Vladmir Maz’ya - “Challenging questions in unrestricted, polyhedral, Lipschitz graph, fractal and convex domains”

Problem statement:

• Some questions are obvious to engineers, but have no satisfactory mathematical analysis. For example, for the Lamé or Stokes equations with appropriate boundary conditions and an applied force, f, that vanishes
at the boundary, is the displacement/velocity bounded? And does the maximum principle hold when the applied force is zero?

• Common sense would suggest yes to both questions, but there is no analysis (at present) that says so.

Suggestions/comments:

• EJH: The interesting case is in a narrowing domain. Could a fluid forced down a narrowing tube attain an unbounded velocity or will the fact that f vanishes at the boundary prevent this?

• JRO: Cusps are sorted out because of research on punctured balls settled by Wiener in 1924.

People interested in follow-up: