C ∞ regularity in semilinear free boundary problems

Area of Science:

• Mathematical analysis
• Partial differential equations
• Geometric measure theory

Background:

• The Alt-Phillips problem involves a semilinear elliptic equation with a power nonlinearity.
• Understanding the regularity of solutions and free boundaries is crucial for analyzing such problems.
• Previous studies have explored various aspects of this problem, but higher regularity remains an active research area.

Purpose of the Study:

• To establish higher regularity for solutions and free boundaries in the Alt-Phillips problem for $\gamma \in (0, 1)$.
• To demonstrate that if free boundaries are $C^{1,\alpha}$, they are in fact $C^{\infty}$.
• To investigate the regularity of specific solution components, namely $u/d^{2/(2-\gamma)}$ and $u^{(2-\gamma)/2}$.

Main Methods:

• Development of fine regularity estimates for solutions to linear equations with boundary-singular Hardy potentials.
• Analysis of the equation $-\Delta v = \kappa v/d^2$ in $\Omega$, where $d$ is the distance to the boundary and $\kappa \leq 1/4$.
• Inclusion of the critical case $\kappa = 1/4$, corresponding to $\gamma = 2/3$.

Main Results:

• The study proves that free boundaries, once $C^{1,\alpha}$, become $C^{\infty}$.
• It is shown that the quantities $u/d^{2/(2-\gamma)}$ and $u^{(2-\gamma)/2}$ are $C^{\infty}$.
• The analysis successfully handles the critical Hardy potential case.

Conclusions:

• The findings contribute to a deeper understanding of the regularity properties of solutions in the Alt-Phillips problem.
• The results have implications for the analysis of free boundary problems in mathematical physics and engineering.
• This work advances the regularity theory for elliptic equations with singular potentials.