Parabolic Boundary Harnack Inequalities with Right-Hand Side

Area of Science:

• Partial Differential Equations
• Harmonic Analysis
• Geometric Measure Theory

Background:

• The parabolic boundary Harnack inequality is crucial for understanding solutions to parabolic partial differential equations.
• Previous results were limited to homogeneous equations (zero right-hand side).
• Flat Lipschitz domains present unique challenges for analytical techniques.

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 the inequality to a broader class of parabolic equations.
• To provide new methods for analyzing free boundary problems.

Main Methods:

• Blow-up techniques adapted for parabolic equations.
• Analysis of solutions to non-divergence form operators with bounded measurable coefficients.
• Investigating the optimal regularity of the quotient of solutions for the heat equation.

Main Results:

• The parabolic boundary Harnack inequality is proven for non-zero right-hand sides in flat Lipschitz domains.
• The method accommodates non-divergence form operators and a range of right-hand side functions.
• Optimal regularity is established for the quotient in the heat equation case.

Conclusions:

• This work generalizes the parabolic boundary Harnack inequality, significantly broadening its applicability.
• The findings offer a novel approach to proving the smoothness of free boundaries in parabolic obstacle and Signorini problems.
• The results contribute to a deeper understanding of parabolic PDEs in irregular domains.