论文标题
非本地的诺伊曼问题
The nonlocal Neumann problem
论文作者
论文摘要
经过经典的本地诺伊曼问题，对此问题进行了充分的研究，通常在Sobolev空间中存在解决方案。在这项工作中，我们专注于可测量的，非负核的非本地诺伊曼问题，其解决方案需要较少的规律性假设。对于这种内核，我们制定和研究非本地诺伊曼问题的弱公式，并研究了Sobolev空间的非局部对应物$ h^{1} $以及由此产生的非局部痕量空间。我们进一步建立了主要用于对称内核的，对于诺伊曼问题的弱解决方案的各种存在结果，我们讨论了相关的必要条件。都考虑了同质和非均匀的诺伊曼边界条件。除此之外，我们还提出了一种新的弱弱表述，在该问题中，我们将罗宾问题重新制定为区域问题。
The classical local Neumann problem is well studied and solutions of this problem lie, in general, in a Sobolev space. In this work, we focus on nonlocal Neumann problems with measurable, nonnegative kernels, whose solutions require less regularity assumptions. For kernels of this kind we formulate and study the weak formulation of the nonlocal Neumann problem and we investigate a nonlocal counterpart of the Sobolev space $H^{1}$ as well as a resulting nonlocal trace space. We further establish, mainly for symmetric kernels, various existence results for the weak solution of the Neumann problem and we discuss related necessary conditions. Both, homogeneous and nonhomogeneous Neumann boundary conditions are considered. In addition to that, we present a new weak formulation of a Robin problem, where we reformulate the Robin problem into a regional problem.
