Laplace-Beltrami operator and exact solutions for branes

Proposed is a new approach to finding exact solutions of nonlinear $p$-brane equations in $D$-dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints $\Delta^{(p)}\vec{x}=0$ and $\Delta^{(p)}\vec{x}=-\Lambda(t,\sigma^r)\vec{x}$ give two sets of exact solutions.