# Global Analysis - Studies and Applications I by Y.G. Borisovich, Y.E. Gliklikh

By Y.G. Borisovich, Y.E. Gliklikh

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Extra resources for Global Analysis - Studies and Applications I

Example text

In this case p* = N/(N-l). Let Some Classical Inequalities 45 e > 0. Define ^ ^ {p(x,dQ)/e, 1, if p(x>dQ) > e if p(x,dQ) J^ da; = |Q| as e —> 0. Further, if £>£ - { x e f i | p(a;,5n) < e}, we have , |v , / 1/e in £>£ ^ = \ o in n\z>e. /n £! £ The limit of the right-hand side is, in fact, the Minkowski content M;v_i(<9ft). Thus, passing to the limit in the Sobolev inequality for 0, we get |0fi|jv_i >Nu^\n\'-^ which is exactly the isoperimetric inequality.

Integrating this equation once and using the definition of /* in terms of the unidimensional decreasing rearrangement (cf. 3), we get N •r"- V(r) = f Jo aN~1f#(u;NaN)da = (NUN)-1 P"" Jo f#(s)di Setting G(r) to be the last term in the above relation and integrating again, we get v(r) = f r1~NG(r)dT, which, on changing the variable to £ = UNTN , yields v(r) = ( i V < ) - 2 / Ju>NrN Z»-2[ / f#(s)ds df. 13), it follows that u#{s) < v#(s) and the proof is complete. 3) (cf. 6)). 8). It is also possible to prove this result just using the Polya - Szego inequality for the case p = 2, which, in turn, can be proved without using the co-area formula or the isoperimetric inequality (cf.

Further, by the classical isoperimetric inequality, we also have PRs({u>t}) >PR*({u*>t}). 1, it follows that / \Vu\dx Jn = / Jo PRN({u>t})dt /•OO •oo > / Jo /•* PRN({U* >t})dt = / Jo. \Vu*\dx Step 2. Let 1 < p < oo. Let u G V(Q) such that u > 0. Let M = maxxe^u(x). 2, it suffices to show that, for almost every t G (0, M), / \Vu\p-xda J{u=t} > f \Vu*\p~lda. 2) Since u is smooth, we can assume, by Sard's theorem, that |Vu| does not vanish on the set {u = t] for almost every t G (0, M). 1 are valid.