Let $M$ be a compact riemannian manifold with boundary. If we consider harmonic functions in this manifold it may be imposible to build a sequence of harmonic functions which will converge (in some sense) to a localized function near a minimal surface, mainly due to the fact that harmonic functions converge to a harmonic function and also mean value property. (I hope this is all not too vague)

My question is now on the possibility of perhaps building a non linear purturbation on the laplacian so as to gain the advantage of having 'localized' solutions near a minimal surface...

To make this perhaps a little less vague, consider for instance 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.

As you may see the problem however is that even though this is an example of a non linear pde it depends on a parameter $\epsilon$ which I want to avoid! I thank you for your response in advance!



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