Regularity of Free Boundaries in Obstacle Problems

Abstract

Free boundary problems are those described by PDE that exhibit a priori unknown (free) interfaces or boundaries. Such type of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations, and Geometric Measure Theory. The main mathematical challenge is to understand the regularity of free boundaries. The Stefan problem and the obstacle problem are the most classical and motivating examples in the study of free boundary problems. A milestone in this context is the classical work of Caffarelli, in which he established for the first time the regularity of free boundaries in the obstacle problem, outside a certain set of singular points. This is one of the main results for which he got the Wolf Prize in 2012 and the Shaw Prize in 2018.

The goal of these notes is to introduce the obstacle problem, prove some of the main known results in this context, and give an overview of more recent research on this topic.

Appendix: Proof of Proposition 12

Proposition 12 will follow from the following.

Lemma 9

For every β > 0 there exists δ > 0 such that the following holds.

Let $E \subseteq \mathbb {R}^n$ such that for each x ∈ E and r ∈ (0, r ₀) there exists a m-dimensional plane L _x,r , passing through x, for which

$$\displaystyle \begin{aligned} E \cap B_{r}(x) \subset \{ y : {\mathrm{dist}}(y, L_{x,r}) < \delta r \} . \end{aligned}$$

Then, $\mathcal {H}^{m+\beta }(A) = 0$.

Proof

By a covering argument, we may assume that E ⊆ B ₁ and 0 ∈ E. By assumption, there exists a plane L _0,1 such that

$$\displaystyle \begin{aligned} E \cap B_1 \subset \{ y : {\mathrm{dist}}(y, L_{0,1}) < \delta \} . \end{aligned}$$

Cover L _0,1 by a finite collection of balls {B _2δ(z _k)}_k=1,2,…,N where z _k ∈ L _0,1 for each k and N ≤ Cδ ^−m. Observe that {B _2δ(z _k)}_k=1,2,…,N covers {y : dist(y, L _0,1) < δ} and thus covers E ∩ B ₁. Throw away the balls B _2δ(z _k) that do not intersect E. For the remaining balls, let x _k ∈ A ∩ B _2δ(z _k). Now {B _4δ(x _k)}_k=1,2,…,N covers E ∩ B ₁, x _k ∈ E, with N ≤ Cδ ^−m, and thus N(4δ)^m+β ≤ Cδ ^β ≤ 1∕2, provided that δ > 0 is small enough.

Now observe that we can repeat this argument with B _4δ(x _k) in place of B ₁ to get a new covering $\{B_{(4\delta )^2}(x_{k,l})\}_{l=1,2,\ldots ,N_k}$ of E ∩ B _4δ(x _k) with N _k(4δ)^m+β < 1∕2. Thus $\{B_{(4\delta )^2}(x_{k,l})\}_{k = 1,2,\ldots ,N, \, l=1,2,\ldots ,N_k}$ covers E with x _k,l ∈ E and $\sum _{k=1}^N N_k (4\delta )^{2 \cdot (m+\beta )} < (1/2)^2$. Repeating this argument for a total of j times, we get a finite covering of E by M balls with centers on E, radii = (4δ)^j, and M(4δ)^j(m+β) < (1∕2)^j. Thus $\mathcal {H}^{m+\beta }(E) \leq C(1/2)^j$ for every integer j = 1, 2, 3, …. Letting j →∞, we get $\mathcal {H}^{m+\beta }(E) = 0$.

Proof (Proof of Proposition 12 )

It follows from Lemma 9 and the definition of Hausdorff dimension.