Green's functions for some highly degenerate elliptic operators
Let \(x\in\mathbb{R}^n\), \(t\in \mathbb{R}^m\), \(n,m,k\in \mathbb{N}\) and consider \(L:= \Delta_x+ b^{2k-2}\Delta_t\), \(b(x,t):= |x|\). Looking for an explicit Green's function \(K\) for \(L\), the authors obtain \(K(x,t; y,s)= R^{-(n+ mk-2)/2k} F(p,\rho)\) where \(R:= |x|^{2k}+|y|^{2k}+ k^2|t-s|^2\), \(\rho:= |x|^k|y|^k R^{-1}\), \(p:= \langle x,y\rangle R^{-1/k}\) and \(F:= F_{nmk}\) is a function given explicitly in the form of an algebraic expression for \(m\) even resp. of an definite integral for \(m\) odd.
scientific article; zbMATH DE number 1319967