Barrier methods for critical exponent problems in geometric analysis and mathematical physics

We consider the design and analysis of numerical methods for approximating positive solutions to nonlinear geometric elliptic partial differential equations containing critical exponents. This class of problems includes the Yamabe problem and the Einstein constraint equations, which simultaneously contain several challenging features: high spatial dimension n >= 3, varying (potentially non-smooth) coefficients, critical (even super-critical) nonlinearity, non-monotone nonlinearity (arising from a non-convex energy), and spatial domains that are typically Riemannian manifolds rather than simply open sets in Rn. These problems may exhibit multiple solutions, although only positive solutions typically have meaning. This creates additional complexities in both the theory and numerical treatment of such problems, as this feature introduces both non-uniqueness as well as the need to incorporate an inequality constraint into the formulation. In this work, we consider numerical methods based on Galerkin-type discretization, covering any standard bases construction (finite element, spectral, or wavelet), and the combination of a barrier method for nonconvex optimization and global inexact Newton-type methods for dealing with nonconvexity and the presence of inequality constraints. We first give an overview of barrier methods in non-convex optimization, and then develop and analyze both a primal barrier energy method for this class of problems. We then consider a sequence of numerical experiments using this type of barrier method, based on a particular Galerkin method, namely the piecewise linear finite element method, leverage the FETK modeling package. We illustrate the behavior of the primal barrier energy method for several examples, including the Yamabe problem and the Hamiltonian constraint.

💡 Research Summary

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This paper addresses the challenging task of computing positive solutions to nonlinear geometric elliptic partial differential equations that involve critical (or super‑critical) exponents. The two canonical examples motivating the study are the Yamabe problem and the Hamiltonian (Lichnerowicz) constraint arising in the constant‑mean‑curvature (CMC) formulation of the Einstein constraint equations. Both problems share a collection of difficult features: they are posed on three‑dimensional (or higher) Riemannian manifolds, the coefficients may be highly irregular, the nonlinearities contain critical Sobolev exponents, the associated energy functionals are generally non‑convex, and only positive solutions have physical or geometric relevance, which imposes an inequality constraint on the unknown. Moreover, the equations may admit multiple solutions, so a numerical method must be able to locate a physically admissible branch without being trapped by spurious solutions.

The authors propose a unified computational framework that combines a Galerkin discretization with a primal barrier interior‑point strategy and a globally convergent inexact Newton method. The key ideas are as follows. First, the continuous problem is written in weak form and interpreted as the stationarity condition of an energy functional (J(u)). When (J) is convex (e.g., non‑negative scalar curvature and Robin coefficient), minimizing (J) over the positive cone yields the desired solution. In the more typical non‑convex case, the solution is a saddle point or a local maximizer, and a pure minimization approach fails.

To enforce the positivity constraint (u>0) while still being able to handle non‑convexity, the authors introduce a logarithmic barrier term and define a family of barrier functionals
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