Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian

TL;DR

This paper establishes a new uniqueness theorem for an obstacle problem involving the fractional Laplacian, extending classical results and employing probabilistic potential theory applicable to various integro-differential operators.

Contribution

It introduces a novel proof technique based on probabilistic potential theory for the fractional Laplacian obstacle problem, broadening applicability beyond classical operators.

Findings

Proved uniqueness for the obstacle problem with fractional Laplacian

Extended classical Laplace operator results to fractional case

Developed a new probabilistic proof method

Abstract

We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-differential operators.

Equations

$$\left\{\begin{array}[]{l}\max\big{\{}-(\Delta^{\alpha/2})_{|D}u-au,u-\mathbb{I}_{D\setminus\overline{D}_{0}}\big{\}}=0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\ u>0,\quad\mbox{on}\quad D,\end{array}\right.$$

$$\mathbb{I}_{D\setminus\overline{D}_{0}}(x)=\left\{\begin{array}[]{l}1,\quad\quad\,\,\quad x\in D\setminus\overline{D}_{0},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ +\infty,\quad\quad x\in\overline{D}_{0}.\end{array}\right.$$

$${\mathcal{E}}_D(u,v)=a(u,v)-{\int }_D\tilde{v}d\nu ,v\in {\tilde{H}}^{\alpha /2}(D),$$

$${\int }_D(\eta -\tilde{u})d\nu =0,$$

$$-({\Delta }^{\alpha /2}{)}_{|D}u=au-\nu ,$$

$$u\cdot \nu =\nu ,$$

$$-({\Delta }^{\alpha /2}{)}_{|D}u=au-u\cdot \nu .$$

$$\big{(}-(\Delta^{\alpha/2})_{|D}+\nu\big{)}u=au.$$

$$|{\Gamma }_x(a)-{\Gamma }_x(b)|\le \lambda |a-b|,a,b\in {\mathbb{R}}^{d-1},$$

$$B(x,r_0)\cap D=B(x,r_0)\cap \{y:y_d>{\Gamma }_x(y_1,y_2,\dots ,y_{d-1})\}.$$

$$D({\Delta }^{\alpha /2})=\{u\in L^2({\mathbb{R}}^d;m):{\int }_{{\mathbb{R}}^d}|x{|}^{2\alpha }|\hat{u}(x){|}^{2}m(dx)<\infty \},$$

$$\widehat{{\Delta }^{\frac{\alpha }{2}}u}(x)=|x{|}^{\alpha }\hat{u}(x),x\in {\mathbb{R}}^d,$$

$$\mathcal{E}(u,v)={\int }_{{\mathbb{R}}^d}\hat{u}(x)\hat{v}(x)|x{|}^{\alpha }m(dx),u,v\in D(\mathcal{E}),$$

$$D(\mathcal{E})=\{u\in L^2({\mathbb{R}}^d;m):{\int }_{{\mathbb{R}}^d}|x{|}^{\alpha }|\hat{u}(x){|}^{2}m(dx)<\infty \}.$$

$$-{\Delta }^{\alpha /2}u(x)=c_{d,\alpha }\underset{r\searrow 0}{\lim }{\int }_{{\mathbb{R}}^d\setminus B(x,r)}\frac{u(x)-u(y)}{|x-y{|}^{d+\alpha }}dy,x\in {\mathbb{R}}^d.$$

$$\text{mbox}\mathrm{Cap}(U)=\inf \{\mathcal{E}(u,u),u\in D(\mathcal{E}),u\ge 1_{U}m\text{mbox}-a.e.\},$$

$$\text{mbox}\mathrm{Cap}(B)=\inf \{\text{mbox}\mathrm{Cap}(U):U\supset B,U\subset {\mathbb{R}}^d,U\text{mbox}open\}.$$

$$D({\mathcal{E}}_D)=\{u\in D(\mathcal{E}):\tilde{u}=0,\text{mbox}q.e.on{\mathbb{R}}^d\setminus D\},{\mathcal{E}}_D(u,v)=\mathcal{E}(u,v),u,v\in D({\mathcal{E}}_D).$$

$$D(A_D)\subset D({\mathcal{E}}_D),{\mathcal{E}}_D(u,v)=(-A_{D}u,v),u\in D(A_D),v\in D({\mathcal{E}}_D).$$

$$({\Delta }^{\alpha /2}{)}_{|D}:=A_D.$$

$$({\Delta }^{\alpha /2}{)}_{|D}u(x)={\Delta }^{\alpha /2}u(x)-{\kappa }_D(x)u(x),x\in D,$$

$${\kappa }_D(x)=c_{d,\alpha }{\int }_{{\mathbb{R}}^d\setminus D}|x-y{|}^{-d-\alpha }dy,x\in D.$$

$$T_{t}f(x)=E_{x}f(X_t):={\int }_{\mathcal{D}}f(X_t(\omega ))d{P}_x(\omega ),x\in {\mathbb{R}}^d,t\ge 0,m\text{mbox}-a.e.$$

$$R_{\beta }f(x)=E_x{\int }_0^{\infty }{e}^{-\beta t}f(X_t)dt,x\in {\mathbb{R}}^d.$$

$$T_t^{D}f(x)=E_x^{D}f(X_t):={\int }_{\mathcal{D}}f(X_t(\omega ))d{P}_x^D(\omega ),x\in D,t\ge 0,m\text{mbox}-a.e.$$

$$P_t^{D}f(x):=E_x^{D}f(X_t)=E_{x}f(X_t)1_{\{t<{\tau }_D\}},x\in D,$$

$${\tau }_D=\inf \{t>0:X_t\in {\mathbb{R}}^d\setminus D\}.$$

$$R_{\beta }^{D}f(x):=E_x^D{\int }_0^{\infty }{e}^{-\beta t}f(X_t)dt=E_x{\int }_0^{{\tau }_D}{e}^{-\beta t}f(X_t)dt,x\in D.$$

$$R_{\beta }f(x)={\int }_{{\mathbb{R}}^d}{G}_{\beta }(x,y)f(y)dyx\in {\mathbb{R}}^d.$$

$$R_{\beta }^{D}f(x)={\int }_D{G}_{D,\beta }(x,y)f(y)dyx\in D.$$

Full text

††footnotetext: Mathematics Subject Classification: Primary 35R35; Secondary 35R11††footnotetext: Keywords: Dirichlet fractional Laplacian, obstacle problem, logistic equation, intrinsic ultracontractivity.

1 Introduction

Let $D\subset{\mathbb{R}}^{d}$, $d\geq 2$, be a bounded Lipschitz domain, $D_{0}\subset D$ be a bounded Dirichlet regular domain (with respect to the fractional Laplacian). For $\alpha\in(0,2)$, we denote by $(\Delta^{\alpha/2})_{|D}$ the Dirichlet fractional Laplacian on $D$ with zero exterior condition (see Section 2 for details). In case $\alpha=2$, by $(\Delta^{\alpha/2})_{|D}$ we mean the classical Laplace operator $\Delta_{D}$ on $D$ with zero boundary condition. In the present paper, we prove a uniqueness result for the following obstacle problem:

[TABLE]

where $a$ is a positive constant, and

[TABLE]

Problem of this type arises in the study of asymptotic behaviour, as $p\rightarrow\infty$, of logistic type equations (see [11, 29] for the case $\alpha=2$ and [22] for equations with $\alpha\in(0,2)$). Problem (1.1) with $\alpha=2$ also arises as the limit of some predator-prey models (see [9, 12]).

From [11] (in case $\alpha=2$) and [22] (in case $\alpha\in(0,2)$) we know that (1.1) has a solution if and only if $a\in[\lambda_{1}^{D},\lambda_{1}^{D_{0}})$, where $\lambda_{1}^{D},\lambda_{1}^{D_{0}}$ stand for the first eigenvalue of $-(\Delta^{\alpha/2})_{|D}$ and $-(\Delta^{\alpha/2})_{|D_{0}}$, respectively. In the present paper, we concentrate on the uniqueness of solutions to (1.1). This problem is quite subtle and difficult. In [10] it is investigated in case $\alpha=2$. Suppose that $D_{0},D$ are smooth and $\overline{D}_{0}\subset D$. The main result of [10] states that (1.1) has at most one solution if $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$. This result is proved by using an equivalent free boundary formulation of (1.1) and tools from the theory of variational inequalities and harmonic functions.

The method used in [10] seems to be suitable only for $\alpha=2$. To deal with nonlocal operators, we propose a new method. It allows us to prove that if $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$, then for any $\alpha\in(0,2]$ there exists at most one solution to (1.1). This generalizes the result of [10] to nonlocal operators but also slightly strengthens the known uniqueness result for $\alpha=2$ because we assume that $D_{0}\subset D$ and not that $\overline{D}_{0}\subset D$ as in [10]. Moreover, we consider less regular than in [10] domains $D_{0},D$ (see comments in [10, Remark (i)]).

In the present paper, we use a definition of solution to (1.1), which is equivalent to, but different from that of [10]. Let ${\cal E}_{D}$ denote the Dirichlet form associated with the operator $(\Delta^{\alpha/2})_{|D}$ (see Section 2). By a solution to (1.1) we mean a strictly positive function $u\in\tilde{H}^{\alpha/2}(D)$ (closure of $C_{c}^{\infty}(D)$ in $H^{\alpha/2}({\mathbb{R}}^{d})$) having the property that $u\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ $m$-a.e. and such that for some bounded smooth nonnegative Borel measure $\nu$ on $D$ (called the reaction measure for $u$) we have

[TABLE]

where $(\cdot,\cdot)$ stands for the usual inner product in $L^{2}(D;m)$ and $\tilde{v}$ is a quasi-continuous $m$-version of $v$ ($m$ stands for the Lebesgue measure on ${\mathbb{R}}^{d}$). Moreover, we require that $u$ satisfies the minimality condition, which says that for every quasi-continuous $\eta$ such that $u\leq\eta\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$ $m$-a.e. we have

[TABLE]

where $\tilde{u}$ is a quasi-continuous $m$-version of $u$. In other words, $u$ is a solution of the following equation with measure data

[TABLE]

where $\nu$ is a positive measure which acts only when $u$ touches the barrier. Our proof of uniqueness is based on the following two crucial observations. The first one is that if $u$ is a solution to (1.1), then

[TABLE]

so in fact,

[TABLE]

Equivalently,

[TABLE]

This shows that any solution to (1.1) is in fact an eigenfunction for the operator $-(\Delta^{\alpha/2})_{|D}+\nu$, i.e. the operator $-(\Delta^{\alpha/2})_{|D}$ perturbed by the smooth bounded measure $\nu$. It is well known that this operator is a self-adjoint nonnegative operator on $L^{2}(D;m)$ generating a Markov $C_{0}$-semigroup of contractions on $L^{2}(D;m)$. The second crucial observation is that $\nu$ has a compact support in $D$. This allows us to prove, by using some results of Hansen [17], that the Green function $G_{D}$ for $-(\Delta^{\alpha/2})_{|D}$ is comparable with the Green function $G^{\nu}_{D}$ for $-(\Delta^{\alpha/2})_{|D}+\nu$. These two facts, when combined with the sub and supersolutions method (generalized in the present paper to the case of our obstacle problem), give the uniqueness result.

2 Potential theory for fractional Laplacian on bounded domain

For fixed $x\in{\mathbb{R}}^{d}$ and $r>0$, we denote $B(x,r)=\{y\in{\mathbb{R}}^{d}:|y-x|<r\}$. In the whole paper, we assume that $D$ is an open bounded Lipschitz domain in ${\mathbb{R}}^{d}$ with Lipschitz character $(r_{0},\lambda)$, $r_{0},\lambda>0$, i.e. open nonempty subset of ${\mathbb{R}}^{d}$ such that: for every $x\in\partial D$ there exists a function $\Gamma_{x}:{\mathbb{R}}^{d-1}\rightarrow{\mathbb{R}}$ such that

[TABLE]

and there exists an orthogonal coordinate system $S_{x}$ such that if $y=(y_{1},y_{2},\dots,y_{d})$ in $S_{x}$, then

[TABLE]

We denote by $m$ the Lebesgue measure on ${\mathbb{R}}^{d}$. By ${\cal B}({\mathbb{R}}^{d})$ (resp. ${\cal B}(D))$, we denote the set of Borel measurable functions on ${\mathbb{R}}^{d}$ (on $D$) with values in ${\mathbb{R}}\cup\{-\infty\}\cup\{\infty\}$. ${\cal B}_{b}(D)$ is the subset of ${\cal B}(D)$ consisting of all bounded functions.

2.1 Dirichlet fractional Laplacian

Let $\alpha\in(0,2)$. To define the Dirichlet fractional Laplacian on $D$, we first set

[TABLE]

and for $u\in D(\Delta^{\alpha/2})$, we set

[TABLE]

where $\hat{u}$ stands for the Fourier transform of $u$. Let us consider the form $({\cal E},D({\cal E}))$ defined as

[TABLE]

where

[TABLE]

By [15, Example 1.4.1], $({\cal E},D({\cal E}))$ is a regular symmetric Dirichlet form on $L^{2}({\mathbb{R}}^{d};m)$. In the language of Sobolev spaces, $D({\cal E})=H^{\frac{\alpha}{2}}({\mathbb{R}}^{d})$ (see [26, page 76]). When $u\in C^{\infty}_{c}({\mathbb{R}}^{d})$ the more explicit formula for $-\Delta^{\alpha/2}u$ is known (see e.g. [15, Example 1.4.1]):

[TABLE]

The capacity Cap$:2^{{\mathbb{R}}^{d}}\rightarrow{\mathbb{R}}^{+}\cup\{\infty\}$ associated with $({\cal E},D({\cal E}))$ is defined as follows: for an arbitrary open set $U\subset{\mathbb{R}}^{d}$, we set

[TABLE]

and then, for an arbitrary $B\subset{\mathbb{R}}^{d}$, we set

[TABLE]

We say that some property holds ${\cal E}$-quasi everywhere (${\cal E}$-q.e. in abbreviation) if it holds except possibly a set of capacity Cap zero. Such sets shall be called ${\cal E}$-exceptional.

Recall that a function $u$ on ${\mathbb{R}}^{d}$ is called ${\cal E}$-quasi-continuous if for every $\varepsilon>0$ there exists a closed set $F_{\varepsilon}\subset{\mathbb{R}}^{d}$ such that Cap$({\mathbb{R}}^{d}\setminus F_{\varepsilon})\leq\varepsilon$ and $u_{|F_{\varepsilon}}$ is continuous. By [15, Theorem 2.1.3], each $u\in D({\cal E})$ admits an ${\cal E}$-quasi-continuous $m$-version. In what follows for given function $u\in{\cal B}({\mathbb{R}}^{d})$, we denote by $\tilde{u}$ its ${\cal E}$-quasi-continuous $m$-version (whenever it exists).

We denote by $({\cal E}_{D},D({\cal E}_{D}))$ the part of $({\cal E},D({\cal E}))$ on $D$. Recall that

[TABLE]

By [15, Theorem 4.4.3], $({\cal E}_{D},D({\cal E}_{D}))$ is a regular symmetric Dirichlet form on $L^{2}(D;m)$. Therefore, by [15, Sections 1.3,1.4], there exists a unique self-adjoint nonpositive definite operator $(A_{D},D(A_{D}))$ on $L^{2}(D;m)$ such that

[TABLE]

The operator $(A_{D},D(A_{D}))$ is called the Dirichlet fractional Laplacian. We put

[TABLE]

By [15, Theorem 4.4.3], $C_{c}^{\infty}(D)$ is a dense subspace of $D({\cal E}_{D})$ in the norm determined by ${\cal E}$. Therefore $D({\cal E}_{D})=\tilde{H}^{\alpha/2}(D)$ (see [26, page 77]). For $u\in C_{c}^{\infty}(D)$,

[TABLE]

where

[TABLE]

It is worth noting here that $A_{D}\neq-(-\Delta_{D})^{\alpha/2}$, where $\Delta_{D}$ is the Laplace operator with zero Dirichlet boundary condition on $D$. The latter operator is called the fractional Dirichlet Laplacian.

By replacing $({\cal E},D({\cal E}))$ by $({\cal E}_{D},D({\cal E}_{D}))$ and ${\mathbb{R}}^{d}$ by $D$ in (2.1) and (2.2), we define the capacity CapD associated with ${\cal E}_{D}$, and then we define the notions of ${\cal E}_{D}$-exceptional sets and ${\cal E}_{D}$-quasi-continuity. By [15, Theorem 4.4.4], Cap and CapD are strongly equivalent on $D$. Therefore, without ambiguity, we may write q.e., exceptional or quasi-continuous instead of ${\cal E}$-q.e., ${\cal E}_{D}$-q.e. ${\cal E}$-exceptional, ${\cal E}_{D}$-exceptional or ${\cal E}$-quasi-continuous, ${\cal E}_{D}$-quasi-continuous.

2.2 Green functions and transition functions

We denote by $(J_{\beta})_{\beta>0}$, $(T_{t})_{t\geq 0}$ the resolvent and the $C_{0}$-semigroup of contractions generated by $A:=\Delta^{\alpha/2}$, respectively, and by $(J^{D}_{\beta})_{\beta>0}$, $(T^{D}_{t})_{t\geq 0}$ the resolvent and the $C_{0}$-semigroup of contractions generated by $(\Delta^{\alpha/2})_{|D}$, respectively.

Let $\partial$ be a one-point compactification of $D$. Let $\cal D$ denote the set of all functions $\omega:[0,\infty)\to{\mathbb{R}}^{d}\cup\{\partial\}$, that are right continuous and possess the left limits for all $t\geq 0$ (cádlágs), and have the property that if $\omega(t)=\partial$, then $\omega(s)=\partial,\,s\geq t$. We equip $\mathcal{D}$ with the Skorokhod topology, see e.g. Section 12 of [1]. Define the canonical process: $X:\mathcal{D}\to\mathcal{D}$, $X_{t}(\omega):=\omega(t)$, $\omega\in\mathcal{D}$, shift operator: $\theta_{s}:\mathcal{D}\to\mathcal{D}$, $\theta_{s}(\omega)(t)=\omega(t+s)$, and life time: $\zeta:\mathcal{D}\to[0,\infty]$, $\zeta(\omega):=\inf\{t>0:X_{t}(\omega)=\partial\}$. It is well known that there exists a rotation invariant $\alpha$-stable Lévy process $\mathbb{X}=(X,(P_{x})_{x\in{\mathbb{R}}^{d}},({\cal F}_{t})_{t\geq 0})$ on ${\mathbb{R}}^{d}$ such that for every nonnegative $f\in{\cal B}({\mathbb{R}}^{d})$,

[TABLE]

We let for $\beta>0$ and $f\in{\cal B}^{+}({\mathbb{R}}^{d})$

[TABLE]

In the whole paper we adopt the convention that any function $f$ defined on a subset of ${\mathbb{R}}^{d}$ equals zero at $\partial$. By [15, Theorem 4.4.2], there exists a Hunt process $\mathbb{X}^{D}=(X,(P^{D}_{x})_{x\in D\cup\{\partial\}},({\cal F}_{t})_{t\geq 0})$ on $D\cup\{\partial\}$ such that

[TABLE]

Here $({\cal F}_{t})_{t\geq 0}$ is a filtration (non-decreasing sequence of $\sigma$-fields of subsets of $\Omega$). By [15, Theorem 4.4.1, Theorem 4.4.2], for all $t\geq 0$ and nonnegative $f\in{\cal B}(D)$,

[TABLE]

where

[TABLE]

Hence

[TABLE]

It is well known (see [15, Exercise 4.2.1, Lemma 4.2.4]) that there is $G_{\beta}\in{\cal B}({\mathbb{R}}^{d})\times{\cal B}({\mathbb{R}}^{d})$ (called the $\beta$-Green function) such that for every $f\in{\cal B}^{+}({\mathbb{R}}^{d})$,

[TABLE]

Similarly, there is $G_{D,\beta}\in{\cal B}(D)\times{\cal B}(D)$ (called the $\beta$-Green function for $D$) such that for every $f\in{\cal B}^{+}(D)$,

[TABLE]

Given a nonnegative Borel measure $\mu$ on $D$, we set

[TABLE]

Note that $R^{D}_{\beta}f=J^{D}_{\beta}f$ $m$-a.e. for every $f\in L^{2}(D;m)$. Moreover, by [15, Theorem 4.2.3], $J^{D}_{\beta}f\in D({\cal E}_{D})$ and $\widetilde{J^{D}_{\beta}f}=R^{D}_{\beta}f$ q.e. for $f\in L^{2}(D;m)$. By [15, Exercise 4.2.1], there exists a transition function $p_{D}:\mathbb{R}^{+}\times D\times D\rightarrow{\mathbb{R}}^{+}$ such that for every $f\in{\cal B}^{+}(D)$,

[TABLE]

It is well known (see e.g. [16]) that $p_{D}(t,x,y)$ is finite and strictly positive for $x,y\in D,t>0$. Given a nonnegative Borel measure $\mu$ on $D$, we set

[TABLE]

By [15, Theorem 4.2.3], $T^{D}_{t}f\in D({\cal E})$ and $\widetilde{T^{D}_{t}f}=P^{D}_{t}f$ q.e. for all $f\in L^{2}(D;m)$ and $t>0$. Moreover,

[TABLE]

It follows that we can set

[TABLE]

It is well known that $(\Delta^{\alpha/2})_{|D}:D((\Delta^{\alpha/2})_{|D})\rightarrow L^{2}(D;m)$ is a bijection. From the above definition of $G_{D}$ it follows that for every $f\in L^{2}(D;m)$,

[TABLE]

By [15, Lemma 2.2.11], $J^{D}f\in D({\cal E}_{D})$ and $\widetilde{J^{D}f}=R^{D}f$ q.e. for every $f\in L^{2}(D;m)$. For a nonnegative Borel measure $\mu$ on $D$, we set

[TABLE]

Recall that a positive Borel measurable function $u$ on $D$ is called excessive if

[TABLE]

By [15, Lemma 4.2.4], for every positive Borel measure $\mu$, $R^{D}\mu$ is an excessive function.

2.3 Smooth measures

An increasing sequence $\{F_{n}\}$ of closed subsets of ${\mathbb{R}}^{d}$ (resp. $D$) is called a generalized ${\cal E}$-nest (resp. generalized ${\cal E}_{D}$-nest) if for every compact $K\subset{\mathbb{R}}^{d}$ (resp. $K\subset D$), Cap$(K\setminus F_{n})\rightarrow 0$ (resp. Cap${}_{D}(K\setminus F_{n})\rightarrow 0$) as $n\rightarrow\infty$. A Borel signed measure $\mu$ on ${\mathbb{R}}^{d}$ (resp. $D$) is called ${\cal E}$-smooth (resp. ${\cal E}_{D}$-smooth) if it charges no set of capacity Cap (resp. CapD) zero and there exists a generalized ${\cal E}$-nest (resp. ${\cal E}_{D}$-nest) such that $|\mu|(F_{n})<\infty$, $n\geq 1$, where $|\mu|$ denotes the variation of $\mu$. Note that if a Borel measure $\mu$ on $D$ is bounded, then it is ${\cal E}$-smooth if and only if it is ${\cal E}_{D}$-smooth. This follows from the fact that Cap and CapD are equivalent.

We denote by ${\cal M}_{0}$ (resp. ${\cal M}_{0}(D)$) the set of all ${\cal E}$-smooth (resp. ${\cal E}_{D}$-smooth) measures on ${\mathbb{R}}^{d}$ (resp. $D$). We also set

[TABLE]

By [23, Proposition 5.13],

[TABLE]

Also note that if $\mu$ is a Borel measure such that $R^{D}|\mu|<\infty$ on $D$, then $\mu\in{\cal M}_{0}(D)$ (see [15, Exercise 4.2.2]). We say that an ${\cal E}_{D}$-smooth measure $\mu$ belongs to the class $S_{0}(D)$ (called the class of measures of finite energy integral) if there exists $c>0$ such that

[TABLE]

By [15, Theorem 2.2.2], for any positive $\mu\in S_{0}(D)$ and $\eta\in D({\cal E}_{D})$,

[TABLE]

2.4 Probabilistic potential theory

Recall that a bounded subset $D$ of ${\mathbb{R}}^{d}$ is called Dirichlet regular (with respect to the fractional Laplacian) if for every $x\in{\mathbb{R}}^{d}\setminus D$,

[TABLE]

By [2, Example VII.3.4.3], each bounded Lipschitz domain is Dirichlet regular. From this and [6] it follows that $(P^{D}_{t})_{t\geq 0}$ is doubly Feller. This means that it is strongly Feller, i.e. $P^{D}_{t}f\in C_{b}(D)$ for every $f\in\mathcal{B}_{b}(D)$, and it is Fellerian, i.e. $P^{D}_{t}f\in C_{0}(D)$ for every $f\in C_{0}(D)$.

By [15, Theorem 5.1.4], there is a one-to-one correspondence between nonnegative ${\cal E}_{D}$-smooth measures and positive continuous additive functionals (PCAFs) of $\mathbb{X}^{D}$ (see [15, Section 5.1]). By $A^{\mu}$ we denote the unique PCAF of $\mathbb{X}^{D}$ associated with $\mu\in{\cal M}_{0}(D)$. By [15, Theorem 5.1.3] and (2.6),

[TABLE]

for q.e. $x\in D$. For a signed measure $\mu\in{\cal M}_{0}(D)$ having a decomopsition $\mu=\mu^{+}-\mu^{-}$ into a positive and negative part, we set $A^{\mu}=A^{\mu^{+}}-A^{\mu^{-}}$. Note also that if $\mu\in{\cal M}_{0,b}^{+}(D)$ and $\sup_{x\in D}R^{D}\mu<\infty$, then there exists a strict PCAF $A^{\mu}$ of $\mathbb{X}^{D}$ such that (2.9) holds for every $x\in D$ (see [15, Theorem 5.1.6]).

Recall that a càdlàg process $M$ adapted to $({\cal F}_{t})_{t\geq 0}$ is called a martingale additive functional (MAF) of $\mathbb{X}^{D}$ iff $M_{0}=0$, $M$ is an additive functional, $E^{D}_{x}|M_{t}|<\infty$ q.e. $t\geq 0$, and $E^{D}_{x}M_{t}=0$ q.e. $t\geq 0$. A MAF $M$ of $\mathbb{X}^{D}$ is called uniformly integrable iff for q.e. $x\in D$, and every stopping time $\alpha\leq\tau_{D}$,

[TABLE]

The following very useful result will be frequently used in the sequel.

Lemma 2.1**.**

Assume that $u\in{\cal B}(D)$ and $\mu\in\mathbb{M}_{0}(D)$. Then

[TABLE]

for q.e. $x\in D$ if and only if there exists a uniformly integrable MAF $M$ of $\mathbb{X}^{D}$ such that for q.e. $x\in D$,

[TABLE]

Moreover, if $\sup_{x\in D}R^{D}|\mu|(x)<\infty$, then ”for q.e. $x\in D$” may be replaced by ”for every $x\in D$” with $A^{\mu}$ being a strict PCAF of $\mathbb{X}^{D}$.

Proof.

First suppose that (2.11) is satisfied. Taking the expectation with respect to $P_{x}$ of both sides of (2.11) with $t=0$, and then using (2.9) with $f=1$ we get (2.10). Now suppose that (2.10) holds. Let $N$ be an exceptional set such that (2.10) holds for $x\in D\setminus N$. By [15, Theorem 4.1.1], we may assume that $P^{D}_{x}(X_{t}\notin N,\,t\geq 0)=1$ for every $x\in D\setminus N$. Hence, by (2.10), additivity of $A^{\mu}$, and strong Markov property, for every $x\in D\setminus N$ and every stopping time $\sigma\leq\tau_{D}$, we have

[TABLE]

Set

[TABLE]

By quasi-continuity of $u$ and [15, Theorem 4.6.1], $M$ is a càdlàg process. Clearly, $M$ is an additive functional and $M_{0}=0$. By (2.12), $M$ is a uniformly integrable martingale on $[0,\tau_{D}]$ under measure $P^{D}_{x}$. Equivalently, by (2.5), $M$ is a uniformly integrable martingale on $[0,\tau_{D}]$ under measure $P_{x}$ for $x\in D\setminus N$. If $\sup_{x\in D}R^{D}|\mu|<\infty$, then the above argument holds true with $N=\emptyset$ with one exception that instead of [15, Theorem 4.6.1] one need to apply [3, Theorem III.5.7]. ∎

2.5 Feynman-Kac formula

For a nonnegative measure $\mu\in{\cal M}_{0,b}(D)$ such that $\sup_{x\in D}R^{D}\mu(x)<\infty$, we define the perturbation of the form $({\cal E}_{D},D({\cal E}_{D}))$ by $\mu$ as follows:

[TABLE]

By [15, Theorem 6.1.2], $({\cal E}^{\mu}_{D},D({\cal E}^{\mu}_{D}))$ is a regular symmetric Dirichlet form on $L^{2}(D;m)$. Let $(T^{D,\mu}_{t})_{t>0}$ be the semigroup generated by ${\cal E}^{\mu}_{D}$ and $A^{\mu}_{D}$ be its generator. We let

[TABLE]

By [15] there exists a Hunt process $\mathbb{X}^{D,\mu}=(X,(P^{D,\mu}_{x})_{x\in D\cup\{\partial\}},({\cal F}_{t})_{t\geq 0})$ on $D\cup\{\partial\}$ such that

[TABLE]

We denote by $G^{\mu}_{D}$ the Green function for $-(\Delta^{\alpha/2})_{|D}+\nu$ (see [15, Exercise 6.1.1]). By [15, Theorem 6.1.1], we have the Feynman-Kac representation formula

[TABLE]

3 Integral supersolutions of the obstacle problem

Let $h:D\rightarrow\mathbb{R}\cup\{\infty\}$ be a measurable strictly positive function and $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Consider the following obstacle problem:

[TABLE]

In this section, we give a definition of an integral solution to (3.1). We also recall widely used in the literature definition of weak solutions via variational inequalities. We show that both definitions are equivalent under regularity assumptions on $u$. However, we prefer to work on integral solutions. The advantage of this notion is that its formulation does not require any regularity from solutions (merely integrability), and it allows us introduce the notion of super and subsolutions. Moreover, by using integral solutions, we may apply probabilistic potential theory, which is very useful in many reasonings. We then show that minimum of two integral supersolutions of (3.1) is again an integral supersolution. This property will be one of the crucial ingredients in the proof of uniqueness of (1.1).

Definition 3.1**.**

We say that a quasi-continuous function $u\in L^{1}(D;m)$ is an integral solution to (3.1) if there exists a nonnegative $\nu\in\mathbb{M}_{0}(D)$ (we call it the reaction measure for $u$) such that

1. (a)

$0<u\leq h$ $m$-a.e. and $R^{D}|f(u)|<\infty$ q.e., 2. (b)

For q.e. $x\in D$,

[TABLE] 3. (c)

For any quasi-continuous function $\eta$ on $D$ such that $u\leq\eta\leq h$ $m$-a.e.,

[TABLE]

Remark 3.2**.**

(i) Take any quasi-continuous function $\hat{h}$ such that $u\leq\hat{h}\leq h$ $m$-a.e. Observe that $u$ is an integral solution to (3.1) if and only if $u$ is an integral solution to (3.1) with $h$ replaced by $\hat{h}$.

(ii) If $h\in D({\cal E}_{D})$ is quasi-continuous, then $u$ is an integral solution to (3.1) if and only if (a), (b) are satisfied and $\int_{D}(h-u)\,d\nu=0$.

(iii) Recall that by [15, Lemma 2.1.4], if $u\leq\eta,\,m$-a.e. and $u,\eta$ are quasi-continuous, then $u\leq\eta$ q.e.

(iv) Note that by [15, Theorem 4.6.1,Theorem A.2.7], functions $R^{D}f^{+}(u)$, $R^{D}f^{-}(u)$, $R^{D}\nu$ are quasi-continuous as all the mentioned functions are excessive and finite q.e. So, if (b) holds $m$-a.e., then by (iii), (b) holds q.e. Clearly, $R^{D}f^{+}(u),R^{D}f^{-}(u)$ does not depend on the version of $u$. Therefore, if we assume in the Definition 3.1 that $u$ is merely a measurable function such that (a), (c) hold, and (b) holds $m$-a.e., then there exists a version $\tilde{u}$ of $u$ such that $\tilde{u}$ is quasi-continuous and (b) holds q.e. with $u$ replaced by $\tilde{u}$: this version is given by $\tilde{u}(x):=R^{D}f(u)(x)-R^{D}\nu(x)$ if $R^{D}|f|(u)(x)+R^{D}\nu(x)<\infty$ and zero otherwise. Clearly, $\tilde{u}=R^{D}f(\tilde{u})-R^{D}\nu$ q.e.

Recall that $(\cdot,\cdot)$ stands for the usual inner product in $L^{2}(D;m)$.

Definition 3.3**.**

We say that a quasi-continuous function $u\in D({\cal E}_{D})$ is a weak solution to (3.1) if

1. (a)

$0<u\leq h$ $m$-a.e. and $f(u)\in L^{2}(D;m)$, 2. (b)

For every $\eta\in D({\cal E}_{D})$ such that $\eta\leq h$ $m$-a.e.,

[TABLE]

Proposition 3.4**.**

Let $u\in D({\cal E}_{D})$ and $f(u)\in L^{2}(D;m)$. Then $u$ is an integral solution to (3.1) iff $u$ is a weak solution to (3.1).

Proof.

Let $u$ be an integral solution to (3.1). Since $f(u)\in L^{2}(D;m)$, $R^{D}f(u)\in D((\Delta^{\alpha/2})_{|D})\subset D({\cal E}_{D})$. Thus, $\nu\in S_{0}(D)$. Therefore, by (2.8) and (c) of Definition 3.1, for every $\eta\in D({\cal E}_{D})$ such that $\eta\leq h\,m$-a.e.,

[TABLE]

Now, let $u$ be a weak solution to (3.1). By [25, Theorem 5.2, Chapter 3] $u_{n}\rightarrow u$ weakly in ${\cal E}_{D}$, where $u_{n}\in D({\cal E}_{D})$ is a unique solution to the following variational equality

[TABLE]

Clearly, $u_{n}=R^{D}f(u)-R^{D}\nu_{n}$, $m$-a.e., where $\nu_{n}=n(u_{n}-h)^{+}\cdot m$. By [21, Theorem 3.8], $u_{n}\searrow w$, where $w$ is an integral solution to

[TABLE]

Since $u=w,\,m$-a.e., we get the desired result (cf. Remark 3.2(iv)). ∎

Definition 3.5**.**

We say that a quasi-continuous function $u\in L^{1}(D;m)$ is an integral supersolution (resp. integral subsolution) to (3.1) if there exists a nonnegative measure $\nu\in\mathbb{M}_{0}(D)$ and a nonnegative (resp. nonpositive) measure $\mu\in\mathbb{M}_{0}(D)$ such that conditions (a) and (c) of Definition 3.1 are satisfied, and moreover, for q.e. $x\in{\mathbb{R}}^{d}$,

[TABLE]

Proposition 3.6**.**

*If $u$ is an integral supersolution and an integral subsolution to (3.1), then $u$ is an integral solution to *(3.1).

Proof.

By assumptions and Definition 3.5, there exist nonnegative measures $\nu_{1},\nu_{2},\mu_{1},\mu_{2}\in\mathbb{M}_{0}(D)$ such that

[TABLE]

[TABLE]

and $\int_{D}(\eta-u)\,d\nu_{1}=\int_{D}(\eta-u)\,d\nu_{2}=0$ for every quasi-continuous $\eta$ on $D$ such that $u\leq\eta\leq h$ $m$-a.e. Thus,

[TABLE]

for q.e. $x\in D$. From this, we conclude that $\nu_{1}-\nu_{2}=\mu_{1}+\mu_{2}$. Therefore, there exist nonnegative $\alpha,\beta,\gamma\in{\cal B}(D)$ such that $\alpha+\beta=1,\gamma\leq 1$ and

[TABLE]

Consequently, for q.e $x\in D$,

[TABLE]

Since $\int_{D}(\eta-u)(\beta+\alpha\gamma)\,d\nu_{1}=0$ for every quasi-continuous $\eta$ on $D$ such that $u\leq\eta\leq h$ $m$-a.e., we see that $u$ is an integral solution to (3.1). ∎

Proposition 3.7**.**

*Let $u_{1},u_{2}$ be integral supersolutions to (3.1). Then $u_{1}\wedge u_{2}$ is a an integral supersolution to *(3.1).

Proof.

By the definition of a an integral supersolution to (3.1) and Lemma 2.1, for q.e. $x\in D$, we have

[TABLE]

$i=1,2$, for some nonnegative $\nu_{1},\nu_{2},\mu_{1},\mu_{2}\in\mathbb{M}_{0}(D)$ and some uniformly integrable MAFs $M^{1},M^{2}$ of $\mathbb{X}^{D}$. By the Tanaka-Meyer formula (see, e.g., [27, IV.Theorem 70]) applied to $u_{1}(X)-(u_{1}(X)-u_{2}(X))^{+}=u_{1}(X)\wedge u_{2}(X)$, there exists an increasing càdlàg process $C$ with $C_{0}=0$ such that for q.e. $x\in D$,

[TABLE]

$t\in[0,\tau_{D}],\,P_{x}\mbox{-a.s.}$ From the above formula, we get, in particular, that $C$ is a positive AF of $\mathbb{X}^{D}$. By [15, Theorem A.3.16] there exists a positive AF $C^{p}$ which is the dual predictable projection of $C$ under measure $P^{D}_{x}$ for q.e. $x\in D$. Since $\mathbb{X}^{D}$ is a Hunt process it is, by the very definition, quasi-left continuous, so it has only totally inaccessible jumps. Therefore, since $u_{1}\wedge u_{2}$ is quasi-continuous, process $(u_{1}\wedge u_{2})(X)$ has only totally inaccessible jumps under measure $P^{D}_{x}$ for q.e. $x\in D$ (see [15, Theorem 4.2.2]). Moreover, by [7, Proposition 2, Proposition 4] every local $({\cal F}_{t})_{t\geq 0}$-martingale has only totally inaccessible jumps. By the definition of dual predictable projection, $C^{p}-C$ is an $({\cal F}_{t})_{t\geq 0}$-martingale. Therefore, by (3), $C^{p}$ has only totally inaccessible jumps under measure $P^{D}_{x}$ for q.e. $x\in D$. However, $C^{p}$ is predictable. Consequently, $C^{p}$ is continuous. So, $C^{p}$ is a PCAF of $\mathbb{X}^{D}$. Hence $C^{p}=A^{\beta}$ for some nonnegative $\beta\in\mathbb{M}_{0}(D)$ (cf. Section 2.4). Define $\mu=\mathbf{1}_{u_{1}>u_{2}}\cdot\mu_{2}+\mathbf{1}_{u_{2}\geq u_{1}}\cdot\mu_{1}+\beta$ and $\nu=\mathbf{1}_{u_{1}>u_{2}}\cdot\nu_{2}+\mathbf{1}_{u_{2}\geq u_{1}}\cdot\nu_{1}$. By Lemma 2.1, for q.e $x\in D$,

[TABLE]

Let $\eta$ be a quasi-continuous function such that $u_{1}\wedge u_{2}\leq\eta\leq h$ $m$-a.e. Observe that

[TABLE]

Clearly, $u_{2}\leq\eta\vee u_{2}\leq h$ $m$-a.e., and $u_{1}\leq\eta\vee u_{1}\leq h$ $m$-a.e. So, by condition (c) of Definition 3.1 applied to $(u_{1},\nu_{1})$ and $(u_{2},\nu_{2})$, we get

[TABLE]

Thus, $\int_{D}(\eta-u_{1}\wedge u_{2})\,d\nu=0$. This combined with (3.4) implies that $u_{1}\wedge u_{2}$ is an integral supersolution to (3.1). ∎

Proposition 3.8**.**

Assume that $f$ is nondecreasing. Let $\underline{u}$ (resp. $\overline{u}$) be a bounded integral subsolution (resp. supersolution) to (3.1) and $\underline{u}\leq\overline{u}$ q.e. Then there exists an integral solution $u$ to (3.1) such that $\underline{u}\leq u\leq\overline{u}$ q.e.

Proof.

Let $u_{0}=\underline{u}$. We first show that for each $n\geq 1$ there exists an integral solution $u_{n}$ to the problem

[TABLE]

and

[TABLE]

where $\nu_{n}$ is the reaction measure for $u_{n}$. Indeed, the existence of $u_{1}$ follows from [21, Theorem 3.8]. By [21, Proposition 3.12], $\underline{u}\leq u_{1}\leq\overline{u}$ q.e. In particular, $R^{D}|f(u_{1})|<\infty$ q.e. Hence, by [21, Theorem 3.8] again, there exists an integral solution $u_{2}$ to (3.5), and by [21, Proposition 3.12] again, $\underline{u}\leq u_{2}\leq\overline{u}$ q.e. and $u_{1}\leq u_{2}$ q.e. Continuing in this fashion, we get the existence of $\{u_{n}\}$ having the desired properties. Moreover, by [21, Proposition 3.12], $\nu_{n}\leq\nu_{n+1},\,n\geq 1$. Let $u=\sup_{n\geq 1}u_{n}$, and

[TABLE]

Observe that

[TABLE]

Therefore, by [23, Lemma 5.4],

[TABLE]

The right-hand side is finite since $\underline{u},\overline{u}$ are bounded and $D$ is bounded. By the Vitali-Hahn-Saks theorem $\nu$ is a bounded Borel measure and

[TABLE]

By (2.7), $R^{D}\nu<\infty$ q.e., and $R^{D}|f(\overline{u})|+R^{D}|f(\underline{u})|<\infty$ q.e. So, by (3.6) and the Lebesgue dominated convergence theorem,

[TABLE]

Thus,

[TABLE]

By Remark 3.2(iv), $u$ is quasi-continuous. Clearly, $u\geq h,\,m$-a.e. Let $\eta$ be an arbitrary quasi-continuous function such that $u\leq\eta\leq h$ $m$-a.e. Then, by the minimality condition (c) of Definition 3.1 applied to $(u_{n},\nu_{n})$, and by (3.6),

[TABLE]

Thus, $u$ is an integral solution to (3.1). ∎

4 Uniqueness result

As in Sections 2 and 3, we assume that $D$ is a bounded Lipschitz domain in ${\mathbb{R}}^{d}$ ($d\geq 2$) and $D_{0}\subset D$ is a bounded Dirichlet regular domain. We assume that $a>\lambda_{1}^{D}$, where $\lambda_{1}^{D}$ is the first eigenvalue for the operator $-(\Delta^{\alpha/2})_{|D}$. By $\varphi_{1}^{D}$ we denote the ground state for $-(\Delta^{\alpha/2})_{|D}$, i.e. a unique strictly positive function $\varphi_{1}^{D}\in D({\cal E}_{D})$ such that $\|\varphi_{1}^{D}\|_{L^{2}}=1$ and

[TABLE]

It is well known that $\lambda_{1}^{D}>0$, and by the regularity of $D$, $\varphi_{1}^{D}\in C_{0}(D)$.

To prove a uniqueness result for (1.1), we shall need some regularity results for integral solutions to (1.1), and the result which compare the Green function $G_{D}$ for $(\Delta^{\alpha/2})_{|D}$ with the Green function $G^{\nu}_{D}$ for $(\Delta^{\alpha/2})_{|D}-\nu$.

It is well known (see e.g. [16]) that the semigroup $(P^{D}_{t})_{t>0}$ is intrinsically ultracontractive, which implies that for every $t>0$ there exist constants $c_{1}(t),\,c_{2}(t)>0$ such that

[TABLE]

Proposition 4.1**.**

Let $u$ be an integral solution to (1.1) and $\nu$ be the reaction measure for $u$. Then $u\cdot\nu=\nu$.

Proof.

Set $w=R^{D}u,\,w_{n}=R^{D}(u\wedge n)$, and

[TABLE]

Clearly, $w_{n}\in D({\cal E}_{D})$ since $u\wedge n\in L^{2}(D;m)$. By Dynkin’s formula (see [15, (4.4.2)]),

[TABLE]

By [15, Lemma 4.3.1], $E_{\cdot}[w_{n}(X_{\tau_{D_{0}}})]$ is an excessive function for $n\geq 1$. So, by [3, Proposition II.2.2], $E_{\cdot}[w(X_{\tau_{D_{0}}})]$ is an excessive function. By the definition of an integral solution, $u\in L^{1}(D;m)$. Therefore, by (2.7), $R^{D}u$ is finite q.e. So, by (4.3), $E_{\cdot}[w(X_{\tau_{D_{0}}})]$ is finite q.e. Consequently, $\rho$ as a difference of excessive functions finite q.e. is quasi-continuous (see comments in Remark 3.2(iv)). Thus, $h$ is quasi-continuous. By regularity of $D_{0}$, $\rho(x)=0,\,x\in D\setminus D_{0}$. Moreover,

[TABLE]

By Dynkin’s formula again

[TABLE]

From this and the fact that $\rho=0$ on $D\setminus D_{0}$, we get that the $n$-th power of $h$ satisfies

[TABLE]

Hence, by the definition of an integral solution to the obstacle problem,

[TABLE]

By (4.4), for every $x\in D_{0}$, $h^{n}(x)\nearrow\infty$ as $n\rightarrow\infty$. It follows from (4.5) that supp$[\nu]\subset D\setminus D_{0}$. By this and (4.5) again, for every nonnegative $\eta\in C_{c}(D)$ we have

[TABLE]

which implies the desired result. ∎

Proposition 4.2**.**

If $u$ is an integral solution to (1.1), then

1. (i)

$u(x)\leq c\|u\|_{L_{1}(D;m)}\varphi_{1}^{D}(x)$* for q.e. $x\in D$.* 2. (ii)

$u\in C_{0}(D)$. 3. (iii)

The reaction measure $\nu$ for $u$ is bounded and $\|\nu\|_{TV}\leq a\|u\|_{L^{1}(D;m)}$.

Proof.

By the definition of an integral solution to (1.1) and Proposition 4.1,

[TABLE]

for q.e $x\in D$. Using Lemma 2.1 and the integration by part formula applied to the product $e^{at-A^{\nu}_{t}}u(X_{t})$ yields

[TABLE]

for q.e. $x\in D$. Therefore, by the ultracontractivity of $(P^{D}_{t})_{t\geq 0}$, for every $t>0$,

[TABLE]

for q.e. $x\in D$, which proves (i). To prove (ii), consider the function $h$ defined by (4.2). By regularity of the set $D_{0}$ and [6], $(P^{D_{0}}_{t})_{t\geq 0}$ is doubly Feller (cf. Section 2.4). Therefore, by (i) $\rho\in C(D_{0})$. By regularity of $D_{0}$, in fact $\rho\in C_{0}(D_{0})$. Thus, $h\in C(\overline{D})$. By the proof of Proposition 4.1,

[TABLE]

So, by Remark 3.2(i), $u$ is an integral solution to (1.1) with $\mathbb{I}_{D\setminus\overline{D}_{0}}$ replaced by $h$. Therefore, $u\in C_{0}(D)$ by [30, Theorem 1], which proves (ii). By the definition of an integral solution to (1.1),

[TABLE]

for q.e. $x\in D$. From (4.6) and [23, Lemma 5.4] we get (iii). ∎

Proposition 4.3**.**

Let $u$ be an integral solution to (1.1) and $\nu$ be its reaction measure. Then

1. (i)

supp$[\nu]$* is a compact subset of $D$.* 2. (ii)

$\sup_{x\in D}R^{D}\nu(x)<\infty$.

Proof.

By Proposition 4.1, supp$[\nu]\subset\{u=1\}$, which when combined with Proposition 4.2(ii) implies (i). Assertion (ii) follows easily from (4.6), Proposition 4.2 and (2.9). ∎

Proposition 4.4**.**

Let $u$ be an integral solution to (1.1) and let $\nu$ be its reaction measure. Then $u\in D({\cal E}_{D})$, $\nu\in S_{0}(D)$ and for every $\eta\in D({\cal E}_{D})$,

[TABLE]

Proof.

By Proposition 4.2, $u\in L^{1}(D;m)$ and $\nu\in{\cal M}_{0,b}(D)$. Therefore, by [24, Theorem 3.5], $u$ is a renormalized solution to (1.1) in the sense defined in [24]. By the definition of a renormalized solution, $u\wedge k\in D({\cal E}_{D})$, $k\geq 1$, which when combined with Proposition 4.2(i) implies that $u\in D({\cal E}_{D})$. Moreover, by the definition of a renormalized solution, there exists a sequence $\nu_{k}\subset{\cal M}_{0,b}(D)$ such that $\|\nu_{k}\|_{TV}\rightarrow 0$ and for every bounded $\eta\in D({\cal E}_{D})$,

[TABLE]

Letting $k\rightarrow\infty$ yields (4.7) for every bounded $\eta\in D({\cal E}_{D})$. Applying now a simple approximation argument, we get that $\nu\in S_{0}(D)$, and (4.7) holds for any $\eta\in D({\cal E}_{D})$. ∎

Let $w$ be a strictly positive excessive function with respect to $(P^{D}_{t})_{t\geq 0}$. We say that $G_{D}$ has $w$-triangle property iff there exists $C>0$ such that

[TABLE]

The above notion was introduced in [17]. Observe that if we set

[TABLE]

then (4.8) is equivalent to

[TABLE]

Clearly, $\rho(x,y)=0$ if and only if $x=y$, and $\rho(x,y)=\rho(y,x)$. Therefore, $w$-triangle property is equivalent to the statement that $\rho$ is a quasi-metric on $D$.

Recall here that at the beginning of Section 2, we introduced $\lambda>0$ and $r_{0}>0$. We shall show that $G_{D}$ has $w$-triangle property for Lipschitz domains, where $w=\phi$ and

[TABLE]

for a fixed $x_{0}\in D$. Here $c_{d,\alpha}=\frac{\Gamma((d-\alpha)/2)}{2^{\alpha}\pi^{d/2}|\Gamma(\alpha/2)|}$. Set $\kappa=1/2\sqrt{1+\lambda^{2}}$, and fix $x_{1}\in D$ such that $|x_{0}-x_{1}|=r_{0}/4$. We let $\delta_{D}(x):=dist(x,\partial D),\,x\in D$.

Proposition 4.5**.**

Green function $G_{D}$ has $\phi$-triangle property.

Proof.

Let $x,y,z,z_{1},z_{2}\in D$. By [18, Theorem 1],

[TABLE]

for $A_{i}\in\mathfrak{B}(x,z_{i}),i=1,2$, where $c_{1},c_{2}$ depend only on $D,d,\lambda,\alpha$. Here, for $r_{i}:=\delta_{D}(x)\vee\delta_{D}(z_{i})\vee|x-z_{i}|\leq r_{0}/32$,

[TABLE]

and $\mathfrak{B}(x,z_{i})=\{x_{1}\}$ for $r_{i}>r_{0}/32$. Suppose that $|x-z|\geq|y-z|$. Set $z_{1}:=y,z_{2}:=z$. By (4.10),

[TABLE]

Observe that $|x-z_{1}|\leq 2|x-z_{2}|$. Therefore, by [18, Lemma 13], there exists $C_{1}>0$ such that $\frac{\phi(A_{1})}{\phi(A_{2})}\leq C_{1}$, where $C_{1}$ depends only on $\alpha$, $d$ and $D$. Consequently, there exists $C>0$ such that

[TABLE]

Equivalently,

[TABLE]

Hence we get easily (4.9). ∎

Let $G$ denote the Green function for ${\mathbb{R}}^{d}$ (and the operator $\Delta^{\alpha/2}$). It is well known that there is $c>0$ such that

[TABLE]

Lemma 4.6**.**

Assume that $\mu\in{\cal M}_{0,b}(D)$ is nonnegative, $K:=\mbox{supp}[\mu]$ is a compact subset of $D$ and $\sup_{x\in D}R^{D}\mu(x)<\infty$. Then $G_{D}^{\mu}\sim G_{D}$, i.e. there exist $c_{1},c_{2}>0$ such that $c_{1}G_{D}^{\mu}\leq G_{D}\leq c_{2}G_{D}^{\mu}$ on $D\times D$.

Proof.

Let $\bar{\mu}$ be an extension of $\mu$ to ${\mathbb{R}}^{d}$ defined as $\bar{\mu}(B)=\mu(D\cap B)$ for any Borel set $B\subset{\mathbb{R}}^{d}$. Let $V$ be an open set such that $K\subset V\subset\overline{V}\subset D$, and let $r=\mbox{dist}(K,\partial V)$. By [5, Theorem 1.2], there is $c>0$ such that

[TABLE]

From this and the assumptions of the lemma it follows that

[TABLE]

Next, for all $x,y\in D$,

[TABLE]

By the 3G Theorem (see [19]),

[TABLE]

This when combined with (4) and (4) shows that there exists $C>0$ such that

[TABLE]

From this we conclude that for every nonnegative Borel measure $\nu$ on $D$,

[TABLE]

Equivalently,

[TABLE]

It is well known (see [14, Theorem 17, page 230]) that each $(P^{D}_{t})_{t\geq 0}$-excessive function is an increasing limit of functions of the form $R^{D}\nu$ for some nonnegative Borel measure $\nu$. Therefore from (4.14) it follows that for every excessive function $e$,

[TABLE]

Taking $e=\phi$ (it is excessive as a minimum of excessive functions), we get

[TABLE]

From this, Proposition 4.5, and [17, Theorem 9.1] we get the desired result. ∎

Theorem 4.7**.**

Assume that $a\in(\lambda_{1}^{D},\lambda_{1}^{D_{0}})$. Then there exists at most one integral solution to (1.1).

Proof.

Let $u_{1},u_{2}$ be two integral solutions to (1.1). We divide the proof into two steps.

Step 1. We shall show that without loss of generality we may assume that $u_{1}\leq u_{2}$. Assume that whenever we know that $w,v$ are integral solutions to (1.1) such that $w\leq v$, then $w=v$. By Proposition 3.8, $u:=u_{1}\wedge u_{2}$ is a an integral supersolution to (1.1). It is clear that for a sufficiently small $c>0$, $c\varphi_{1}^{D}\leq\mathbb{I}_{D\setminus\overline{D}_{0}}$. Therefore, since

[TABLE]

and $a>\lambda_{1}^{D}$, we see that $c\varphi_{1}^{D}$ is a an integral subsolution to (1.1) (cf. (2.8)). By the definition of an integral supersolution to (1.1), there exist nonnegative measures $\mu,\nu\in\mathbb{M}_{0}(D)$ such that

[TABLE]

and $\nu$ is the reaction measure for $u$. By Proposition 4.1, the above equation may be equivalently rewritten as

[TABLE]

By Lemma 2.1 there exists a MAF $M$ such that for any $x\in D$,

[TABLE]

By the integration by parts formula applied to the product $e^{-A_{t}}u(X_{t})$, we get

[TABLE]

$P_{x}\mbox{-a.s.}$ Taking expectation $E_{x}$ of both sides of the above equation with $t=0$ yields

[TABLE]

Therefore, by (2.15),

[TABLE]

By Proposition 4.2, Proposition 4.3 and Lemma 4.6, $G^{\nu}_{D}\sim G_{D}$. This when combined with the above equation and (4.1) gives

[TABLE]

so $u\geq c\varphi_{1}^{D}$ for some $c>0$. Hence, for a sufficiently small $c>0$, $c\varphi_{1}^{D}$ is a an integral subsolution to (1.1) such that $c\varphi_{1}^{D}\leq u$. By Proposition 3.8, there exists an integral solution $v$ to (1.1) such that $c\varphi_{1}^{D}\leq v\leq u$. Hence $v\leq u_{1}$ and $v\leq u_{2}$. By the assumption of Step 1, $u_{1}=v=u_{2}$.

Step 2. Assume that $u_{1}\leq u_{2}$. Let $\nu_{1},\nu_{2}$ be the reaction measures for $u_{1}$ and $u_{2}$, respectively. Then, by Proposition 4.4,

[TABLE]

Hence

[TABLE]

From this and Proposition 4.1 we conclude that

[TABLE]

We can regard $u_{i}$ as a solution to the following obstacle problem

[TABLE]

where $g_{i}=au_{i},\,i=1,2$. Since $u_{1}\leq u_{2}$, we have $g_{1}\leq g_{2}$. Applying [21, Proposition 3.12] yields $d\nu_{1}\leq d\nu_{2}$. This when combined with (4.17) and the fact that $u_{1},u_{2}$ are strictly positive implies that $\nu_{1}=\nu_{2}$. Therefore, we have

[TABLE]

Thus,

[TABLE]

We have assumed that $u_{2}-u_{1}\geq 0$. Striving for a contradiction, suppose that $(u_{2}-u_{1})(x)>0$ for some $x\in D$. Then continuity of $u_{1},u_{2}$ and (4.18) would imply that $u_{2}-u_{1}$ is strictly positive on $D$, in contradiction with the fact that $a>\lambda_{1}^{D}$. ∎

Remark 4.8**.**

All the results of the paper hold for $\alpha=2$. In case $\alpha=2$ the proofs run in the same way as in case $\alpha\in(0,2)$, the only difference being in the proof of Proposition 4.5 and Lemma 4.6. In case $d\geq 3$, Proposition 4.5 follows from [28, Theorem 3.1], and in case $d=2$ it follows from [17, Corollary 9.6] - we need however $D$ to be finitely connected. As to the proof of Lemma 4.6, in case $d=2$, to get (4.13), we use [8, Theorem 6.24], and in case $d\geq 3$, we use [8, Theorem 6.5]. Instead of [5, Theorem 1.2], we use [8, Lemma 6.7].

Ultracontractivity of the semigroup generated by $\Delta_{D}$ follows from [13, Theorem 9.3]. That Lipschitz bounded domains are Dirichlet regular is well known (see, e.g., [2, page 350]).

Acknowledgements

This work was supported by Polish National Science Centre (Grant No. 2017/25/B/ST1/00878).

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