Short Communication: A Probabilistic Approachto a Nonlinear Differential Equationon a Riemannian Manifold
Evgenii Borisovich Dynkin · Theory of Probability and Its Applications · 1998
We investigate the minimal solution of the problem $$ \eqalign{ Lu&=u^\ap\q\hbox{in}\ D, \cr u&=f\q\hbox{on}\ O, \cr} $$ where $1<\ap\l 2,\ D$ is an open subset f a Riemannian manifold, O is a regular relatively, open subset of $\po D$, and f is a mapping from $\po D$ to $[0,\iy]$ which is continuous on O and vanishes on $\po D\bs O$. An explicit formula for such a solution is given in terms of the $(L,\ap)$-superdiffusion.