# DE seminar: Mitchell Sutton, UTK

Title: New Families of Fractional PDEs Arising from Fractional Calculus of Variations.

Abstract: In this presentation we shall explore two new families of fractional PDEs obtained as Euler-Lagrange equations of fractional calculus of variations problems. Several new fractional differential operators will be introduced, including the fractional $p$-Laplacian, Laplacian, and Neumann boundary operator. In each family of problems, we consider one-sided differentiation as well as differentiation in each direction; both in the weak sense. The first family of problems connects minimization problems with prescribed boundary conditions to associated fractional PDEs via the calculus of variations. The second family of problems establishes the connection between minimization problems with natural boundary conditions and fractional PDEs with Neumann boundary data. We prove the existence and uniqueness of weak solutions in the newly developed fractional Sobolev space(s) $\leftidx{^{\pm}}{W}{^{\alpha,p}}$. We also consider fractional PDEs for which there is no associated minimization problem. In addition to proving existence and uniqueness of solutions, we discuss the issue of choosing appropriate initial conditions and our interpretation of an initial value problem.

Thursday, March 5, 2020 at 2:10pm to 3:25pm

Ayres Hall, 111
1403 Circle Drive, Knoxville, TN 37996

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