Optimal decay exponent and fractional weak Harnack inequality for solutions of strongly nonlocal equations with decaying source (Dipierro-Figalli-Valdinoci, 2014)

Dipierro, Figalli and Valdinoci (Comm. Partial Differential Equations 39 (2014) 2351-2387) reduced the passage from the atomic Peierls-Nabarro dislocation model to its macroscopic particle dynamics in the strongly nonlocal regime $s\in(0,1/2)$ to a pointwise decay estimate (their Theorem 1.3) for solutions of the linear nonlocal equation $-L_s v+c(x)v=g$ on $\mathbb{R}$, proved with a constructive exponent from their formula (5.4). The paper explicitly leaves two problems open: (A) determine the optimal decay exponent $\vartheta$, i.e. the sharp rate at which bounded solutions driven by a decaying source vanish at infinity, in the general setting of their Proposition 5.1 ($n\ge1$, $s\in(0,1)$, source decay $\alpha$); and (B) establish the fractional analogue of the weak Harnack inequality that, in classical theory, converts integral (average) decay estimates into pointwise ones, whose absence forced the bespoke barrier and sliding arguments of the paper.

Background

In the Peierls-Nabarro model of crystal plasticity, the atom dislocation of a one-dimensional crystal is described by a function $v(t,x)$ solving the semilinear evolution equation $v_t=L_sv-W'(v)+\sigma_\varepsilon(t,x)$, where $W$ is a 1-periodic multi-well potential describing the crystal's energy landscape, $\sigma_\varepsilon$ is a small external stress, and $L_s$ is the fractional Laplacian of order $2s\in(0,2)$, defined for smooth bounded $\phi$ by

$$L_s\phi(x)=\frac{1}{2}\int_{\mathbb{R}}\frac{\phi(x+y)+\phi(x-y)-2\phi(x)}{|y|^{1+2s}}\,dy.$$

This is a nonlocal operator: the value $L_s\phi(x)$ depends on $\phi$ over the whole line, and the smaller $s$, the more the contributions of far-away points dominate. Dipierro, Figalli and Valdinoci (2014) treated the remaining parameter range $s\in(0,1/2)$, the strongly nonlocal regime, proving that under the parabolic rescaling $v_\varepsilon(t,x)=v(t/\varepsilon^{1+2s},x/\varepsilon)$ the dislocation function converges, as $\varepsilon\to0$, to a superposition of Heaviside steps whose jump points evolve by a repulsive ODE system driven by the external stress. This completed a program carried out for $s=1/2$ by Gonzalez and Monneau (2012) and for $s\in(1/2,1)$ by Dipierro, Palatucci and Valdinoci (2015); in those ranges the analysis rests on harmonic-extension methods for the fractional Laplacian, which lose their force when $s<1/2$ because far-field energy contributions dominate.

The analytic cornerstone of the 2014 paper is a decay estimate at infinity for a linear nonlocal equation (Theorem 1.3 there): if $s\in(0,1/2)$, $v\in L^\infty(\mathbb{R})\cap C^2(\mathbb{R})$ tends to $0$ at $\pm\infty$, and

$$-L_sv+c(x)v=g(x)\quad\text{in }\mathbb{R},$$

with $c\in L^\infty(\mathbb{R})$ bounded below by a positive constant and $|g(x)|\le C(1+|x|^{4s})^{-1}$, then $|v(x)|\le C(1+|x|)^{-\vartheta}$ with $\vartheta\in(2s,1+2s]$ depending only on $s$. The proof of the underlying Proposition 5.1 of that paper (stated in $\mathbb{R}^n$ for all $s\in(0,1)$ and sources decaying like $(1+|x|)^{-\alpha}$) proceeds in two steps: first an average (integral) decay estimate $\int_{B_1(x)}|u(y)|\,dy\le C^*|x|^{-\vartheta}$, obtained by test-function identities, and then an upgrade to the pointwise bound by explicit barriers and a sliding argument. The constructive exponent is

$$\vartheta=\frac{\min\{\,n+2s-(n-2\alpha)^{+},\,2\alpha\,\}}{2},$$

which in the dislocation case $n=1$, $\alpha=4s$ equals $4s$ for $s\le1/6$ and $(1+2s)/2$ for $s\ge1/6$. In the classical local theory, the upgrade from integral to pointwise decay is routine via the weak Harnack inequality (Theorem 4.8(2) in the book of Caffarelli and Cabre): for a nonnegative supersolution in a ball, the infimum over a smaller ball is bounded below by a constant multiple of a local $L^p$-average of the solution. For nonlocal operators this fails as stated, because remote negative values of the function enter the equation at every point through the far-field tail; the authors state that, to the best of their knowledge, no fractional analogue suited to their regime was known, and that its absence is what forced the bespoke barrier and sliding machinery. The paper leaves two problems open, which are the subject of this record: (A) the optimal value of the decay exponent $\vartheta$ in the general setting, and (B) a fractional weak Harnack inequality able to convert integral decay estimates into pointwise ones when far-field contributions dominate.

Problem Statement

The source states two open problems serving one common objective: sharp pointwise asymptotic control of solutions of nonlocal equations in the regime where far-field contributions dominate, obtained by conceptual tools rather than bespoke barrier arguments.

(A) Optimal decay exponent. In the general setting of Proposition 5.1 of the source paper, let $s\in(0,1)$, $n\ge1$, and let $u\in L^\infty(\mathbb{R}^n)\cap C^2(\mathbb{R}^n)$ solve

$$-L_su+c(x)u=g(x)\quad\text{in }\mathbb{R}^n,$$

where $c$ is measurable with $0<c_0\le c(x)\le c_0^{-1}$ and $|g(x)|\le C(1+|x|)^{-\alpha}$ for some $\alpha>0$. Determine the optimal decay exponent: the largest $\vartheta^*(n,s,\alpha)$ - in particular $\vartheta^*(1,s,4s)$ for $s\in(0,1/2)$, the case used for dislocation dynamics - such that every such solution satisfies $|u(x)|\le C'(1+|x|)^{-\vartheta}$ for every $\vartheta\le\vartheta^*$, with $C'$ depending only on the data ($n$, $s$, $\alpha$, $c_0$, $\|c\|_{L^\infty}$, the constant in the bound on $g$, and $\|u\|_{L^\infty}$); an answer must include admissible data showing that no strictly larger exponent enjoys this uniform property. The source proves the estimate with the constructive exponent of its formula (5.4), $\vartheta=\min\{n+2s-(n-2\alpha)^{+},2\alpha\}/2$, and asks for its optimal value in a general setting.

(B) Fractional weak Harnack inequality for the strongly nonlocal regime. Establish or refute the existence of an analogue of the classical weak Harnack inequality (Caffarelli-Cabre, Theorem 4.8(2)) for supersolutions of linear nonlocal equations $-L_su+c(x)u=g$ (a supersolution being a function $u$ satisfying $-L_su+c(x)u\ge g$ in the weak, test-function sense) that remains valid when far-field contributions dominate, in particular for $s\in(0,1/2)$ on the whole space, without assuming local positivity of the supersolution and with the influence of the sign-changing far field controlled through explicit correction (tail) terms or an equivalent device. In the affirmative case, demonstrate that the tool performs the passage for which it is intended: deduce the pointwise decay bound of the source's Theorem 1.3 from an average decay estimate such as $\int_{B_1(x)}|u(y)|\,dy\le C^*|x|^{-\gamma}$, thereby replacing the paper's barrier and sliding arguments.

An answer that resolves either part in full constitutes a solution of this problem as posed. Resolving (A) in full requires both the optimal exponent $\vartheta^*(n,s,\alpha)$ and the admissible data showing that no strictly larger exponent enjoys the uniform property; resolving (B) in full requires settling the existence question and, in the affirmative case, demonstrating that the tool performs the intended passage. The two parts are linked by the shared objective above, the second being the structural tool that the first would naturally use.

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