The inverse mean curvature flow and $p$-harmonic functions

Abstract

We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of p-harmonic functions and give a new proof for the existence of weak solutions.

Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

“…(5) Moser [25] showed that weak solutions in the sense of Huisken-Ilmanen can also be obtained by regularizing (?) with the help of p-harmonic functions.…”

The Dirichlet problem for a singular elliptic equation arising in the level set formulation of the inverse mean curvature flow

“…This differential operator appears in the level set formulation of the inverse mean curvature flow ( [12], see also [11], [17] and [18]). …”

“…Afterwards, Huisken and Ilmanen in [13] have proved regularity results for the inverse mean curvature flow and as consequence that every weak solution is regular after the first instant where a level set is star shaped. A different proof for the existence of weak solution of problem (1.3) is given in [17], which is based on the observation that for p > 1, a logarithmic change of dependent variable transforms the approximating equation div(|∇u| p−2 ∇u) = |∇u| p to the homogeneous p-Laplace equation.…”

The Hele-Shaw injection problem for an extremely shear-thinning fluid

“…Identifying β 0 with ω we see that (3.24) corresponds to (3.20) in the special case G(κ) = −1/κ of flow by the reciprocal of curvature, that is to to the motion of a two-dimensional curve evolving with the velocity law This result has been remarked upon by Moser [27] who noted that (3.20) is the level set formulation of (3.25). Formulating the problem in terms of dependent variable Y and independent variables x and β 0 , where (x, y) = (x, Y (x, β 0 )), so that β 0 plays the role of a time variable, and gives (on matching to region I) the initial condition…”

“…The boundary condition is obtained from the consideration of (3.21), 27) and is equivalent to requiring that the level sets of β 0 meet the boundary of the fluid filled region, ∂Ω f , normally. We remark further that it is possible to linearise (3.25) to the heat equation (cf.…”