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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
##### Keywords:
symmetrization; isoperimetric inequality; rearrangement
Full Text:
##### 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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