Optimal Regularity and the Free Boundaryin the Parabolic Signorini Problem

Abstract

We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.2000 Mathematics Subject Classification. Primary 35R35, 35K85.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

“…Indeed, it is not difficult to verify that the flat parabolic Signorini problem (see the book by Duvaut and Lions [14, Section 2.2.1], also [13]) is equivalent to the obstacle problem for (∂ t − ∆) 1/2 . Similarly, the diffusion model for biological invasions introduced in [4] is equivalent to a local-nonlocal system coupling a classical heat equation with an equation for (∂ t − ∆) 1/2 .…”

“…(4.4)], [4], [9], [13], [14] and [23]), our novel ideas and results open the way to consider anisotropic equations of the form (∂ t − L) s u = f on Riemannian manifolds, where L is an elliptic operator with bounded measurable coefficients that may depend on t and x. Observe that it is not clear at all how to define these nonlocal parabolic operators or how to obtain pointwise expressions for them.…”

The Two-Phase Parabolic Signorini Problem

“…For the parabolic Signorini problem, the regularity of solutions was studied in [14] and [2] where a method for obtaining Hölder continuity of the gradient is given. Optimal regularity of solutions to the parabolic Signorini problem was recently proven in [7]. Rather than attempt to adapt and utilize the arguments in [14], [2], and [7] to a two-phase problem, we obtain the same regularity results quickly by first proving the separation of the phases and reducing the regularity problem to that of a one-phase problem.…”

“…The first consequence is that the study of the local properties of the free boundaries of two-phase boundary temperature control problem (1.1) is completely reduced to studying the free boundary in the parabolic Signorini problem (1.3)-(1.4). For the latter problem, when the free boundary is on a flat portion of ∂Ω × (0, T ), the optimal regularity of the solution and the structure of the free boundary are studied in [7]. Therefore, as a corollary to Theorem 1.1, when Γ ± are contained on a flat portion of ∂Ω × (0, T ), we obtain results for the structure of the free boundary.…”

“…If the distance from (x, t) to a free boundary is less than or equal to d/4, then either u ∓ λ ± x + n is a solution to (1.3)-(1.4) in Q + d/4 (x, t) and we utilize the optimal regularity result in [7] to conclude (5.4) for u in Q Remark 5.7. The above regularity result is optimal since Re(x n−1 + ix n ) 3/2 + λ + x n is a time-independent solution to (1.1) in Q r (0, 0) for r sufficiently small and λ + sufficiently large.…”