Nonlocal problems with Neumann boundary conditions
By Serena Dipierro, Xavier Ros-Oton, Enrico Valdinoci. 2014
265 citations recorded by the source corpus.
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 $$ \int_\Omega \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \ {\mbox{ for }} x\in\R^n\setminus\overline{\Omega}.$$
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/50c8ee961605b2754ac8ca9b238703c85323003c
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In Applied Mathematics
- Yi-Hsuan Lin
- Tuhin Ghosh
- Zhiping Mao
- Mark Ainsworth
- Xiaoning Zheng
- G. Em Karniadakis
- Fangying Song
- Christian Glusa
In Computational Theory and Mathematics
- Yi-Hsuan Lin
- G. Em Karniadakis
- Guofei Pang
- Wei Cai
- Tuhin Ghosh
- Xiaoning Zheng
- Mark Ainsworth
- Zhiping Mao
In Mathematical Physics
- Yi-Hsuan Lin
- Tuhin Ghosh
- Zhiping Mao
- Mark Ainsworth
- G. Em Karniadakis
- Mark M. Meerschaert
- Gerd Grubb
- Guofei Pang
Citations in the corpus
- Linear theory for a mixed operator with Neumann conditions (2020, unknown): “More specifically, in [37], we have introduced a new set of nonlocal Neumann conditions, extending those previously set forth in [38], with the aim of dealing with a mathematical problem motivated by ethology and biology.”
- What is the fractional Laplacian? A comparative review with new results (2018, background): “∫ Rd u(x)−u(y) |x−y|d+α dy, for C = 2 α ( α 2 + 2 ) πd/2 | (− 2 )| Functions on Rd Exterior (Rd \ ) Stopped α-stable motion [22] Various conditions/processes proposed [65,66]”
- Regularity of spectral fractional Dirichlet and Neumann problems (2014, methodology): “[CK02], Bogdan, Burdzy and Chen [BBC03]. Note that (p0(u,u)+ kuk2) 1 2 is equivalent with the norm in Ha 2 (Ω). The other choice p1 has recently been introduced in Dipierro, Ros-Oton and Valdinoci in [DRV14] (formulated for real functions), where it is shown how it defines an operator r+(−∆)a applied”
- The Calderón problem for variable coefficients nonlocal elliptic operators (2017, methodology): “We introduce the anisotropic nonlocal Neumann operator Ns analogues to the Neumann operator which is initiated in [14] for the fractional Laplacian operator (−1)s.”
- Global uniqueness for the fractional semilinear Schrödinger equation (2017, unknown): “) has a unique solution, the inverse problem is to recover q(x,u) from the DN map. For more details about the nonlocal Neumann derivative Ns q , DN map Λq, and their connection, we refer to (2.2) and [3, 6]. In this paper, we focus on the Caldero´n problem for the fractional semilinear Schro¨dinger ”