Combination and mean width rearrangements of solutions to elliptic equations in convex sets

Salani, Paolo

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 763-783.

We introduce a method to compare solutions of different equations in different domains. As a consequence, we define a new kind of rearrangement which applies to solution of fully nonlinear equations $F(x,u,Du,D^{2}u)=0$, not necessarily in divergence form, in convex domains and we obtain Talenti's type results for this kind of rearrangement.

Mots clés : Rearrangements, Elliptic equations, Infimal convolution, Power concave envelope, Minkowski addition of convex sets

[1] O. Alvarez, J.M. Lasry, P.L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. 76 (1997), 265 -288 | MR | Zbl

[2] A. Alvino, G. Trombetti, Elliptic equations with lower-order terms and reordering, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 66 (1979), 194 -200 | MR

[3] A. Alvino, G. Trombetti, P.L. Lions, Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990), 37 -65 | EuDML | Numdam | MR | Zbl

[4] A. Alvino, G. Trombetti, P.L. Lions, S. Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999), 167 -188 | EuDML | Numdam | MR | Zbl

[5] M. Bianchini, P. Salani, Power concavity for solutions of nonlinear elliptic problems in convex domains, R. Magnanini, et al. (ed.), Geometric Properties for Parabolic and Elliptic PDEs, Springer INdAM Ser. vol. 2 (2013), 35 -48 | MR | Zbl

[6] C. Borell, Convex set functions in d-space, Period. Math. Hung. 6 (1975), 111 -136 | MR | Zbl

[7] C. Borell, Capacitary inequalities of the Brunn–Minkowski type, Math. Ann. 263 (1983), 179 -184 | EuDML | MR | Zbl

[8] H.J. Brascamp, E.H. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366 -389 | MR | Zbl

[9] X. Cabré, L.A. Caffarelli, Fully Nonlinear Elliptic Equations, Colloq. Publ. – Am. Math. Soc. vol. 43 , Am. Math. Soc., Providence, RI (1995) | MR | Zbl

[10] A. Colesanti, Brunn–Minkowski inequalities for variational functionals and related problems, Adv. Math. 194 (2005), 105 -140 | MR | Zbl

[11] A. Colesanti, P. Salani, The Brunn–Minkowski inequality for p-capacity of convex bodies, Math. Ann. 327 (2003), 459 -479 | MR | Zbl

[12] A. Colesanti, P. Cuoghi, P. Salani, Brunn–Minkowski inequalities for two functionals involving the p-Laplace operator, Appl. Anal. 85 (2006), 45 -66 | MR | Zbl

[13] M.G. Crandall, H. Ishii, P.L. Lions, User's guide to viscosity solution of second order elliptic PDE, Bull. Am. Math. Soc. 27 (1992), 1 -67 | MR

[14] P. Cuoghi, P. Salani, Convexity of level sets for solutions to nonlinear elliptic problems in convex rings, Electron. J. Differ. Equ. 124 (2006) | EuDML | MR | Zbl

[15] V. Ferone, B. Kawohl, Remarks on a Finsler–Laplacian, Proc. Am. Math. Soc. 137 (2009), 247 -253 | MR | Zbl

[16] R.J. Gardner, The Brunn–Minkowski inequality, Bull. Am. Math. Soc. 39 (2002), 355 -405 | MR | Zbl

[17] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge (1959) | MR | Zbl

[18] K. Ishige, P. Salani, Parabolic quasi-concavity for solutions to parabolic problems in convex rings, Math. Nachr. 283 (2010), 1526 -1548 | MR | Zbl

[19] K. Ishige, P. Salani, Parabolic power concavity and parabolic boundary value problems, Math. Ann. 358 (2014), 1091 -1117 | MR | Zbl

[20] B. Kawohl, Geometrical properties of level sets of solutions to elliptic problems, Nonlinear Functional Analysis and Its Applications, Berkeley, CA, 1983, Proc. Symp. Pure Math. vol. 45, Part 2 , Am. Math. Soc., Providence, RI (1986), 25 -36 | MR

[21] B. Kawohl, Rearrangements and Convexity of Level Sets in P.D.E., Lect. Notes Math. vol. 1150 , Springer, Berlin (1985) | MR

[22] B. Kawohl, A remark on N. Korevaar's maximum principle, Math. Methods Appl. Sci. 8 (1986), 93 -101 | MR | Zbl

[23] A.U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), 687 -704 | MR | Zbl

[24] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, MSJ Memoirs vol. 13 , Mathematical Society of Japan, Tokyo (2004) | MR | Zbl

[25] N.J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), 603 -614 | MR | Zbl

[26] P. Juutinen, Concavity maximum principle for viscosity solutions of singular equations, Nonlinear Differ. Equ. Appl. 17 (2010), 601 -618 | MR | Zbl

[27] K.-A. Lee, J.L. Vázquez, Parabolic approach to nonlinear elliptic eigenvalue problems, Adv. Math. 219 (2008), 2006 -2028 | MR | Zbl

[28] P. Liu, X.N. Ma, L. Xu, A Brunn–Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain, Adv. Math. 225 (2010), 1616 -1633 | MR | Zbl

[29] X.N. Ma, L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in ${ℝ}^3$ , J. Funct. Anal. 255 (2008), 1713 -1723 | MR | Zbl

[30] L.G. Makar-Limanov, The solution of the Dirichlet problem for the equation $\Delta =-1$ in a convex region, Mat. Zametki 9 (1971), 89 -92 , Math. Notes 9 (1971), 52 -53 | MR | Zbl

[31] G. Pòlya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Stud. vol. 27 , Princeton University Press, Princeton, NJ (1951) | MR | Zbl

[32] C. Pucci, Operatori Ellittici Estremanti, Ann. Mat. Pura Appl. (4) 72 (1966), 141 -170 | MR | Zbl

[33] R.T. Rockafellar, Convex Analysis, Princeton Math. Ser. vol. 28 , Princeton University Press, Princeton, NJ (1970) | MR | Zbl

[34] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 14 (1987), 403 -421 | EuDML | Numdam | MR | Zbl

[35] P. Salani, A Brunn–Minkowski inequality for the Monge–Ampère eigenvalue, Adv. Math. 194 (2005), 67 -86 | MR | Zbl

[36] P. Salani, Convexity of solutions and Brunn–Minkowski inequalities for Hessian equations in $R^3$ , Adv. Math. 229 (2012), 1924 -1948 | MR | Zbl

[37] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encycl. Math. Appl. vol. 44 , Cambridge University Press, Cambridge (1993) | MR | Zbl

[38] T. Strömberg, The operation of infimal convolution, Diss. Math. 352 (1996) | EuDML | MR | Zbl

[39] G. Talenti, Elliptic equations and rearrangements, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 3 (1976), 697 -718 | EuDML | Numdam | MR | Zbl

[40] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4) 120 (1979), 160 -184 | MR | Zbl

[41] G. Trombetti, J.L. Vazquez, A symmetrization result for elliptic equations with lower-order terms, Ann. Fac. Sci. Toulouse 7 (1985), 137 -150 | EuDML | Numdam | MR | Zbl

[42] K. Tso, On symmetrization and Hessian equation, J. Anal. Math. 25 (1989), 94 -106 | MR | Zbl

[43] G. Wang, C. Xia, A Brunn–Minkowski inequality for a Finsler–Laplacian, Analysis (Munich) 31 (2011), 103 -115 | MR | Zbl

[44] Y. Ye, Power convexity of a class of elliptic equations involving the Hessian operator in a 3-dimensional bounded convex domain, Nonlinear Anal. 84 (2013), 29 -38 | MR | Zbl