Optimal Regularity in Free Boundary Problems

Free boundary problems arise in diverse areas of science and engineering wherever an interface or phase boundary is not known a priori but is part of the solution. The classical obstacle problem, tumour growth models, melting and solidification phenomena, and fluid–structure interactions all lead to questions about the smoothness—or regularity—of both the solution and its free boundary. Optimal regularity refers to the highest level of differentiability and Hölder continuity that one can generally expect under minimal hypotheses. Achieving and proving optimal regularity is central to qualitative understanding, numerical approximation and stability analysis of free boundary problems. Over the past decade, breakthroughs have been made in establishing C¹,α and higher regularity at free boundary points, developing boundary Harnack principles that allow for non-zero right-hand sides, and extending methods to degenerate and nonlocal operators. These advances not only deepen theoretical insight but also underpin practical algorithms in material science, biomedical modelling and fluid mechanics.

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Recent work has extended boundary Harnack inequalities to parabolic settings with non-zero forcing terms, proving that solutions to heat-type obstacle problems admit optimal C¹,ε regularity of the quotient and that flat Lipschitz free boundaries become C¹,α. In evolutionary p-Laplace obstacle problems, new intrinsic geometries yield optimal C¹,α regularity at free boundary points in the degenerate regime p > 2, up to the critical exponent and across full cylinders without assumptions on time derivatives of the obstacle. In tumour growth models with incompressible cell density, comparison principles and L¹-contraction estimates establish C¹,α boundary regularity of evolving tumour patches in the absence of nutrient diffusion, providing a sharp contrast to models with diffusive or decay terms.

Optimal Regularity in Free Boundary Problems publication trend

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Technical terms

Free boundary problem: A differential equation problem in which part of the domain boundary is unknown and must be determined as part of the solution.

Optimal regularity: The maximal smoothness (differentiability and Hölder continuity) attainable by solutions and their free boundaries under given structural conditions.

Obstacle problem: A prototypical free boundary problem where a membrane or elastic surface is constrained to lie above a given obstacle.

Parabolic boundary Harnack inequality: A fundamental comparison principle for positive solutions of parabolic equations near boundaries, extended to include non-zero right-hand sides and used to infer regularity of the free boundary.

p-Laplace operator: A nonlinear generalisation of the Laplacian, defined by div(|∇u|ⁿ⁻²∇u) for p > 1, governing degenerate or singular diffusion phenomena.

References

1. Parabolic Boundary Harnack Inequalities with Right-Hand Side. Archive for Rational Mechanics and Analysis (2024).
2. Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems. Journal de Mathématiques Pures et Appliquées (2024).