Flat level set regularity of p-Laplace phase transition by Enrico Valdinoci
By Enrico Valdinoci
We turn out a Harnack inequality for point units of $p$-Laplace part transition minimizers. specifically, if a degree set is integrated in a flat cylinder, then, within the inside, it truly is incorporated in a flatter one. The extension of a outcome conjectured via De Giorgi and lately confirmed by way of the 3rd writer for $p=2$ follows.
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Example text
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Second mic rolocalization since A is positive ddinite. Then T L n (iTL) = dimension n. h is totally real an u if Xo E L, there is a holomorphic change of coordinates in a neighborhood of Xo such that L is transformed into the su bmanifold Imx = O. r and Xo = 0, one just has to take the diffeomorphism Re x + i [m x -+ M Re x + iM [m x where 111 is a linear isomorphism from Illn over L. [n general, we may thus assume that ToL = R n and so that there is a real analytic fUllcti on h in a neighbor hood of 0 in an, satisfying h(O) =.
The assump tions we do below may seem quite complicated and technical. We make them because they will appear to be the natural hypothesis in the applications we will treat in Chapter IV. Let us first introduce the geometric data. We assume given: - Z a closed real analytic suhmanifold of R", - Ej, j = 1, ... , k, k real analytic hypersurfaces in Z x ]0, 1], which are subanalytic , _ ]0, 1], in Z x [0I], and transverse to th e fibres of the second projection Z x ]0 I] - E a real analytic submallifold of Z x JO, ],I su banalytic in Z x 0,[ I], transverse to the 6bres of the second projection Z x ]0,I] _ ]0, I].