Schwarz rearrangement does not decrease the energy for the pseudo p-laplacian operator. Key Words:Schwarz symmetrization, pseudo p-laplacian operator.
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1 Bol. Soc. Paran. Mat. (3s. v. 9 1 (11: c SPM ISSN on line ISSN in ress SPM: doi:1.569/bsm.v9i1.148 Schwarz rearrangement does not decrease the energy for the seudo -Lalacian oerator Mohammed MOUSSA abstract: It is well known that the Schwarz symmetrization decrease the energy for the -Lalacian oerator, i.e u dx u dx. where u is the Schwarz rearranged function of u, for aroriate u and. In this note, we shall roof that the Schwarz rearrangement does not decrease the energy for the seudo -Lalacian oerator, that is, there exist a bounded domain R N and a function u W 1, ( such that x dx i x dx. i Key Words:Schwarz symmetrization, seudo -Lalacian oerator. Contents 1 Introduction A catalogue of rearrangement Some results for Schwarz symmetrization Main result Introduction The rearrangement method is defined by relacing a given function u by a related function u wich has some roerties like monotonicity or symmetry. The function u can be reconstructed from its level sets c {x u(x c} A catalogue of rearrangement. In the litterature we find many tye of rearrangement, 1. circular and sherical symmetrization,. monotone decreasing rearrangement in direction y, 3. radial symmetrization, 4. Schwarz symmetrization, Mathematics Subject Classification: 35HXX 49 Tyeset by B S P M style. c Soc. Paran. de Mat.
2 5 M. Moussa 5. Steiner symmetrization in direction y. more details of rearrangement can be found in [3,4,5,7]. In this note we are intersted only by Schwarz symmetrization (the most frequently used kind symmetrization. Hence, for a Lebesgue measurable set D R n we define the Schwarz symmetrization D of D by { D B(,R if D if D where B(,R is a ball of R n with center in the origin with same n dim. Lebesgue measure and for a Lischitz continuous function u, the rearranged function u is defined as follows u (x su{c R x c} for x 1.. Some results for Schwarz symmetrization. One of the first owerful alications of Schwarz symmetrization was the roof of the Krahn-Faber inequality [6] : Among all fixed membranes of given area, the circular one has the lowest rincial eigenvalue. This was shown by looking at λ 1 ( min u W 1, (\{} u u. one can easily conclude, with the two following roositions, that λ 1 ( λ 1 ( Proosition 1.1 ([1,5,7] For every continuous maing F : R + R and every nonnegative function u : R + then, F(udx F(u dx. Proosition 1. ([1,5,7] For u W 1, ( and > 1, we have E (u 1 u dx 1 u dx E (u Then, Schwarz symmetrization decrease the otentiel energy E (u of -Lalacian oerator u defined by u div ( u u. The same question is osed for the so called seudo -Lalacian oerator defined by ( n u div x i. x i We show, in this aer, that the answer is negative by exhibiting an exlicit function with Schwarz s symmetrization does not decrease the energy for seudo -Lalacian oerator.
3 The seudo -Lalacian oerator 51. Main result Theorem.1 The Schwarz rearrangement does not decrease the energy for the seudo -Lalacian oerator, that is, there exist a bounded domain R n and a function u W 1, ( such that x i dx x i dx. (1 where and u are the Schwarz rearrangement of and u resectively. Proof. Let {(x 1,x R x 1 + x } and u(x 1,x ( x 1 + x, u is a Lischitz continuous function that u W 1, (. Then the Schwarz rearrangement of is B(,. Level sets of u are given by c {(x 1,x R x 1 + x < c}. Then, c meas( c ( c, so Now we have then, u (x 1,x (x 1 +x 1. x 1 1 x, x i in the other hand, x 1 x 1 (x 1 +x 1 dx meas( 4 ( and by assing to olar coordinates we obtain ( ( x i dx cosθ dθ ( ( cosθ dθ + ( ( ( ( 4 ( 8 x x (x 1 +x 1 cosθ dθ + rdr+ sinθ dθ [ 1 sinθ dθ sinθ dθ cosθ dθ + sinθ dθ ( sinθ dθ cosθ dθ r ] rdr sinθ dθ
4 5 M. Moussa we have used the following equality so, cosθ dθ sinθ dθ sinθ dθ + sinθ dθ sinθ dθ (ut in the second integral θ θ ( sinθ dθ + sinθ dθ ( sinθ dθ + sinθ dθ (ut in the second integral θ θ 4 sinθ dθ, x i ( dx 8 sinθ dθ (3 the integral in the right member of equation (3 is given by the well known Wallis formula [] (age : 15 sinθ Γ( +1 dθ Γ( +1. Finally, x i ( Γ ( +1 dx 4 Γ( +1 (4 the function Γ(α is increasing for α 3 and Γ(α+1 αγ(α for all α > 1 so, Γ ( +1 Γ( +1 Γ ( Γ(, alying equation (4 to get then the inequality is verified if x i x i dx 8 dx ( 8 ( 4 x i dx
5 The seudo -Lalacian oerator 53 which is equivalent to (. (5 An elementary study of the function (x ( x f(x x shows that f is strictly increasing in [ 1 ln,+ [. Equation (5 is then established if f(, that is c where c is defined by f( c. Mean value theorem shows that c [9,1]. Consequently, for all 1 x i dx x i Conclusion : the Schwarz rearrangement does not decrease the energy for the seudo -Lalacian oerator like it does for -Lalacian oerator. References 1. A. Alvino, P.L. Lions, G. Trombetti, comarison results for ellitic and arabolic via Schwarz symmetrization, Annales de l I.H.P, section C, tome 7, Nř, (199, E. Azoulay and J. Avignant, Mathématiques 3. Analyse, McGraw-Hill, ( C. Bandle, Isoerimetric inequalities and alications, Pitman, London, ( B. Kawohl, Rearrangement and convexity of level sets in PDE, Lecture Notes in Mathematics, Sringer Verlag, Berlin, Heidelberg, New York, Tokyo, 115 ( S. Kesavan, Symmetrization and alications, Series in Analysis, Vol. 3, (6. 6. E. Krahn, Über eine von Rayleigh formulierte minimalegenschaft des Kreises, Math. Ann. 94 ( G. Polya, G. Szegö, Isoerimetric inequalities in mathematical hysics, Ann. Math. Stud. Princeton, University Press, 7 (195. dx. Mohammed MOUSSA Déartement de mathématiques Université Ibn Tofail, Kénitra Maroc mohammed.moussa9@gmail.com
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