Nonlocal problems with Neumann boundary conditions
Nonlocal problems with Neumann boundary conditions
复制标题
DOI：
10.4171/rmi/942
复制
发表时间：
2014-07
期刊：
影响因子：
--
通讯作者：
S. Dipierro;Xavier Ros-Oton;E. Valdinoci
中科院分区：
文献类型：
--
作者：
S. Dipierro;Xavier Ros-Oton;E. Valdinoci
我们从简单的概率考虑中引入了分数拉普拉斯的新诺伊曼问题，并讨论了该模型的基本属性。我们可以在任何域中考虑椭圆方程和抛物线方程。此外，我们还用非齐次诺依曼条件以及混合狄利克雷和诺依曼条件来表述问题，所有这些都具有清晰的概率解释。我们证明，具有齐次诺伊曼条件的分数热方程的解具有以下自然属性：$\Omega$ 内的质量守恒、能量递减以及收敛到常数 $t\to \infty$。此外，对于椭圆情况，我们给出了问题的变分公式，并建立了解的存在性。我们还研究了这种非局部诺伊曼条件引起的极限性质和边界行为。具体而言，我们可以认为经典诺伊曼条件 ~$\partial_\nu u=0$ on~$\partial\Omega$ 的非局部类比包含在非局部处方 $$ \int_\Omega \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} 中x\in\R^n\setminus\overline{\Omega}.$$
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside $\Omega$, decreasing energy, and convergence to a constant as $t\to \infty$. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition~$\partial_\nu u=0$ on~$\partial\Omega$ consists in the nonlocal prescription $$ \int_\Omega \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} x\in\R^n\setminus\overline{\Omega}.$$