We prove an integral representation formula for the distributional normal derivative of solutions of

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} - \Delta u + V u &{}= \mu &{}&{} \text {in }\Omega , \\ u &{}= 0 &{}&{} \text {on }\partial \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$

where \(V \in L_\mathrm {loc}^1(\Omega )\) is a nonnegative function and \(\mu \) is a finite Borel measure on \(\Omega \). As an application, we show that the Hopf lemma holds almost everywhere on \(\partial \Omega \) when \(V\) is a nonnegative Hopf potential.

中文翻译：

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分布正态导数的表示公式

我们证明了解的分布正态导数的积分表示公式

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ begin {aligned}-\ Delta u + V u＆{} = \ mu＆{}＆{} \ text {in} \欧米茄，\\ u＆{} = 0＆{}＆{} \ text {on} \ partial \ Omega，\ end {aligned} \ end {array} \ right。} \ end {aligned} $$

其中\（V \ in L_ \ mathrm {loc} ^ 1（\ Omega）\）是非负函数，\（\ mu \）是\（\ Omega \）的有限Borel度量。作为应用，我们证明了当\（V \）为非负Hopf势时，Hopf引理几乎在\（\ partial \ Omega \）上的所有地方都有效。