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Energy, Pointwise, and Free Boundary Approximation of the Obstacle Problem for Nonlocal Operators
首页 - 学术活动
首页 - 学术活动报告人：
Abner J. Salgado, Professor, University of Tennessee, Knoxville, United States
邀请人：
Wenbo Li, Associate Professor
题目：
Energy, Pointwise, and Free Boundary Approximation of the Obstacle Problem for Nonlocal Operators
时间地点：
9:00-10:00 December 7 (Wednesday) ZOOM ID: 4120194771 Admission Password: mcm1234
摘要：
We consider the obstacle problem for a nonlocal elliptic operator, like the integral fractional Laplacian of order $s \in (0,1)$. We derive regularity results in weighted Sobolev spaces, where the weight is a power of the distance to the boundary. These are then used to obtain optimal error estimates in the energy norm.
Next, we consider the case of an obstacle problem where the operator is a combination of a fractional Laplacian and a regular Laplacian. We derive regularity results in classical Sobolev spaces, and use these to obtain error estimates.
Finally, we consider a two-scale discretization of the operator, which is monotone and naturally leads to max-norm error estimates for the linear problem. We extend these to the obstacle problem and indicate how, from these, free boundary estimates can be obtained; provided a nondegeneracy condition takes place. As an application of this result, we sketch ongoing work regarding the numerical approximation of a class of fully nonlinear, convex, integrodifferential operators.
This presentation is based on several works in collaboration with: A. Bonito (Texas A&M), J.P. Borthagaray (Montevideo), W. Lei (SISSA), R.H. Nochetto (UMD), and C. Torres (UMD).
Next, we consider the case of an obstacle problem where the operator is a combination of a fractional Laplacian and a regular Laplacian. We derive regularity results in classical Sobolev spaces, and use these to obtain error estimates.
Finally, we consider a two-scale discretization of the operator, which is monotone and naturally leads to max-norm error estimates for the linear problem. We extend these to the obstacle problem and indicate how, from these, free boundary estimates can be obtained; provided a nondegeneracy condition takes place. As an application of this result, we sketch ongoing work regarding the numerical approximation of a class of fully nonlinear, convex, integrodifferential operators.
This presentation is based on several works in collaboration with: A. Bonito (Texas A&M), J.P. Borthagaray (Montevideo), W. Lei (SISSA), R.H. Nochetto (UMD), and C. Torres (UMD).