Eventual regularity for the parabolic minimal surface equation

We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a (strictly positive) finite time.

Problem (1.1) corresponds to the L 2 -gradient flow [9] of the convex lower semicontinuous functional F : L 2 (Ω) → [0, +∞], defined as where, for u ∈ BV (Ω), we write its distributional derivative Du as Here, D s u is the singular part of the measure Du, with respect to the Lebesgue measure (see [20]), and |D s u| stands for its total variation. Equation (1.1) arises in several models of physical systems describing, for instance, the motion of capillary surfaces and the motion of grain boundaries in annealing metals (see [8]). For such reasons, this evolution problem has been already considered in the mathematical literature. In particular, Lichnewski and Temam [21] showed existence of generalized solutions, while Gerhardt [19] and Ecker [15] proved esimates on u, u t , |Du| similar to the ones we present in this paper (see Lemma 3.1).
We point out that equation (1.1) should not be confused with the mean curvature flow for graphs, which has been deeply studied in [16,17], and reads as ∂u ∂t = 1 + |∇u| 2 div ∇u 1 + |∇u| 2 .
Let us state our main result.
Theorem 1.1. Let u 0 ∈ L 2 (Ω) and let u be the unique solution of (1.1). Suppose that one of the two following cases hold: 2) N > 1, and the generalized graph of u 0 1 , is a compact hypersurface of class C 1,1 , meeting orthogonally (∂Ω)× R. 1 Roughly, the graph of u0 with the addition of the "vertical" parts.
Then there exists T > 0 such that Namely, u(t) is analytic in Ω, for any t ∈ (T, +∞). Moreover, u(t) converges to the mean value u 0 : The assumption on graph(u 0 ) when N > 1 is technical, but we are presently not able to remove it. Notice that from this condition it follows that and u 0 is of class C 1,1 in a neighbourhood of ∂Ω, with Neumann boundary condition on ∂Ω. Theorem 1.1 states that the solution u(t) to (1.1) becomes smooth after some time T , which in general is strictly positive. This behaviour is somewhat different from the usual regularity results for parabolic partial differential equations. Indeed, for this problem there is no instantaneous regularization of the solution, which holds for uniformly parabolic equations but does not hold in general for (1.1).
Another well-known degenerate parabolic problem which shares this property is the so-called total variation flow, which has relevant applications in image analysis and denoising (see, e.g., [24,6,3,12,14]) 2 . Example 1.1. As an example of eventual but not instantaneous regularization of u, we consider the following situation: N = 1, Ω = (0, 2), c a positive constant, and Then u 0 is discontinuous at x = 1, and graph(u 0 ) is a curve of class C 1,1 consisting of two quarters of unit circles (hence with constant curvature equal to 1) and a vertical segment of length c, with the correct boundary condition. Then which is still discontinuous at x = 1, because the upper quarter of circle as unit negative vertical velocity, while the lower quarter of circle as unit positive vertical velocity. Hence the time necessary to let the jump disappear is T := c/2, and one checks that the solution becomes smooth in (T, +∞) × (0, 2).
Acknowledgements. This problem has been proposed to us by our friend and collegue Vicent Caselles, who prematurely died in August 2013. Without his contribution and insight this project would not have been possible.
We are deeply indebted with him, and we dedicate this work to his memory.

Notation and preliminary results
We denote by ∂F the subdifferential of F in the sense of convex analysis, which defines a maximal monotone operator in L 2 (Ω). A characterization of ∂F is given in Remark 2.2.

Remark 2.2. Following [3], inclusion (2.1) can be equivalently written as
Note that, in the expression of z, only the absolutely continuous part of the spatial gradient of u is involved.
Then there exists a unique solution u of (1.1). Moreover: it follows that u(t) converges in L 2 (Ω), as t → +∞, to a minimizer of F , that is, to a constant in Ω. The value of this constant is fixed from where the last equality follows from the Gauss-Green Theorem [4].
The maximum and minimum principles ensure the following result.
The following approximation result is proved in [9, Theorem 3.16].
Let u be the solution to (1.1), and let u n be the solution to the first two equations of (1.1), and with u n (0, ·) = u 0n (·). Then, for all T > 0 we have

Proof of the main result
We start with the following estimates, which have been shown in [19,15]. It can be useful to have a detailed proof, which we include here for completeness.
variables, in the normal direction to Γ ∞ at (x ∞ , 0). Therefore, by regularity of minimizers of the prescribed curvature functional [23], the hypersurfaces graph( u n ) ∩ B ρ (x ∞ , 0) are uniformly (with respect to n ∈ N) of class C 1,α for all α ∈ (0, 1), thus leading to a contradiction.
We conclude the paper with an example showing that, in contrast with the one-dimensional case, in higher dimensions there is no instantaneous regularization of graph(u(t)) 7 . Hence and therefore v is a subsolution of (2.2). By comparison principle (see [3]) it follows that u ≥ v almost everywhere in (0, +∞) × Ω. As a consequence, graph(u(t)) is not of class C 1 (Ω) for t ∈ [0, σ).