Fractional De Giorgi Classes and Applications to Nonlocal Regularity Theory

Abstract

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.

This note is mostly based on a talk given by the author at a conference held in Bari on May 29–30, 2017, as part of the INdAM intensive period “Contemporary research in elliptic PDEs and related topics”.

Appendix A: An Explicit Example

It is easy to see that the characteristic function of a sufficiently smooth subset E of $\mathbb {R}^n$ is contained in the fractional Sobolev space W ^{s, p}, provided sp < 1. In this appendix we show that, in dimension n = 1 and under this assumption on s and p, a step function may also belong to a weak fractional De Giorgi class $\widetilde {\mbox{ DG}}{\mathstrut }^{s,p}$—but never to a strong class DG^{s, p}. From this, it follows that the C ^α estimates of Theorem 2.5 and the Harnack inequality of Theorem 2.9—both valid for the elements of the smaller class DG^{s, p}—cannot be extended to $\widetilde {\mbox{ DG}}{\mathstrut }^{s,p}$.

Proposition A.1

Let n = 1 and sp < 1. Then,

$$\displaystyle \begin{aligned} \chi_{(0, +\infty)} \in \widetilde{\mathit{\mbox{ DG}}}{\mathstrut}^{s,p}((-1, 1); 0, H, 0) \end{aligned} $$

(A.1)

for some constant $H \geqslant 1$ . Furthermore,

$$\displaystyle \begin{aligned} \chi_{(0, +\infty)} \notin \mathit{\mbox{ DG}}_-^{s,p}((-1, 1); d, H, \lambda) \end{aligned} $$

(A.2)

for every $d, \lambda \geqslant 0$ and $H \geqslant 1$.

Proof

We begin by showing that (A.1) holds true. We only check that u := χ _(0,+∞) belongs to the class $\widetilde {\mbox{ DG}}{\mathstrut }_-^{s,p}$, as the verification of its inclusion in $\widetilde {\mbox{ DG}}{\mathstrut }_+^{s,p}$ is analogous.

Fix any x ₀ ∈ (−1, 1), 0 < r < R < 1 −|x ₀|, and $k \in \mathbb {R}$. In order to check the validity of the inequality defining $\widetilde {\mbox{ DG}}{\mathstrut }_-^{s,p}$, we clearly can restrict ourselves to considering the case of k > 0, since otherwise (u − k)₋≡ 0. For shortness, we only deal with k ∈ (0, 1], the case k > 1 being similar. We first estimate from above the left-hand side of (2.2):

$$\displaystyle \begin{aligned} \begin{aligned}{}[(u - k)_-]_{W^{s, p}((x_0 - r, x_0 + r))}^p & = \int_{x_0 - r}^{x_0 + r} \int_{x_0 - r}^{x_0 + r} \frac{|(u(x) - k)_- - (u(y) - k)_-|{}^p}{|x - y|{}^{1 + s p}} \, dx dy \\ & = 2 k^p \chi_{(|x_0|, +\infty)}(r) \int_{x_0 - r}^0 \int_0^{x_0 + r} \frac{dx dy}{|x - y|{}^{1 + s p}} \\ & \leqslant \frac{2 (r - |x_0|)_+^{1 - s p} k^p}{s p (1 - s p)}. \end{aligned} \end{aligned} $$

(A.3)

In view of this, it suffices to estimate from below the right-hand side of (2.2) when r > |x ₀|. In this case, also R > |x ₀| and therefore such right-hand side is larger than

$$\displaystyle \begin{aligned} H \frac{R^{(1 - s) p}}{(R - r)^p} \| (u - k)_- \|{}_{L^p(x_0 - R, x_0 + R)}^p = H \frac{R^{(1 {-} s) p}}{(R - r)^p} k^p \int_{x_0 - R}^0 dx {\geqslant} H (R - |x_0|)^{1 {-} s p} k^p. \end{aligned} $$

As R > r, the latter quantity controls the one appearing on the last line of (A.3), provided H is sufficiently large (in dependence of s and p only). Consequently, u belongs to the class $\widetilde {\mbox{ DG}}{\mathstrut }_-^{s,p}((-1, 1); 0, H, 0)$.

We now turn our attention to (A.2). We point out that, arguing by contradiction, its validity could be inferred from Theorem 2.10. Nevertheless, we present here a proof of it based on a direct computation, for we show that inequality (2.5) does not hold when x ₀ = 0 and R = 2r, with k, r > 0 suitably small. Indeed, under these assumptions the left-hand side of (2.5) is larger than

$$\displaystyle \begin{aligned} \int_{-r}^{r} (u(x) - k)_- \left\{ \int_{\mathbb{R}} \frac{ (u(y) - k)_+^{p - 1}}{|x - y|{}^{1 + s p}} \, dy \right\} dx & \geqslant \int_{- r}^{0} k \left\{ \int_{0}^{x + r} \frac{(1 - k)^{p - 1}}{(y - x)^{1 + s p}} \, dy \right\} dx \\ & = \frac{r^{1 - s p} k (1 - k)^{p - 1}}{1 - s p}. \end{aligned} $$

On the other hand, it is easy to check that the right-hand side of (2.5) is bounded above by CH(r ^1+λ d ^p + r ^1−sp k ^p), for some constant $C \geqslant 1$ depending only on s and p. By taking r and k smaller and smaller (but positive), it follows that the latter quantity cannot control the one displayed above, no matter how large H is. Hence, (A.2) holds true. □