Traversing the Lavrentiev Gap for Manià's functional
A talk in the BI.discrete series by
Joshua Siktar from Texas A&M University
| Abstract: | The Manià functional, an innocent-looking one-dimensional integral functional, is famous for being one of the simplest examples of an integral functional that possesses the Lavrentiev Gap phenomenon. That is, standard conforming finite element methods fail to approximate the minimizer of the continuous problem. To circumvent this, we introduce a class of cut-off functionals to control the pointwise values of the gradient, and prove that these cut-off functionals $\Gamma$-converge to the classical Manià functional in the strong $W^{1, p}(0, 1)$ topology for any $1 < p < \infty$. The key ingredient in the proof is the introduction of an interpolant that has quantitative convergence properties in the $W^{1, 1}(0, 1)$ norm even for functions that do not have a second-order weak derivative. We conclude this talk with a discussion of how the introduction of nonlocal gradients can potentially remove the Lavrentiev Gaps from the Manià functional, and on the role of our new interpolant in the study of nonlocal problems. Within the CRC this talk is associated to the project(s): A7, B7 |