In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

中文翻译：

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多值非单调动态边界条件

在本文中，我们引入了一类新的半变分不等式，称为动态边界半变分不等式，这反映了控制算子在边界上也是活跃的。在我们的上下文中，它涉及具有温兹尔（动态）边界条件的拉普拉斯算子，该边界条件受到以Clarke次微分表示的多值非单调算子的干扰。我们表明，人们可以重新提出问题，从而可以应用标准技术。我们使用完善的边界半变分不等式理论来证明，在增长和一般符号条件下，动态边界半变分不等式允许一个弱解。此外，在以局部有界整数表示功能的情况下，在不连续点处使用“填补空白”程序来表征乘积空间上的次微分。最终，我们证明，在增长条件下以及最终变小的条件下，Faedo-Galerkin逼近序列收敛到所需的解。