2020/04/25 by José C. Bellido, Bellido, José C., Alejandro López Ortega +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2004.12160
In this work we introduce volume constraint problems involving the nonlocal\noperator (-\Δ)\δs, closely related to the fractional Laplacian\n(-\Δ)s, and depending upon a parameter \δ>0 called horizon. We\nstudy the associated linear and spectral problems and the behavior of these\nvolume constraint problems when \δ\→0+ and \δ\→+\∞. Through\nthese limit processes on (-\Δ)\δs we derive spectral\nconvergence to the local Laplacian and to the fractional Laplacian as\n\δ\→ 0+ and \δ \→ +\∞ respectively, as well as we prove the\nconvergence of solutions of these problems to solutions of a local Dirichlet\nproblem involving (-\Δ) as \δ\→0+ or to solutions of a nonlocal\nfractional Dirichlet problem involving (-\Δ)s as \δ\→+\∞.\n