American Institute of Mathematical Sciences

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

Optimization problems for the energy integral of p-Laplace equations

Antonio Greco 1, and  Giovanni Porru 1,

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 $\alpha$.
Keywords: optimization, regularity., Energy integral, rearrangements.
Mathematics Subject Classification: Primary: 35J20, 35J92; Secondary: 49K20, 52A4.
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.

[2]

G. R. Burton., Rearrangements of functions, maximization of convex functionals and vortex rings. Math. Ann. 276 (1987), 225-253.

[3]

G. R. Burton., Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré 6 (1989), 295-319.

[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.

[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.

[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.

[7]

F. Cuccu and G. Porru., Optimization in problems of heat conduction. Adv. Math. Sci. Appl. 12 (2002), 245-255.

[8]

F. Cuccu, G. Porru and S. Sakaguchi., Optimization problems on general classes of rearrangements. Nonlinear Analysis 74 (2011), 5554-5565.

[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.

[10]

J.I. Diaz., Nonlinear partial differential equations and free boundaries. Volume 1. Elliptic equations, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985.

[11]

J. Heinonen, T. Kilpeläinen and O. Martio., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, New York, Tokyo, 1993.

[12]

B. Kawohl., Rearrangements and convexity of level sets in PDE's, Springer, Lectures Notes in Mathematics, n. 1150, 1985.

[13]

B. Kawohl, M. Lucia and S. Prashanth., Simplicity of the first eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12 (2007), 407-434.

[14]

E.H. Lieb and M. Loss., Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001

[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.

[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.

[17]

M. Marras., Optimization in problems involving the p-Laplacian. Electron. J. Differential Equations 2010, No. 02, 10 pp.

[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.

[19]

P. Tolksdorf., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126-150.

[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.

[21]

Xu-Jia Wang., A class of fully nonlinear equations and related functionals. Indiana Univ. Math. J. 43 (1994), 25-54.

show all references

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.

[2]

G. R. Burton., Rearrangements of functions, maximization of convex functionals and vortex rings. Math. Ann. 276 (1987), 225-253.

[3]

G. R. Burton., Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. Henri Poincaré 6 (1989), 295-319.

[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.

[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.

[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.

[7]

F. Cuccu and G. Porru., Optimization in problems of heat conduction. Adv. Math. Sci. Appl. 12 (2002), 245-255.

[8]

F. Cuccu, G. Porru and S. Sakaguchi., Optimization problems on general classes of rearrangements. Nonlinear Analysis 74 (2011), 5554-5565.

[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.

[10]

J.I. Diaz., Nonlinear partial differential equations and free boundaries. Volume 1. Elliptic equations, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985.

[11]

J. Heinonen, T. Kilpeläinen and O. Martio., Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, New York, Tokyo, 1993.

[12]

B. Kawohl., Rearrangements and convexity of level sets in PDE's, Springer, Lectures Notes in Mathematics, n. 1150, 1985.

[13]

B. Kawohl, M. Lucia and S. Prashanth., Simplicity of the first eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12 (2007), 407-434.

[14]

E.H. Lieb and M. Loss., Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001

[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.

[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.

[17]

M. Marras., Optimization in problems involving the p-Laplacian. Electron. J. Differential Equations 2010, No. 02, 10 pp.

[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.

[19]

P. Tolksdorf., Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126-150.

[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.

[21]

Xu-Jia Wang., A class of fully nonlinear equations and related functionals. Indiana Univ. Math. J. 43 (1994), 25-54.

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