Parabolic Boundary Harnack Inequalities with Right-Hand Side.

Area of Science:

• Partial Differential Equations
• Geometric Analysis
• Potential Theory

Background:

• The parabolic boundary Harnack inequality is crucial for studying solutions to parabolic equations.
• Previous results were limited to homogeneous equations (zero right-hand side).
• Flat Lipschitz domains are a key class of domains in analysis.

Purpose of the Study:

• To establish the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains with a non-zero right-hand side.
• To extend the applicability of Harnack inequalities to a broader class of equations and domains.
• To provide new tools for analyzing free boundary problems.

Main Methods:

• Utilizing blow-up techniques.
• Analyzing solutions to non-divergence form parabolic equations with bounded measurable coefficients.
• Investigating the regularity of quotients of solutions.

Main Results:

• The parabolic boundary Harnack inequality is proven for non-zero right-hand sides in parabolic flat Lipschitz domains.
• Optimal regularity is shown for the quotient of solutions in the heat equation case.
• A new method for proving flatness of free boundaries in parabolic obstacle and Signorini problems is developed.

Conclusions:

• The findings extend the scope of the parabolic boundary Harnack inequality.
• The results offer a novel approach to studying parabolic free boundary problems.
• This work contributes to a deeper understanding of parabolic partial differential equations in complex domains.