Symmetrization & applications by S. Kesavan

By S. Kesavan

The research of isoperimetric inequalities contains a desirable interaction of study, geometry and the idea of partial differential equations. a number of conjectures were made and whereas many were resolved, a multitude nonetheless stay open.One of the imperative instruments within the examine of isoperimetric difficulties, particularly whilst round symmetry is concerned, is Schwarz symmetrization, that's sometimes called the spherically symmetric and lowering rearrangement of services. the purpose of this ebook is to offer an advent to the idea of Schwarz symmetrization and learn a few of its applications.The booklet supplies an sleek and up to date therapy of the topic and comprises a number of new effects proved lately. attempt has been made to maintain the exposition as basic and self-contained as attainable. an information of the lifestyles idea of vulnerable suggestions of elliptic partial differential equations in Sobolev areas is, even if, assumed. except this and a normal mathematical adulthood on the graduate point, there are not any different must haves

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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.

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