关于积分微分方程下解的Hopf型引理
Hopf type lemmas for subsolutions of integro-differential equations
摘要：
本文对一类Lévy型积分微分算子$A$，有界可测函数$c:D\to[0,+\infty)$和整环$D\subset \BR^d$，证明了方程$-Au+cu=0$在$D$中弱下解的Hopf引理的推广.更准确地说，我们证明，如果{ \em强极大值原理}（SMP）持有$A$，那么存在一个Borel函数$ψ:D\to(0,+\infty)$，只取决于系数的运营商$A$，$c$和$D$，使任何子解决方案$u(\cdot)$可以找到一个常数$a>0$（这在一般情况下取决于$u$），其中$\sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)-u(x)\ge aψ(x)$，$x\in D$。这里${\rm cl}D$是$D$的闭包。集合${\cal S}(D)$ -称为$A$在$D$上的非局部性范围-由与$A$相关的Levy跳跃测度的支持度确定。这类结果我们称之为{ \em广义Hopf引理}。证明了$A$的预解式的不可约性蕴涵了SMP。反之亦然，只要我们对预解式做一些额外的（相当弱的）假设。对于某些类型的算子，我们可以承认常数$a$等于$ \sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)$。我们将这种类型的结果称为霍普夫引理的{ \em定量版本}。最后，我们制定了一个必要和充分条件$A$ -表示在遍历性质的预解式-这确保了下界$ψ\in {\rm span}(φ_D)$，其中$φ_D$是一个非负的本征函数的$A$在$D$。我们发现，上述遍历性质是隐含的与$A$半群的内在超压缩性。
英文摘要：
In the paper we prove a generalization of the Hopf lemma for weak subsolutions of the equation: $-Au+cu=0$ in $D$, for a wide class of Lévy type integro-differential operators $A$, bounded and measurable function $c:D\to[0,+\infty)$ and domain $D\subset \BR^d$. More precisely, we prove that if {\em the strong maximum principle} (SMP) holds for $A$, then there exists a Borel function $ψ:D\to(0,+\infty)$, depending only on the coefficients of the operator $A$, $c$ and $D$ such that for any subsolution $u(\cdot)$ one can find a constant $a>0$ (that in general depends on $u$), for which $\sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)-u(x)\ge aψ(x)$, $x\in D$. Here ${\rm cl}D$ is the closure of $D$. The set ${\cal S}(D)$ - called the range of non-locality of $A$ over $D$ - is determined by the support of the Levy jump measure associated with $A$. This type of a result we call the {\em generalized Hopf lemma}. It turns out that the irreducibility property of the resolvent of $A$ implies the SMP. The converse also holds, provided we assume some additional (rather weak) assumption on the resolvent. For some classes of operators we can admit the constant $a$ to be equal to $ \sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)$. We call this type of a result a {\em quantitative version} of the Hopf lemma. Finally, we formulate a necessary and sufficient condition on $A$ - expressed in terms of ergodic properties of its resolvent - which ensures that the lower bound $ψ\in {\rm span}(φ_D)$, where $φ_D$ is a non-negative eigenfunction of $A$ in $D$. We show that the aforementioned ergodic property is implied by the intrinsic ultracontractivity of the semigroup associated with $A$.
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