American Institute of Mathematical Sciences

2013, 2013(special): 301-310. doi: 10.3934/proc.2013.2013.301

Optimization problems for the energy integral of p-Laplace equations

 1 Department of Mathematics and Informatics, Via Ospedale 72, 09124 Cagliari, Italy, Italy

Received  August 2012 Revised  November 2012 Published  November 2013

We study maximization and minimization problems for the energy integral of a sub-linear $p$-Laplace equation in a domain $\Omega$, with weight $\chi_D$, where $D\subset\Omega$ is a variable subset with a fixed measure $\alpha$. We prove Lipschitz continuity for the energy integral of a maximizer and differentiability for the energy integral of the minimizer with respect to $\alpha$.
Citation: Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
References:
 [1] F. Brock., Rearrangements and applications to symmetry problems in PDE. Handbook of differential equations: stationary partial differential equations. Vol. IV, 1-60, Elsevier/North-Holland, Amsterdam, 2007.  Google Scholar [2] G. R. Burton., Rearrangements of functions, maximization of convex functionals and vortex rings. Math. Ann. 276 (1987), 225-253.  Google Scholar [3] G. R. Burton., Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré 6 (1989), 295-319.  Google Scholar [4] G.R. Burton and J.B. McLeod., Maximisation and minimisation on classes of rearrangements. Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 287-300.  Google Scholar [5] S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi., Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214 (2000), 315-337.  Google Scholar [6] F. Cuccu, B. Emamizadeh and G. Porru., Optimization of the first eigenvalue in problems involving the p-Laplacian. Proc. Amer. Math. Soc. 137 (2009), 1677-1687.  Google Scholar [7] F. Cuccu and G. Porru., Optimization in problems of heat conduction. Adv. Math. Sci. Appl. 12 (2002), 245-255.  Google Scholar [8] F. Cuccu, G. Porru and S. Sakaguchi., Optimization problems on general classes of rearrangements. Nonlinear Analysis 74 (2011), 5554-5565.  Google Scholar [9] F. Cuccu, G. Porru and A. Vitolo., Optimization of the energy integral in two classes of rearrangements. Nonlinear Stud. 17 (2010), no. 1, 23-35.  Google Scholar [10] J.I. Diaz., Nonlinear partial differential equations and free boundaries. Volume 1. Elliptic equations, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985.  Google Scholar [11] J. Heinonen, T. Kilpeläinen and O. Martio., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, New York, Tokyo, 1993.  Google Scholar [12] B. Kawohl., Rearrangements and convexity of level sets in PDE's, Springer, Lectures Notes in Mathematics, n. 1150, 1985.  Google Scholar [13] B. Kawohl, M. Lucia and S. Prashanth., Simplicity of the first eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12 (2007), 407-434.  Google Scholar [14] E.H. Lieb and M. Loss., Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001  Google Scholar [15] P. Lindqvist., On the equation div$(\nabla u^{p-2} \nabla u)+\lambda|u|^{p-2}u=0$. Proc. Amer. Math. Soc. 109 (1990), 157-164.  Google Scholar [16] P. Lindqvist., Addendum: "On the equation div$(\nabla u^{p-2} \nabla u)+\lambda|u|^{p-2}u=0$" [Proc. Amer. Math. Soc. 109 (1990), no. 1, 157-164] Proc. Amer. Math. Soc. 116 (1992), no. 2, 583-584.  Google Scholar [17] M. Marras., Optimization in problems involving the p-Laplacian. Electron. J. Differential Equations 2010, No. 02, 10 pp.  Google Scholar [18] M. Marras, G. Porru and S. Vernier-Piro., Optimization problems for eigenvalues of p-Laplace equations. J. Math. Anal. Appl. 398 (2013), 766-776. Google Scholar [19] P. Tolksdorf., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126-150.  Google Scholar [20] N.S. Trudinger., On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. on Pure and Applied Math. Vol. XX (1967), 721-747.  Google Scholar [21] Xu-Jia Wang., A class of fully nonlinear equations and related functionals. Indiana Univ. Math. J. 43 (1994), 25-54.  Google Scholar

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References:
 [1] F. Brock., Rearrangements and applications to symmetry problems in PDE. Handbook of differential equations: stationary partial differential equations. Vol. IV, 1-60, Elsevier/North-Holland, Amsterdam, 2007.  Google Scholar [2] G. R. Burton., Rearrangements of functions, maximization of convex functionals and vortex rings. Math. Ann. 276 (1987), 225-253.  Google Scholar [3] G. R. Burton., Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré 6 (1989), 295-319.  Google Scholar [4] G.R. Burton and J.B. McLeod., Maximisation and minimisation on classes of rearrangements. Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 287-300.  Google Scholar [5] S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi., Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214 (2000), 315-337.  Google Scholar [6] F. Cuccu, B. Emamizadeh and G. Porru., Optimization of the first eigenvalue in problems involving the p-Laplacian. Proc. Amer. Math. Soc. 137 (2009), 1677-1687.  Google Scholar [7] F. Cuccu and G. Porru., Optimization in problems of heat conduction. Adv. Math. Sci. Appl. 12 (2002), 245-255.  Google Scholar [8] F. Cuccu, G. Porru and S. Sakaguchi., Optimization problems on general classes of rearrangements. Nonlinear Analysis 74 (2011), 5554-5565.  Google Scholar [9] F. Cuccu, G. Porru and A. Vitolo., Optimization of the energy integral in two classes of rearrangements. Nonlinear Stud. 17 (2010), no. 1, 23-35.  Google Scholar [10] J.I. Diaz., Nonlinear partial differential equations and free boundaries. Volume 1. Elliptic equations, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985.  Google Scholar [11] J. Heinonen, T. Kilpeläinen and O. Martio., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, New York, Tokyo, 1993.  Google Scholar [12] B. Kawohl., Rearrangements and convexity of level sets in PDE's, Springer, Lectures Notes in Mathematics, n. 1150, 1985.  Google Scholar [13] B. Kawohl, M. Lucia and S. Prashanth., Simplicity of the first eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12 (2007), 407-434.  Google Scholar [14] E.H. Lieb and M. Loss., Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001  Google Scholar [15] P. Lindqvist., On the equation div$(\nabla u^{p-2} \nabla u)+\lambda|u|^{p-2}u=0$. Proc. Amer. Math. Soc. 109 (1990), 157-164.  Google Scholar [16] P. Lindqvist., Addendum: "On the equation div$(\nabla u^{p-2} \nabla u)+\lambda|u|^{p-2}u=0$" [Proc. Amer. Math. Soc. 109 (1990), no. 1, 157-164] Proc. Amer. Math. Soc. 116 (1992), no. 2, 583-584.  Google Scholar [17] M. Marras., Optimization in problems involving the p-Laplacian. Electron. J. Differential Equations 2010, No. 02, 10 pp.  Google Scholar [18] M. Marras, G. Porru and S. Vernier-Piro., Optimization problems for eigenvalues of p-Laplace equations. J. Math. Anal. Appl. 398 (2013), 766-776. Google Scholar [19] P. Tolksdorf., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126-150.  Google Scholar [20] N.S. Trudinger., On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm. on Pure and Applied Math. Vol. XX (1967), 721-747.  Google Scholar [21] Xu-Jia Wang., A class of fully nonlinear equations and related functionals. Indiana Univ. Math. J. 43 (1994), 25-54.  Google Scholar
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