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PDEs & Applications Seminar

Friday, February 7, 2025

Time: 10:30 a.m.  Place: Jeffery Hall, Room 319

Speaker: Jerin Tasnim Farin (¾ÅÐãÖ±²¥)

Title: Regularity of solutions to the Navier equations with mixed boundary conditions

Abstract: In this talk, I will present some regularity results for the partial differential equations that model the deformations of an elastic material. These equations are known as the Navier equations of linear elasticity. We impose mixed (Dirichlet and Neumann-type) boundary conditions on the solid's boundary. More precisely, we assume that the boundary of the solid is constituted by two disjoint components: an “interior" one subject to a non-zero displacement (a Dirichlet boundary condition), and an “exterior" one subject to a traction-free boundary condition (a Neumann-type condition). We will demonstrate existence and uniqueness of the solution to the relevant initial-boundary value problem in a weak sense and derive some additional regularity of the traction vector on the boundary. The classical energy estimates do not yield such a regularity result; it is in the spirit of the "hidden regularity" shown for solutions to the wave equation with Dirichlet boundary conditions.