Project Grant 2554417

This $200,000 project grant, awarded by the National Science Foundation (NSF) Division of Mathematical Sciences under the Mathematical and Physical Sciences program (CFDA 47.049), funds fundamental research on nonminimizing and min-max solutions to free boundary problems from September 1, 2026, through August 31, 2029. Free boundary problems model systems where an unknown interface or shape evolves according to governing equations, with direct applications to fluid dynamics, petroleum engineering, phase transitions, and combustion phenomena. The research will develop mathematical approaches and analytical tools to study moving interfaces and non-minimizing steady-state solutions, extending beyond the current state-of-the-art methods that primarily address energy-minimizing solutions in steady-state configurations.

The project deliverables include compactness theorems for critical points of Bernoulli-type free boundaries with applications to gravity water waves and min-max constructions; general solutions to free boundary problems arising from semilinear elliptic equations; and analysis of parabolic free boundaries for which traditional minimality concepts do not apply. Through rigorous approximation schemes, perturbation stability analysis, and development of qualitative solution properties, the research will advance mathematical understanding with practical implications for industrial applications. The award also supports graduate and undergraduate student training in applied mathematics with significant real-world relevance. The work is conducted at Rutgers, The State University in Piscataway, New Jersey, a major research institution that serves as the prime recipient for approximately 4,900 federal awards across 252 agencies.