The role of minimal surfaces in the study of the Allen-Cahn equation
Frank Pacard · Contemporary mathematics - American Mathematical Society · 2012
In these lectures, we review some recent results on the existence of solutions of the Allen-Cahn equation ε 2 Δ g u + u − u 3 = 0 \varepsilon ^2 \, \Delta _g u + u - u^3 = 0 defined in a given Riemannian manifold ( M , g ) (M,g) . In the case where the ambient manifold is compact, we give a simple complete proof of the existence of solutions whose zero set is close to a given minimal hypersurface.