Two-Phase Free Boundary Problems: From Existence to Smoothness

Abstract

We describe the theory we developed in recent times concerning two-phasefree boundary problems governed by elliptic operators with forcing terms.Our results range from existence of viscosity solutions to smoothness ofboth solutions and free boundaries. We also discuss some open questions,possible object of future investigation.

Keywords

• two-phase free boundary problem
• free boundary regularity
• perron solutions, elliptic systems
• 35r65
• 35j57
• 35b65