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Convex symmetrization and applications. (English) Zbl 0877.35040
The paper deals with the following problem: How to replace a function \(u\in W^1_0(\mathbb{R}^n)\) with a “symmetrized” function \(u^*\) in order to have \[ \int(H(Du))^2dx\geq \int(H(Du^*))^2dx, \] where \(H:\mathbb{R}^n\to[0,\infty)\) is a convex function positively homogeneous of degree 1. It is well-known that, when \(H(\xi)=|\xi|\) then one can use Schwarz symmetrization to decrease the corresponding integral. For a general \(H(\xi)\), the authors consider the convex set \(K=\{x: H(x)\leq 1\}\), with \(H\) normalized so that the measure of \(K\) equals that of the unit ball in \(\mathbb{R}^n\). For a function \(u\in W^1_0(\mathbb{R}^n)\), its rearrangement with respect to \(H\) is defined as a function \(u^*\) whose level sets have the same measure of the corresponding level sets of \(u\) and are homothetic to \(K^0\), the polar set to \(K\). In this kind of “symmetrization”, the convex set \(K^0\) plays the role of a ball in the Schwarz symmetrization. In the first part of the paper, a generalized perimeter \(P_H(E;\mathbb{R}^n)\) of a set \(E\) with respect to \(H\) is introduced. Then, the isoperimetric inequality \[ P_H(E;\mathbb{R}^n)\geq n|K^0|^{{1/n}}|E|^{1-{1/n}} \] is proved. In the second part, the Pólya-Szegö principle is obtained in the following form: for \(u\in W^{1,p}_0(\mathbb{R}^n)\), \(p\geq 1\), one has \[ \int H^p(Du)dx\geq \int H^p(Du^*)dx. \] In the last part, some applications are included. One example is the following. Let \(u\in W^1_0(\Omega)\) be a solution to the equation \(-(a_i(Du))_{x_i}= f\) in \(\Omega\), with \(a_i(\xi)\xi_i\geq (H(\xi))^2\), and let \(w\) satisfy \(-(H(Dw)H_{\xi_i}(Dw))_{x_i}= f^*\) in \(\Omega^*\), \(w=0\) on \(\partial\Omega^*\), where \(f^*\) is the rearrangement of \(f\) with respect to \(H\) and \(\Omega^*\) is the set homothetic to \(K^0\) and with the same measure as \(\Omega\). Then, the solution \(u(x)\), after symmetrization with respect to \(H\), satisfies \(u^*(x)\leq w(x)\).
Reviewer: G.Porru (Cagliari)

MSC:
35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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References:
[1] Alvino, A.; Díaz, J.I.; Lions, P.-L.; Trombetti, G., Elliptic equations and Steiner symmetrization, C.R. acad. sci. Paris, Vol. 314, 1015-1020, (1992) · Zbl 0795.35022
[2] Alvino, A.; Lions, P.-L.; Trombetti, G., Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. inst. Henri Poincaré, analyse nonlineaire, Vol. 7, 37-65, (1990) · Zbl 0703.35007
[3] Amar, M.; Bellettini, G., A notion of total variation depending on a metric with discontinuous coefficients, Ann. inst. Henri Poincaré, analyse nonlineaire, Vol. 11, 91-133, (1994) · Zbl 0842.49016
[4] Bandle, C., On symmetrizations in parabolic equations, J. anal. math., Vol. 30, 98-112, (1976) · Zbl 0331.35036
[5] Betta, M.F.; Ferone, V.; Mercaldo, A., Regularity for solutions of nonlinear elliptic equations, Bull. sci. math., Vol. 118, 539-567, (1994) · Zbl 0842.35014
[6] Burago, Yu.D.; Zalgaller, V.A., Geometric inequalities, (1988), Springer-Verlag Berlin · Zbl 0633.53002
[7] Buseman, H., The isoperimetric problem for Minkowski area, Amer. J. math., Vol. 71, 743-762, (1949) · Zbl 0038.10301
[8] De Giorgi, E., Su una teoria generale Della misura (r − 1)-dimensionale in uno spazio ad r dimensioni, Ann. mat. pura e appl., Vol. 36, 191-213, (1954) · Zbl 0055.28504
[9] Ferone, V.; Posteraro, M.R., Symmetrization results for elliptic equations with lower-order terms, Atti sem. mat. fis. univ. modena, Vol. 39, 47-61, (1991) · Zbl 0796.35034
[10] Ferone, V.; Posteraro, M.R.; Volpicelli, R., An inequality concerning rearrangements of functions and Hamilton-Jacobi equations, Arch. rat. mech. anal., Vol. 125, 257-269, (1993) · Zbl 0787.35020
[11] Giarrusso, E.; Nunziante, D., Symmetrization in a class of first-order Hamilton-Jacobi equations, Nonlinear analysis T.M.A., Vol. 8, 289-299, (1984) · Zbl 0543.35014
[12] Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhäuser Basel · Zbl 0545.49018
[13] Lay, S.R., Convex sets and their applications, (1982), J. Wiley and Sons New York · Zbl 0492.52001
[14] Lions, P.-L., Generalized solutions of Hamilton-Jacobi equations, (1982), Pitman London
[15] Maz’ja, V.G., Sobolev spaces, (1985), Springer-Verlag Berlin
[16] Mossino, J.; Rakotoson, J.M., Isoperimetric inequalities in parabolic equations, Ann. scuola norm. sup. Pisa, Vol. 13, 51-73, (1986) · Zbl 0652.35053
[17] Pólya, G.; Szegö, G., Isoperimetric inequalities in mathematical physics, (), No. 27 · Zbl 0101.41203
[18] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0229.90020
[19] Talenti, G., Best constant in Sobolev inequality, Ann. mat. pura e appl., Vol. 110, 353-372, (1976) · Zbl 0353.46018
[20] Talenti, G., Elliptic equations and rearrangements, Ann. scuola norm. sup. Pisa, Vol. 3, 4, 697-718, (1976) · Zbl 0341.35031
[21] Talenti, G., Nonlinear elliptic equations, rearrangements of function and Orlicz spaces, Ann. mat. pura appl., Vol. 120, 4, 159-184, (1979) · Zbl 0419.35041
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