By Silvanus P. Thompson

ISBN-10: 3871447390

ISBN-13: 9783871447396

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The prefix 'iso' means same in Greek. Today an isoperimetric inequality refers to an inequality obtained when we optimize some domain dependent functional keeping some geometric parameter (like the measure, for instance) of the domain fixed. 1) are available. We refer the reader to the survey article of [Osserman (1978)] for a nice proof. The inequality can also easily be proved using Fourier series. We will defer a proof of this inequality to later in this section, when we will give a simple proof (at least for smooth domains) in all space dimensions.

We now prove another particular case of the co-area formula. 2 Let Q C RN be an open set and let u £ £>(fi). 9) J where M = max a . G ^u(x). 10) Proof: Step 1. Since u is smooth, by Sard's thoerem, for almost every t in the range of u, we have that |Vu| ^ 0 on the level set {« = £}. Thus, {u = t} will be an (N — 1) - dimensional surface and, further {u = t} = d{u > t). Also, \{u* = t}\ = |{w = t}| = 0. Step 2. Let 2 < p < oo. Define f = -div{\Vu\p~2Vu). Vvdx Jo, = J fvdx. 11) Let t > 0. Set v = (u-1)+ € WQ'P{Q).

Finally, we consider the rearrangement u* of u. Let R be the radius of Q*. Let us write (by abuse of notation) u*(x) = u*(\x\) and thus consider it as a function of a single variable, for convenience. dx = / * \u*'(r)\pNu>NrN-1dr = Jo* \u*' {r^N^r"'1^™*' (r))dr

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