The Wiener Criterion for Nonlocal Dirichlet Problems

Abstract

We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.

Appendix A. Algebraic Inequalities

In this section, we provide some algebraic inequalities that are used in the Caccioppoli-type estimates. Let $p \in (1, \infty )$ and $\beta , \gamma \in \mathbb {R}$ be such that $\beta = \gamma -(p-1)$. Recall that, in the local case, we use the following inequalities for the proof of Caccioppoli-type estimates [31, Theorem 8.25 and 8.26]: assume $\beta \ne 0$, $\gamma \ne 0$, and let $l \ge 0$. Then, there exists constants $c, C > 0$, depending only on p, such that

$$\begin{aligned} \begin{aligned} |\nabla u|^{p-2} \nabla u \cdot \nabla (f(u) \eta ^p)&\ge |\nabla F(u)|^p \eta ^p - \frac{|\gamma |}{|\beta |} |\nabla F(u)|^{p-1} \eta ^{p-1} |F(u)| |\nabla \eta | \\&\ge \frac{1}{p} \left( |\nabla F(u)|^p \eta ^p - \left( \frac{|\gamma |}{|\beta |} \right) ^p |F(u)|^{p} |\nabla \eta |^p \right) \\&\ge c \, |\nabla (F(u) \eta )|^p - C \left( 1+\left( \frac{|\gamma |}{|\beta |} \right) ^p \right) |F(u)|^p |\nabla \eta |^p, \end{aligned} \end{aligned}$$

where $f, F: I \rightarrow [0, \infty )$ are defined by

$$\begin{aligned} f(t) = \frac{1}{\beta } \left( (t+d)^{\beta } - (l+d)^{\beta } \right) \quad \text {and}\quad F(t) = \frac{p}{\gamma } (t+d)^{\gamma /p}. \end{aligned}$$

(A.1)

Here, $I = [l, \infty )$ when $\beta > 0$ and $I = [0, l]$ when $\beta < 0$. As a discrete version, we prove the following algebraic inequality.

Lemma A.1

Assume that $\beta \ne 0$ and $\gamma \ne 0$. There exist $c, C > 0$, depending only on p, such that

$$\begin{aligned} \begin{aligned}&|a-b|^{p-2}(a-b) (f(a) \eta _1^p - f(b) \eta _2^p) \\&\quad \ge c \left| F(a) \eta _1 - F(b) \eta _2 \right| ^p - C \left( 1+ \left( \frac{|\gamma |}{|\beta |} \right) ^p \right) \max \lbrace |F(a)|, |F(b)| \rbrace ^p |\eta _1-\eta _2|^p \end{aligned} \end{aligned}$$

for any $a, b \in I$ and $\eta _1, \eta _2 \ge 0$.

Some inequalities similar to Lemma A.1 are known for almost all values of $\beta$. See, for instance, [5] for $\beta \ge 1$ and [10] for $\beta < -(p-1)$. Lemma A.1 can be proved in a similar way, but let us provide a proof to make the paper self-contained. To this end, we first prove the following two lemmas as intermediate steps.

Lemma A.2

If $\beta \ne 0$ and $\gamma \ne 0$, then

$$\begin{aligned} |a-b|^{p-2}(a-b)(f(a)-f(b)) \ge |F(a)-F(b)|^p \end{aligned}$$

and

$$\begin{aligned} |a-b|^{p-1} \min \left\{ F'(a), F'(b) \right\} ^{p-1} \le |F(a)-F(b)|^{p-1} \end{aligned}$$

for any $a, b \in I$.

Proof

We may assume that $a>b$. Then, we have

by Jensen’s inequality, and

since $F'$ is positive on $[0, \infty )$. $\square$

Lemma A.3

Let $A, B \in \mathbb {R}$ and $\eta _1, \eta _2 \ge 0$, then

$$\begin{aligned} \begin{aligned} |A-B|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p&\ge 2^{1-p} |A \eta _1 - B \eta _2|^p - \max \lbrace |A|, |B| \rbrace ^p |\eta _1-\eta _2|^p \quad \text {and} \\ |A-B|^p \max \lbrace \eta _1, \eta _2 \rbrace ^p&\le 2^{p-1} |A \eta _1 - B \eta _2|^p + 2^{p-1} \max \lbrace |A|, |B| \rbrace ^p |\eta _1-\eta _2|^p. \end{aligned} \end{aligned}$$

Proof

We may assume that $\eta _1 \ge \eta _2$. Then, the desired inequalities follow from the equalities

$$\begin{aligned} A\eta _1 - B\eta _2 = (A-B) \eta _2 + A (\eta _1-\eta _2) = (A-B) \eta _1 + B (\eta _1-\eta _2) \end{aligned}$$

and the triangle inequality. $\square$

We prove Lemma A.1 by using Lemmas A.2 and A.3.

Proof of Lemma A.1

We may assume that $a > b$. Then, we have

$$\begin{aligned}&(a-b)^{p-1} (f(a) \eta _1^p - f(b) \eta _2^p) \nonumber \\ {}&= (a-b)^{p-1}(f(a) - f(b))\eta _1^p + (a-b)^{p-1} f(b)(\eta _1^p-\eta _2^p) \end{aligned}$$

(A.2)

$$\begin{aligned} \nonumber \\ {}&= (a-b)^{p-1}(f(a) - f(b))\eta _2^p + (a-b)^{p-1} f(a)(\eta _1^p-\eta _2^p). \end{aligned}$$

(A.3)

We apply Lemma A.2 to (A.2) if $\beta \ge 1$ or to (A.3) if $\beta \le 1$. Then, we obtain

$$\begin{aligned} \begin{aligned} J :=&~ (a-b)^{p-1} (f(a) \eta _1^p - f(b) \eta _2^p) \\ \ge&~ |F(a)-F(b)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\ {}&-|F(a)-F(b)|^{p-1} \max \left\{ \frac{|f(a)|}{F'(a)^{p-1}}, \frac{|f(b)|}{F'(b)^{p-1}} \right\} |\eta _1^p-\eta _2^p|. \end{aligned} \end{aligned}$$

By using

$$\begin{aligned} \frac{|f(t)|}{F'(t)^{p-1}} \le \frac{|\gamma |}{p|\beta |} |F(t)|, \quad |\eta _1^p-\eta _2^p| \le p |\eta _1-\eta _2|\max \lbrace \eta _1, \eta _2 \rbrace ^{p-1}, \end{aligned}$$

and Young’s inequality, we deduce

$$\begin{aligned} \begin{aligned} J&\ge |F(a)-F(b)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\&\quad - \frac{1}{2^{2p-1}} |F(a)\!-\!F(b)|^p \max \lbrace \eta _1, \eta _2 \rbrace ^p \!-\! C \left( \frac{|\gamma |}{|\beta |} \right) ^p \max \lbrace |F(a)|, |F(b)| \rbrace ^p |\eta _1 \!-\! \eta _2|^p, \end{aligned} \end{aligned}$$

where $C = C(p) > 0$. Applying Lemma A.3 with $A = F(a)$ and $B = F(b)$ finishes the proof. $\square$

The case $\gamma = 0$, or equivalently $\beta = -(p-1)$, is treated in the following lemma, which is a discrete version of

$$\begin{aligned} |\nabla u|^{p-2} \nabla u \cdot \nabla (f(u) \eta ^p) \ge c |\nabla \log (u+d)|^p \eta ^p - C |\nabla \eta |^p, \end{aligned}$$

where f is given by (A.1) with $\beta = -(p-1)$ and $c, C > 0$ are constants depending only on p.

Lemma A.4

Assume that $\gamma = \beta + p-1 = 0$. There exist $c, C > 0$, depending only on p, such that

$$\begin{aligned} \begin{aligned}{}&{} |a-b|^{p-2}(a-b) (f(a)\eta _1^p - f(b)\eta _2^p)\\{}&{} \quad \ge c\, |\log (a+d)- \log (b+d)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p - C |\eta _1-\eta _2|^p \end{aligned} \end{aligned}$$

for any $a, b \in I$ and $\eta _1, \eta _2 \ge 0$.

Proof

We may assume that $a > b$. By applying [14, Lemma 3.1] to $\eta _1$ and $\eta _2$ with

$$\begin{aligned} \varepsilon = \delta \frac{a-b}{a+d} \in (0,1), \quad \delta \in (0, 1), \end{aligned}$$

we have

$$\begin{aligned} \eta _1^p - \eta _2^p \le C \varepsilon \eta _2^p + C \varepsilon ^{1-p} |\eta _1-\eta _2|^p, \end{aligned}$$

where $C = C(p) > 0$. Thus, it follows from (A.3) and $f(a) \ge \frac{1}{1-p}(a+d)^{1-p}$ that

$$\begin{aligned} \begin{aligned} K :=&~ (a-b)^{p-1}(f(a)\eta _1^p - f(b)\eta _2^p) \\ \ge&~ (a-b)^{p-1} (f(a)-f(b)) \eta _2^p - C\delta \left( \frac{a-b}{a+d} \right) ^p \eta _2^p - C \delta ^{1-p} |\eta _1-\eta _2|^p. \end{aligned} \end{aligned}$$

By using the same arguments as in the proof of Lemma A.2, we obtain

$$\begin{aligned} \left( \frac{a-b}{a+d} \right) ^{p} \le |\log (a+d) - \log (b+d)|^p \le (a-b)^{p-1}(f(a)-f(b)). \end{aligned}$$

Therefore, we deduce that

$$\begin{aligned} K \ge (1-C\delta ) |\log (a+d) - \log (b+d)|^p \eta _2^p - C \delta ^{1-p} |\eta _1-\eta _2|^p, \end{aligned}$$

from which we conclude the lemma by taking $\delta$ sufficiently small. $\square$

We also provide the following algebraic inequality, which is similar to Lemma A.1.

Lemma A.5

Let $a, b \in \mathbb {R}$, $\eta _1, \eta _2 \ge 0$, and assume that $\gamma$ satisfies (5.3). Let $\tau = \gamma /(p-1)$ and $q = p\gamma /(p-\gamma /(p-1))$, then

$$\begin{aligned} \begin{aligned}&|a-b|^{p-2}(a-b)(g(a_+) \eta _1^p - g(b_+) \eta _2^p) \\&\quad \ge c\, |G(a_+) \eta _1 - G(b_+)\eta _2|^p - C \max \lbrace (1+a_+)^{\gamma }, (1+b_+)^{\gamma } \rbrace \textbf{1}_{\lbrace a>0 \rbrace } \textbf{1}_{\lbrace b>0 \rbrace } |\eta _1 - \eta _2|^p, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} g(t) = \frac{1}{\tau -1}(1-(1+t)^{1-\tau }), \quad G(t) = \frac{q}{\gamma }((1+t)^{\gamma /q}-1), \end{aligned}$$

and $c, C > 0$ are constants depending only on p.

Proof

We may assume that $a>b$. The same arguments as in the proof of Lemma A.2 show that

$$\begin{aligned} (a_+-b_+)^{p-1} (g(a_+) - g(b_+)) \ge |G(a_+) - G(b_+)|^p \end{aligned}$$

(A.4)

and

$$\begin{aligned} (a_+-b_+)^{p-1} g'(a_+)^{\frac{p-1}{p}} \le |G(a_+)-G(b_+)|^{p-1}. \end{aligned}$$

The case $b \le 0$ follows from (A.4). Thus, we assume $b > 0$. Since $a-b \ge a_+ - b_+$, $(a-b)^{p-1} g(b_+) = (a_+ - b_+)^{p-1} g(b_+)$, and $g(b_+) \le 1$, we obtain

$$\begin{aligned} \begin{aligned} L&:= (a-b)^{p-1} (g(a_+) \eta _1^p - g(b_+) \eta _2^p) \\&= (a-b)^{p-1} ((g(a_+) - g(b_+)) \eta _1^p + g(b_+) (\eta _1^p - \eta _2^p)) \\&\ge (a_+-b_+)^{p-1} ((g(a_+) - g(b_+)) \eta _1^p + g(b_+) (\eta _1^p - \eta _2^p)) \\&\ge |G(a_+) - G(b_+)|^p \eta _1^p - p(a_+-b_+)^{p-1} |\eta _1-\eta _2| \max \lbrace \eta _1, \eta _2 \rbrace ^{p-1} \\&\ge |G(a_+) - G(b_+)|^p \eta _1^p - p |G(a_+)-G(b_+)|^{p-1} \\ {}&\quad \max \lbrace \eta _1, \eta _2 \rbrace ^{p-1} g'(a_+)^{-\frac{p-1}{p}} |\eta _1-\eta _2|. \end{aligned} \end{aligned}$$

By using Young’s inequality, Lemma A.3, and $G(t)^p \le C (1+t)^{\gamma }$, we conclude that

$$\begin{aligned} \begin{aligned} L&\ge |G(a_+) - G(b_+)|^p \min \lbrace \eta _1, \eta _2 \rbrace ^p \\&\quad - \frac{1}{2^{2p-1}} |G(a_+)-G(b_+)|^{p} \max \lbrace \eta _1, \eta _2 \rbrace ^{p} - C \frac{1}{g'(a_+)^{p-1}} |\eta _1-\eta _2|^p \\&\ge \frac{1}{2^{p}} |G(a_+)\eta _1 - G(b_+)\eta _2|^p - C \max \lbrace (1+a_+)^{\gamma }, (1+b_+)^{\gamma } \rbrace |\eta _1 - \eta _2|^p, \end{aligned} \end{aligned}$$

where $C = C(p) > 0$. $\square$

Let us finish the section with one more algebraic inequality.

Lemma A.6

Let $a, b > 0$ and assume that $\gamma < p$, $\gamma \ne 0$. Then,

$$\begin{aligned} \max \lbrace a, b \rbrace ^{\gamma -p} |a-b|^p \le \left( \frac{p}{|\gamma |} \right) ^p |a^{\gamma /p} - b^{\gamma /p}|^p. \end{aligned}$$

Proof

We may assume without loss of generality that $a > b$. Then,

$$\begin{aligned} |a^{\gamma /p}-b^{\gamma /p}|^p = \left| \int _b^a \frac{\gamma }{p} t^{\gamma /p-1} \,\textrm{d}t \right| ^p \ge \left| \frac{\gamma }{p} \right| ^p a^{\gamma -p} (a-b)^p, \end{aligned}$$

where we use the monotonicity of $t \mapsto t^{\gamma /p-1}$. $\square$