x_i\in \mathbb R_+\setminus\{0\}\text{ when }\gamma_i\neq 0\}\). Then, the authors consider the operator \(\Delta_{B_\gamma}\) such that for \(u\) a twice continuously differentiable function in \(\mathbb R^n_+\), \(\displaystyle\Delta_{B_\gamma}u(x)=\sum_{1\le i\le n}\frac{\partial^2u(x)}{\partial x_i^2}+\sum_{1\le i\le n}\frac{\gamma_i}{x_i}\frac{\partial u(x)}{\partial x_i}\). The purpose of the authors is to find a positively symmetric quadratic form metric \(\displaystyle ds^2=\sum_{1\le i,j\le n}g_{ij}dx_idx_j\) such that the standard Laplace operator would coincide with \(\Delta_{B_\gamma}\).\N\NFor the entire collection see [Zbl 1531.35009].