UBC Mathematics Department
http://www.math.ubc.ca
A striking new inequality for the Laplacian was proven in 1990 by a student in our department, Wensheng Xie. The inequality and its proof are highly original, very beautiful, and based on only elementary prerequisites.
Xie had undertaken to prove this inequality as a model problem. His ambition was to prove an analogous inequality for the hydrodynamic Stokes operator, for what would be an extremely important application to the Navier-Stokes equations. However, at one point in his proof he resigned himself to the use of the maximum principle. The Stokes equations are an elliptic system for which there is no maximum principle. Aside from this one point, Xie showed that his proof carries over to the Stokes operator. The desired application to the Navier-Stokes equations has also been worked out by Xie and myself, and published in the context of the vector Burgers equation.
Xie used the maximum principle to infer that the L^2 norm of the Green's function for the spectral Laplacian is less than the L^2 norm of the corresponding fundamental singularity. ``Xie's conjecture" is that a similar relation holds for the spectral Stokes operator.
I offer this lecture as an invitation to my friends and colleagues here to give serious thought to one of my favorite problems. I think that the heart of the matter can still be approached in the context of the Laplacian, by restricting oneself to methods that can be expected to generalize. Thus this problem may offer someone the chance of making a big splash in the Navier-Stokes world, without even knowing the equations.
I will begin this lecture with the statement of Xie's inequality, along with the core of its proof, and an explanation of the difficulty in carrying it over to the Stokes operator. Then, as time permits, I will explain its importance to the Navier-Stokes theory, and also offer a second equivalent conjecture that may provide an easier handle on the problem.