A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue
A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue
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非局部双调和算子及其与经典模拟的联系
DOI：
10.1007/s00205-016-1047-2
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发表时间：
2014
影响因子：
2.5
通讯作者：
J. Trageser
中科院分区：
文献类型：
--
作者：
P. Radu;Daniel Toundykov;J. Trageser
我们认为奇异积分算子是薄板理论中双调和算子的自然推广。该算子建立在（Math Models Methods Appl Sci 23（03）：493-540，2013）中定义的非局部演算框架中，并与最近的周波理论相联系。这个框架使我们能够考虑四阶椭圆边值问题的非光滑近似。对于这些系统，我们引入非局部配方的夹紧和铰链的边界条件，即使是不规则的域定义良好。我们证明了这些非局部问题的解的存在性和唯一性，并证明了它们的L2-强收敛函数在W2，2的非局部相互作用视界趋于零。对于正则域，我们将这些极限确定为相应经典椭圆边值问题的弱解。作为证明的一部分，我们还建立了光滑函数的非局部拉普拉斯算子是Lipschitz连续的。
We consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in (Math Models Methods Appl Sci 23(03):493–540, 2013) and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary-value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their L2-strong convergence to functions in W2,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary-value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.