consider the 'Allen-Cahn Equations' given by $ \epsilon^2\Delta_g u + u - u^3=0 $ It is known that if $\Gamma$ is a non degenerate minimal surface then there exists a sequence of solutions $u_\epsilon$ to above equation converging to $1_{M_+} - 1_{M_-} $ in compact subsets of M where $ M_{+} \bigcup M_{-} = M \backslash \Gamma $ and furthermore the energy functional associated to $u_\epsilon$ converges to Area of that minimal surface...
what I am interested is to know given a riemannian manifold (let's assume for simplicity that the dimension is 3) with boundary, and given a given closed curve on the boundary. can we 'cook up' specific dirichlet data on the boundary $ h_{\epsilon}$ such that the energy of solutions to the allen cahn equation with that boundary data converge to the area of the corresponding minimal surface enclosing that curve..
Thanks in advance and I hope this is not too vague
