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

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