2024 Theses Doctoral

On some semi-linear equations related to phase transitions: Rigidity of global solutions and regularity of free boundaries

Zhang, Chilin

In this thesis, we study minimizers of the energy functional 𝐽 (𝑢,Ω) = ∫_Ω |∇𝑢|²/2 + 𝑊(𝑢) 𝑑𝑥 for two different potentials 𝑊(𝑢).

In the first part we consider the Allen-Cahn energy, where 𝑊(𝑢) = (1 − 𝑢²)² is a doublewell potential which is relevant in the theory of phase transitions and minimal interfaces. We investigate the rigidity properties of global minimizers in low dimensions. In particular we extend a result of Savin on the De Giorgi’s conjecture to include minimizers that are not necessarily bounded, and that can have subquadratic growth at infinity.

In the second part we consider potentials of the type 𝑊(𝑢) = 𝑢⁺ which appear in obstacletype free boundary problems. We establish higher order estimates and the analyticity of the regular part of the free boundary. Our method relies on developing higher order boundary Harnack estimates iteratively and deducing them from Schauder estimates for certain elliptic equations with degenerate weights.

Finally we consider similar regularity questions of the free boundary in the Signorini problem which also known as the thin obstacle problem. We develop 𝐶²^𝛼 estimates of the free boundary under sharp assumptions on the coefficients and the data.