On the Large $\Lambda$ Asymptotics of the One Phase Bernoulli Free Boundary Problem

Abstract

In this paper, we investigate the asymptotic behavior of minimizers $u_{\Lambda }$ of the Bernoulli functional ${\int }_D|\nabla v{|}^{2}dx+\Lambda |{\Omega }_v|$ for large $\Lambda$ with a boundary datum $g\ge 0$. In particular, in the case $g>0$, we show graphicality of the free boundary over $\partial D$ and obtain a second order asymptotic expansion in $\Lambda$ of the graph.