On the strong maximum principle for nonlocal operators

Abstract

In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form

$$\begin{aligned} Iu=c(x) u \qquad \text { in }\,\Omega , \end{aligned}$$

where $\Omega \subset \mathbb {R}^N$ is a domain, $c\in L^{\infty }(\Omega )$ and I is an operator of the form

$$\begin{aligned} Iu(x)=P.V.\int \limits _{\mathbb {R}^N}(u(x)-u(y))j(x-y)\ dy \end{aligned}$$

with a nonnegative kernel function j. We formulate minimal positivity assumptions on j corresponding to a class of operators, which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $\mathbb {R}^N$. Our results extend to the regional variant of the operator I and, under weak additional assumptions, also to the case of x-dependent kernel functions.

Appendix

In this section we give the proof of Lemmas 5.5 and 5.6. So in the following, we let $v_1,\ldots , v_N \in \mathbb {R}^N$ be linearly independent, and we let $G:=\sum \nolimits _{k=1}^{N}\mathbb {Z}v_k \subset \mathbb {R}^N$ be the corresponding lattice. We recall Lemma 5.5.

Lemma A.1

Let $x_0 \in \mathbb {R}^N$ and $r>\frac{1}{2}\sum \nolimits _{k=1}^N|v_k|$. Then $B_r(x_0) \cap G\ne \emptyset$.

Proof

Consider the fundamental domain $\Pi := \bigl \{\sum \nolimits _{k=1}^N \alpha _k v_k \,:\, -\frac{1}{2} \le \alpha _k \le \frac{1}{2}\,\, \hbox {for}\,\, k=1,\dots ,N \bigr \}.$ Since the translates $v + \Pi$, $v \in G$ cover the whole space $\mathbb {R}^N$, there exists $v \in G$ with $x_0 -v \in \Pi$. Hence $x_0 -v = \sum \nolimits _{k=1}^N \alpha _k v_k$ with some $\alpha _k \in [-\frac{1}{2},\frac{1}{2}]$, $k=1,\dots ,N$ and therefore $|x_0 - v| \le \frac{1}{2} \sum \nolimits _{k=1}^N |v_k|<r$. Consequently, $v \in G \cap B_r(x_0)$, so that $G \cap B_r(x_0) \not = \varnothing .$$\square$

We now turn to the proof of Lemma 5.6. It will be convenient to use the following definition.

Definition A.2

Let $A \subset \mathbb {R}^N$ be an arbitrary subset, and let $x, x' \in A$ such that $x-x' \in G$. In the following, a G-path in A from x’ to x is defined as an ordered set of points $w_0,\ldots , w_n\in A$ such that $w_0=x'$, $w_n=x$ and

$$\begin{aligned} w_{\ell }-w_{\ell -1} \in G_*:=\{ \pm v_1,\dots , \pm v_N\} \qquad \text {for }\,\ell = 1,\dots ,N. \end{aligned}$$

With this definition, we may now reformulate Lemma 5.6 as follows.

Lemma A.3

For every $\rho \ge \sum \nolimits _{k=1}^N |v_k|$ and every $x,x' \in G$ with $|x-x'|<\rho$ there exists a G-path in $B_{4^{N-1} \rho }(x')$ from $x'$ to x.

Proof

Without loss, we may assume that $x'=0$ in the following. We consider the affine subspaces

$$\begin{aligned} {{\mathscr {W}}}_0:= \{0\},\qquad {{\mathscr {W}}}_j:=\sum \limits _{i=1}^{j} \mathbb {R}v_i , \qquad j=1,\dots ,N \end{aligned}$$

and the sublattices

$$\begin{aligned} G_{j}:= \sum _{k=1}^{j} \mathbb {Z}v_k = G \cap {{\mathscr {W}}}_j,\qquad j=1,\dots ,N. \end{aligned}$$

We then prove the following claim by induction on j.

Claim A

Let $j \in \{1,\dots ,N\}$. For every $\rho \ge \sum \nolimits _{k=1}^N |v_k|$ and every $x \in G_j$ with

$$\begin{aligned} |x|< \rho + \frac{1}{2} \sum _{i=1}^{j-1}|v_j|\qquad \text {and}\qquad \text {dist}(x,{{\mathscr {W}}}_{j-1})< \rho \end{aligned}$$

there exists a G-path in $B_{4^{j-1} \rho }(0)$ from 0 to x.

This is true for $j=1$, since in this case $x= k v_1$ for some $k \in \mathbb {Z}$, $|x|= \text {dist}(x,{{\mathscr {W}}}_0) < \rho$, and there is an obvious one-dimensional G-path in $B_{\rho }(0)$ from 0 to x.

We now assume that Claim A is true for some fixed $j \in \{1,\dots ,N-1\}$, i.e.

$$\begin{aligned} \left\{ \begin{aligned}&\hbox {For every} \rho \ge \sum \limits _{k=1}^N |v_k| \hbox {and every} y \in G_{j}\hbox { with }|y|< \rho + \frac{1}{2} \sum _{i=1}^{j-1}|v_i|\\&\text { and }\text {dist}(y,{{\mathscr {W}}}_{j-1})< \rho \hbox { there exists a }G\hbox {-path in }B_{4^{j-1} \rho }(0)\hbox { from 0 to }y. \end{aligned} \right. \end{aligned}$$

(A.1)

We fix $\rho \ge \sum \nolimits _{i=1}^{N}|v_i|$, and we suppose by contradiction that Claim A is false for $j+1$ and this choice of $\rho$. Then there exists $x= y + k v_{j+1} \in G_{j+1}$ with $y \in G_{j}$ and $k \in \mathbb {Z}$ such that

$$\begin{aligned} |x|< \rho + \frac{1}{2} \sum _{i=1}^{j}|v_i|, \qquad \text {dist}(x,{{\mathscr {W}}}_{j})< \rho , \end{aligned}$$

(A.2)

and such that there does not exist a G-path in $B_{4^{j}\rho }(0)$ from 0 to x. Without loss we may assume that $k \ge 0$, and that k is chosen minimally with this property. In the case $k=0$ we have $x=y$ and thus

$$\begin{aligned} |x|< \rho + \frac{1}{2} \sum _{i=1}^{j}|v_i| \le 2\rho , \qquad \text {dist}(x,{{\mathscr {W}}}_{j-1}) \le |x|< 2\rho , \end{aligned}$$

so that by (A.1)—applied with $2 \rho$ in place of $\rho$—there exists a G-path in $B_{4^{j-1} 2\rho }(0) \subset B_{4^j \rho }(0)$ from 0 to x. This contradicts our choice of x, and thus we have $k>0$.

Let $x_1=y+(k-1)v_{j+1}$, and let $x_1^* \in {{\mathscr {W}}}_{j}$ be the orthogonal projection of $x_1$ on ${{\mathscr {W}}}_{j}$, so that

$$\begin{aligned} |x_1^*| \le |x_1| \qquad \text {and}\qquad |x_1-x_1^*| =\text {dist}(x_1,{{\mathscr {W}}}_{j}) \le |x_1|. \end{aligned}$$

(A.3)

Since the sets $\bigl \{y_*+ \sum \nolimits _{i=1}^{j}\alpha _iv_i\;:\; -\frac{1}{2} \le \alpha _i\le \frac{1}{2}\hbox { for}\ i=1,\ldots , j\bigr \}$, $y_*\in G_{j}$ cover ${{\mathscr {W}}}_{j}$, there exists $y^* \in G_{j}$ and $\alpha _i \in [-\frac{1}{2},\frac{1}{2}]$ with $x_1^* = y^* + \sum \nolimits _{i=1}^{j} \alpha _i v_i$. The point $x':=x_1- y^*$ then satisfies

$$\begin{aligned} \text {dist}(x',{{\mathscr {W}}}_{j})=\text {dist}(x_1,{{\mathscr {W}}}_{j})&= (k-1) \, \text {dist}(v_{j+1},{{\mathscr {W}}}_{j}) \nonumber \\&\le k \,\text {dist}(v_{j+1},{{\mathscr {W}}}_{j})= \text {dist}(x,{{\mathscr {W}}}_{j}) < \rho \end{aligned}$$

(A.4)

and, by (A.3) and (A.4),

$$\begin{aligned} |x'|= & {} |x_1-y^*| \le |x_1-x_1^*| + |x_1^*-y^*| \le \text {dist}(x_1,{{\mathscr {W}}}_{j}) \nonumber \\&+\, \frac{1}{2}\sum _{i=1}^{j}|v_i| < \rho + \frac{1}{2}\sum _{i=1}^{j}|v_i|. \end{aligned}$$

(A.5)

By the minimality property of k, this implies the existence of a G-path $\Gamma _1$ from 0 to $x'$ in $B_{4^{j}\rho }(0)$. Moreover, by (A.2) and (A.3),

$$\begin{aligned}&|y^*| \le |x_1^*| + \frac{1}{2}\sum _{i=1}^{j}|v_i| \le |x_1| + \frac{1}{2}\sum _{i=1}^{j}|v_i| \le |x|+|v_{j+1}| + \frac{1}{2}\sum _{i=1}^{j}|v_i| < \rho \\&\quad +\, \sum _{i=1}^{j+1}|v_j| \le 2 \rho , \end{aligned}$$

and thus also $\text {dist}(y^*,{{\mathscr {W}}}_{j-1}) < 2\rho$. Thus (A.1)—applied with $2 \rho$ in place of $\rho$—yields a G-path from 0 to $y_*$ in $B_{4^{j-1}2\rho }(0)$. By mere translation, this path gives rise to G-path $\Gamma _2$ from $x' = x_1 - y^*$ to $x_1$ in $B_{4^{j-1}2\rho }(x')$, whereas

$$\begin{aligned} B_{4^{j-1}2\rho }(x') \subset B_{4^{j} \rho }(0) \qquad \text {since }|x'| \le 2 \rho \,\text { by }\,(\mathrm{A.5}). \end{aligned}$$

Composing $\Gamma _1$ and $\Gamma _2$, we then get a G-path from 0 to $x_1$ in $B_{4^{j}\rho }(0)$. Simply adding $x=y+ k v_j$ as an endpoint and using that $x \in B_{4^{j}\rho }(0)$ by assumption, we finally obtain a G-path from 0 to x in $B_{4^{j}\rho }(0)$. This contradicts our assumption and finishes the proof of Claim A for $j+1$.

By induction, the proof of Claim A is thus finished. Applying Claim A with $j=N$ and noting that $\text {dist}(x,{{\mathscr {W}}}_{N-1}) \le |x|$ for every $x \in G$, we finally deduce the claim of the lemma.

$\square$