A counterexample to the fractional Michael-Simon inequality

Introduction

The classical Michael-Simon-Allard inequality states that, for a smooth $m$-dimensional submanifold $M$ without boundary in Euclidean space and $1 \le p < m$,

for every compactly supported smooth function $u$ on $M$. Here $H$ is the mean curvature of $M$. This inequality was proved independently by Allard [1] and Michael and Simon [12], and the optimal constant in the case $p=1$ in codimension at most two is due to Brendle [4].

We consider the nonlocal counterpart for boundaries of sets in $\mathbb{R}^n$, $n \ge2$. Let $s \in(0,1)$, for a measurable set $E \subset\mathbb{R}^n$, the fractional perimeter $P_s(E)$ is given by

see [7]. If $E$ is bounded and open with $C^2$ boundary, its fractional mean curvature $H_s^E(x)$ at $x \in\partial E$ is

where p.v. stands for principal value. As customary, we denote by $P(E)$ the perimeter of $E$. For bounded Lipschitz domains it holds $P(E)=\mathcal{H}^{n-1}(\partial E)$. We also write $\omega_n=|B_1|$. The natural fractional analogue of (MS) of order $(s,p)$ for $0<s<1$ and $1 \le p < \infty$ such that $sp<n-1$ reads as, for a constant $C=C(n,s,p)$ independent of $E$,

for $u \in C^\infty(\partial E)$ when $\partial E$ is bounded and $u \in C^\infty_c(\partial E)$ in general see e.g. [6], where $p_s^*=\frac{(n-1)p}{n-1-sp}$ and

2020 Mathematics Subject Classification. Primary 26D10; Secondary 46E35, 49Q20, 53A07.

Key words and phrases. Michael–Simon inequality, Allard inequality, fractional perimeter, nonlocal mean curvature.

Funding. V. Julin was supported by the Academy of Finland grant 347550. G. Palatucci has been supported by the INdAM project 2026 “Parabolic equations with drift: well posedness, gradient estimates and boundary effects” – CUP_E53C25002010001. see [10]. Related fractional Sobolev inequalities are known for nonlocal minimal surfaces ([5 Proposition 5.2]) and, more recently, for sets with vanishing fractional mean curvature ([14]). Eventually, Cabré, Cozzi, and Csató proved $(\mathrm{MS}_p^s)$ when $E$ is convex in an even more general formulation allowing the exponent in the curvature disagreeing with that of the fractional energy, in [6 Theorem 1.3] they use convexity in a pointwise estimate that combines the fractional Sobolev kernel with $H_\alpha^E$. Csató and Roy subsequently extended this result to a fractional Caffarelli–Kohn–Nirenberg family on convex hypersurfaces [9 Theorem 1.1]. A weaker fractional Michael–Simon inequality on convex hypersurfaces under a subcritical fractional Willmore energy bound was obtained by Blatt, Giacomin, Schikorra, and Scheuer; see [2]. The extension of $(\mathrm{MS}_p^s)$ beyond convexity is explicitly left open in [6 p. 190, after Formula (1.13)], [9 Section 1, before Theorem 1.1] and [14 Section 1, after Formula (1.7)].

Observe that by considering $(\mathrm{MS}_p^s)$ with the choice $u \equiv1$, the inequality becomes

In particular, if $(\widetilde{\mathrm{MS}}_p^s)$ would hold then a uniform bound on $\lVert H_s^E\rVert_{L^p(\partial E)}$ would imply uniform bound on the perimeter. In this note we prove that the preceding inequality, and therefore $(\widetilde{\mathrm{MS}}_p^s)$, fails for general smooth sets whenever $0<s<(n-1)/(np)$. The failure occurs even though $H_s^E\geq0$ at every point of the boundary. This gives a negative answer to the extension problem in this range. It also precludes extending the whole family of Csató and Roy [9 Theorem 1.1] to all smooth boundaries with constants independent of the hypersurface: the choice $a=1$, $\alpha=s$, $\gamma=0$ and $\tau=p^*_s$ in their notation gives indeed $(\mathrm{MS}_p^s)$.

For the construction of our counterexample, we maximize the fractional perimeter over a compact family of radial configurations, allowing coincident radii. Each configuration admits an open representative whose boundary is a finite union of concentric spheres. Homogeneity implies that the outermost boundary sphere of a maximizer has the prescribed radius. Radial variations give zero fractional mean curvature on every internal sphere, while a dilation computes the curvature on the outer sphere. The use of concentric annuli is inspired by the work of Cesaroni and Novaga ([8]). In Theorem 2.3 there, they balance the fractional mean curvature through the nonlocal interactions between concentric spheres to construct radial self-shrinkers. We use a similar type of interaction for a different purpose: we select the radii through a different variational problem to obtain the loss of coercivity between the classical perimeter and the fractional quantities.

The counterexample

Unless otherwise specified, all constants in this section depend only on $n$, $s$, and $p$.

In order to prove the result above, we need some preliminary results.

Proof. The inequality (3) is well-known, see e.g. [3 Corollary 4.3] or [13 Formula (1.4)]. Indeed, [3 Proposition 4.2] gives the following general interpolation inequality,

for any $u \in BV(\mathbb{R}^{n}) \cap L^{1}(\mathbb{R}^{n})$. Inequality (3) then follows by applying (4) to $u=\chi_A$ and using $[\chi_A]_{W^{s,1}}=2P_s(A)$.

For the last assertion, lower semicontinuity of total variation gives $P(A)<\infty$. Applying (4) to $u_j=\chi_{A_j}-\chi_A$, whose total variations are uniformly bounded, we obtain

Fix $R\in(0,1]$ and an even integer $N\ge4$. Motivated by the concentric argument of Cesaroni and Novaga in [8 Theorem 2.3], we define

and to any $\mathbf{r}\in\mathbf{J}^{(N)}_R$ we associate the set $A(\mathbf{r})$ defined by

Degenerate configurations are regarded as equivalence classes modulo negligible sets. Almost everywhere,

Proof. By (6), for $\mathbf{r},\mathbf{t}\in\mathbf{J}^{(N)}_R$,

The perimeters are uniformly bounded by (6). Lemma 2.2 therefore gives the continuity of $\mathbf{r}\mapsto P_s(A(\mathbf{r}))$. Since $\mathbf{J}^{(N)}_R$ is compact, a maximum is attained.

Let $\mathbf{r}$ be a maximizing vector and set $A=A_{N,R}:=\operatorname{int}(A(\mathbf{r}))$. This representative coincides with $A(\mathbf{r})$ almost everywhere.

Notice that, since $B_R$ belongs to the family modulo negligible sets, it holds

The radial characteristic function has finitely many jumps, so

In particular, $\partial A$ is $C^\infty$.

We claim that $\rho_m=R$. Otherwise, choose $1<\lambda<R/\rho_m$ and set

Then $\tilde{r} \in J_R^{(N)}$ and, almost everywhere,

Consequently,

which is a contradiction in view of the maximality of $A$.

We are left to prove the equality (7). To that aim we fix one of the internal spheres $\partial B_{\rho_j}$, choose a function $\psi\in C^1(\partial A)$ such that $\psi=1$ on $\partial B_{\rho_j}$ and $\psi=0$ on $\partial A\setminus\partial B_{\rho_j}$. We construct a radial vector field $X\in C_0^1(U;\mathbb{R}^n)$, where $U$ is a neighborhood of $\partial B_{\rho_j}$ and which does not intersect the set $\partial A\setminus\partial B_{\rho_j}$, such that $X\cdot\nu_A=\psi$ on $\partial A$. Consider the associated flow of diffeomorphisms $\Phi_t$, i.e., $\frac{\partial}{\partial t}\Phi_t=X(\Phi_t)$ and $\Phi_0(x)=x$ for all $x\in\mathbb{R}^n$. When $|t|$ is small the sets $\Phi_t(A)$ are admissible and the first-variation Formula (6.12) in [11], Theorem 6.1 implies

By the radial symmetry $H_s^A$ is constant on $\partial B_{\rho_j}$ and therefore we deduce that $H_s^A=0$ on $\partial B_{\rho_j}$. This proves the claim. □

We are now in the position to complete the construction of our counterexample.

Proof of Theorem 2.1. Let $N\geq4$ be even and let $A_{N,R}$ be the open representative of a maximizer given by Lemma 2.3.

Step 1: Estimates on the perimeters. We first show that there are constants $c_*,C_*>0$, depending only on $n$ and $s$, such that the family (5) contains a competitor $A^*$ satisfying

and every member $A$ of the family satisfies

so that, as a consequence, any maximizer $A_{N,R}$ in Lemma 2.3 satisfies

Set $h=\frac{1}{2}\frac{R}{N-1}$ and choose

Let

For $1\leq j\leq N/2-1$, set $q_j=r_{2j-1}^*$ and

For $x=r\theta\in X_j$, with $\theta\in S^{n-1}=\partial B_1$, define

and

Since all these radii lie between $R/2$ and $R$, polar coordinates give

Moreover, for $x=r\theta\in X_j$ and $y=t\varphi\in Y_j(x)$,

The sets $X_j$ are pairwise disjoint and $N/2-1\ge N/4$. Hence

as desired.

For the upper bound, notice that, for any member $A$ of the family, (3) and (6) give

Maximality then yields the estimates in (8).

Finally, such a maximizer $A_{N,R}$ satisfies

Indeed, by construction, $c_*N^sR^{n-s}\le CR^{n(1-s)}P(A_{N,R})^s$, and thus taking the $s$th-root gives the lower bound in (10); whereas the upper bound is already contained in (6).

Step 2: Estimates on the nonlocal curvature. Write $A=A_{N,R}$. By radial symmetry and (7), there is a constant $C_{N,R}$ such that

We choose the velocity vector field $X\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ such that $X(x)=x$ in $B_{2R}$. Then for $x\in A$ and for $|t|$ sufficiently small it holds $\Phi_t(x)=e^tx$. Therefore, it holds

Since $X(x)\cdot\nu_A=R$ on $\partial B_R$ and $H_s^A=0$ on $\partial A\setminus\partial B_R$, the first-variation Formula (6.12) in [11 Theorem 6.1] and (11) imply

Thus $C_{N,R}>0$ and $H_s^A\geq0$ on $\partial A$. Moreover, for every $1\leq p<\infty$ we have by the above

Step 3: Completion of the proof. For each even $N\geq4$, take $A_N=A_{N,1}$ from Lemma 2.3. Then $A_N$ is open, is contained in $B_1$, and has a $C^\infty$ boundary consisting of at most $N$ concentric spheres. Steps 1 and 2 yield

where the constants may also depend on $p$. Then, up to relabeling the constants,

for $0<s<(n-1)/(np)$, as claimed. □

(V. Julin) Department of Mathematics and Statistics, University of Jyväskylä, P. O. Box 35, FI-40014 University of Jyväskylä, Finland

Email address: [email protected]

(G. Palatucci) Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

Email address: [email protected]