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Mikil Foss, University of Nebraska–Lincoln

Local and Nonlocal Poincare Inequalities

4:00 pm–4:50 pm
Avery Hall Room: 115
David Pitts,
In 1890, Henri Poincare established an inequality providing a bound on the size of a function in terms of a size of its derivative. The inequality is a key component in proofs for existence of solutions to PDEs and variational problems. In the differential framework the Poincare inequality is connected to the isoperimetric inequality (in 2D), estimates of eigenvalues for operators (Sturm-Liouville problems), and provides tools for convergence results in numerical approximations. Recently, in the nonlocal framework of integral operators, the inequality has been a critical tool in establishing wellposedness of “rough” (even discontinuous) solutions for nonlocal systems. He will conclude with some recently obtained results in which kernels for nonlocal operators are allowed to be non-symmetric and inhomogeneous–features that arise naturally in nonlocal models of dynamic fracture.

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This event originated in Math Colloquia.