Abtract：We consider on arbitrary Riemannian manifolds the Leibenson equation:$\partial _{t}u=\Delta _{p}u^{q}.$This equation comes from hydrodynamics where it describes filtration of a turbulent compressible liquid in porous medium. Here $u(x,t)$ is the fraction of the volume that the liquid takes in porous medium at time $t$ at point $x$. The parameter $p$ characterizes the turbulence of a...
Abstract：I will discuss some new bounds on the spectra of Laplacian operators on hyperbolic 3-manifolds. One example of such a bound is that the spectral gap of the Laplace-Beltrami operator on a closed orientable hyperbolic 3-manifold must be less than 47.32, or less than 31.57 if the first Betti number is positive. The bounds are derived using two approaches, both of which employ linear prog...
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