2007 # 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.

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“…(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.…”

confidence: 99%

“…(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.…”

confidence: 99%

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

confidence: 99%

“…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.…”

confidence: 99%

“…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…”

confidence: 63%

“…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.…”

confidence: 99%