Regularity and singularities of stable critical points and higher critical points in variational problems

The Euler-Lagrange equation of the functional \[ \int [|\nabla v|^2 + F(v)] \] is $2\Delta u = F'(u)$. At large scale, solutions to these equations for suitable functions $F$ resemble variational solutions associated with one or the other of two natural, scale-invariant, so-called singular limits, namely, the Alt-Caffarelli energy functional for free boundaries or the area functional for minimal surfaces. In these lectures, we will describe the relationship between the functionals and characterize the property of stability for each by finding the second variation. We will then pursue the analogy between minimal surfaces and free boundaries, already a powerful device in work in 1980 of Alt and Caffarelli. James Simons's regularity theory for stable minimal surfaces will lead us to a regularity theorem for stable free boundaries in dimensions 3 and 4. The analogy will guide our investigation into what can be said about minimizers in high dimensions and higher critical points in dimensions 2 and 3

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