American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo"
  • DCDS Home
  • This Issue
  • Next Article
    Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems
doi: 10.3934/dcds.2020391

An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth

Claudia Lederman , and  Noemi Wolanski

IMAS - CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina

* Corresponding author: clederma@dm.uba.ar

Received  October 2020 Published  December 2020

We consider an optimization problem with volume constraint for an energy functional associated to an inhomogeneous operator with nonstandard growth. By studying an auxiliary penalized problem, we prove existence and regularity of solution to the original problem: every optimal configuration is a solution to a one phase free boundary problem—for an operator with nonstandard growth and non-zero right hand side—and the free boundary is a smooth surface.

Keywords: optimal design, minimization problem, free boundary problem, variable exponent spaces, inhomogeneous problem, nonstandard growth, regularity of the free boundary, shape optimization.
Mathematics Subject Classification: Primary: 35R35, 35J20; Secondary: 35J70, 35J60, 49Q10, 49K20, 35B65.
Citation: Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020391
References:
[1]

R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.  doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

A. Acker, An extremal problem involving current flow through distributed resistance, SIAM J. Math. Anal., 12 (1981), 169-172.  doi: 10.1137/0512017.  Google Scholar

[3]

N. Aguilera, H. W. Alt and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control Optim., 24 (1986), 191-198.  doi: 10.1137/0324011.  Google Scholar

[4]

N. Aguilera, L. A. Caffarelli and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 355-387.   Google Scholar

[5]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, Jour. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[6]

L. A. Caffarelli, C. Lederman and N. Wolanski, Uniform estimates and limits for a two phase parabolic singular perturbation problem, Indiana Univ. Math. J., 46 (1997), 453-490.  doi: 10.1512/iumj.1997.46.1470.  Google Scholar

[7]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

[8]

D. Danielli and A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations, 23 (2005), 97-124.  doi: 10.1007/s00526-004-0294-5.  Google Scholar

[9]

D. Danielli, A. Petrosyan and H. Shahgholian, A singular perturbation problem for the $p$-Laplace operator, Indiana Univ. Math. J., 52 (2003), 457-476.  doi: 10.1512/iumj.2003.52.2163.  Google Scholar

[10]

L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heielberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.   Google Scholar
[12]

X. Fan, Global $C^{1, \alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.  doi: 10.1016/j.jde.2007.01.008.  Google Scholar

[13]

X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318.  doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

[14]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[15]

J. Fernandez Bonder, J. D. Rossi and N. Wolanski, Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant, SIAM J. Control Optim., 44 (2005), 1614-1635.  doi: 10.1137/040613615.  Google Scholar

[16]

J. Fernandez Bonder, S. Martinez and N. Wolanski, An optimization problem with volume constraint for a degenerate operator, J. Differential Equations, 227 (2006), 80-101.  doi: 10.1016/j.jde.2006.03.006.  Google Scholar

[17]

J. Fernandez Bonder, S. Martinez and N. Wolanski, A free boundary problem for the $p(x)$-Laplacian, Nonlinear Anal., 72 (2010), 1078-1103.  doi: 10.1016/j.na.2009.07.048.  Google Scholar

[18]

M. Flucher, An asymptotic formula for the minimal capacity among sets of equal area, Calc. Var. Partial Differential Equations, 1 (1993), 71-86.  doi: 10.1007/BF02163265.  Google Scholar

[19]

P. Harjulehto and P. Hasto, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.  Google Scholar

[20]

A. Henrot and M. Pierre, Shape Variation and Optimization. A Geometrical Analysis, EMS Tracts in Mathematics, vol. 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178.  Google Scholar

[21]

O. Kováčik and J. Rákosník, On spaces ${L}^{p(x)}$ and ${W}^{k, p(x)}$, Czechoslovak Math. J, 41 (1991), 592-618.   Google Scholar

[22]

C. Lederman, An optimization problem in elasticity, Differential Integral Equations, 8 (1995), 2025-2044.   Google Scholar

[23]

C. Lederman, A free boundary problem with a volume penalization, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 249-300.   Google Scholar

[24]

C. Lederman and N. Wolanski, Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 27 (1998), 253-288.   Google Scholar

[25]

C. Lederman and N. Wolanski, An inhomogeneous singular perturbation problem for the $p(x)$-Laplacian, Nonlinear Anal., 138 (2016), 300-325.  doi: 10.1016/j.na.2015.09.026.  Google Scholar

[26]

C. Lederman and N. Wolanski, Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian, Interfaces Free Bound., 19 (2017), 201-241.  doi: 10.4171/IFB/381.  Google Scholar

[27]

C. Lederman and N. Wolanski, Inhomogeneous minimization problems for the $p(x)$-Laplacian, J. Math. Anal. Appl., 475 (2019), 423-463.  doi: 10.1016/j.jmaa.2019.02.049.  Google Scholar

[28]

S. Martinez, An optimization problem with volume constraint in Orlicz spaces, J. Math. Anal. Appl., 340 (2008), 1407-1421.  doi: 10.1016/j.jmaa.2007.09.061.  Google Scholar

[29]

S. Martinez and N. Wolanski, A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman, SIAM J. Math. Anal., 41 (2009), 318-359.  doi: 10.1137/070703740.  Google Scholar

[30]

K. Oliveira and E. Teixeira, An optimization problem with free boundary governed by a degenerate quasilinear operator, Differential Integral Equations, 19 (2006), 1061-1080.   Google Scholar

[31] V. D. Radulescu and D. D. Repovs, Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.  Google Scholar
[32]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[33]

E. Teixeira, The nonlinear optimization problem in heat conduction, Calc. Var. Partial Differential Equations, 24 (2005), 21-46.  doi: 10.1007/s00526-004-0313-6.  Google Scholar

[34]

E. Teixeira, Optimal design problems in rough inhomogeneous media. Existence theory, Amer. J. Math., 132 (2010), 1445-1492.   Google Scholar

[35]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar

[36]

N. Wolanski, Local bounds, Harnack inequality and Hölder continuity for divergence type elliptic equations with non-standard growth, Rev. Un. Mat. Argentina, 56 (2015), 73-105.   Google Scholar

show all references

References:
[1]

R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.  doi: 10.1016/j.camwa.2008.01.017.  Google Scholar

[2]

A. Acker, An extremal problem involving current flow through distributed resistance, SIAM J. Math. Anal., 12 (1981), 169-172.  doi: 10.1137/0512017.  Google Scholar

[3]

N. Aguilera, H. W. Alt and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control Optim., 24 (1986), 191-198.  doi: 10.1137/0324011.  Google Scholar

[4]

N. Aguilera, L. A. Caffarelli and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 355-387.   Google Scholar

[5]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, Jour. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[6]

L. A. Caffarelli, C. Lederman and N. Wolanski, Uniform estimates and limits for a two phase parabolic singular perturbation problem, Indiana Univ. Math. J., 46 (1997), 453-490.  doi: 10.1512/iumj.1997.46.1470.  Google Scholar

[7]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.  doi: 10.1137/050624522.  Google Scholar

[8]

D. Danielli and A. Petrosyan, A minimum problem with free boundary for a degenerate quasilinear operator, Calc. Var. Partial Differential Equations, 23 (2005), 97-124.  doi: 10.1007/s00526-004-0294-5.  Google Scholar

[9]

D. Danielli, A. Petrosyan and H. Shahgholian, A singular perturbation problem for the $p$-Laplace operator, Indiana Univ. Math. J., 52 (2003), 457-476.  doi: 10.1512/iumj.2003.52.2163.  Google Scholar

[10]

L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heielberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.   Google Scholar
[12]

X. Fan, Global $C^{1, \alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.  doi: 10.1016/j.jde.2007.01.008.  Google Scholar

[13]

X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318.  doi: 10.1016/S0362-546X(97)00628-7.  Google Scholar

[14]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[15]

J. Fernandez Bonder, J. D. Rossi and N. Wolanski, Regularity of the free boundary in an optimization problem related to the best Sobolev trace constant, SIAM J. Control Optim., 44 (2005), 1614-1635.  doi: 10.1137/040613615.  Google Scholar

[16]

J. Fernandez Bonder, S. Martinez and N. Wolanski, An optimization problem with volume constraint for a degenerate operator, J. Differential Equations, 227 (2006), 80-101.  doi: 10.1016/j.jde.2006.03.006.  Google Scholar

[17]

J. Fernandez Bonder, S. Martinez and N. Wolanski, A free boundary problem for the $p(x)$-Laplacian, Nonlinear Anal., 72 (2010), 1078-1103.  doi: 10.1016/j.na.2009.07.048.  Google Scholar

[18]

M. Flucher, An asymptotic formula for the minimal capacity among sets of equal area, Calc. Var. Partial Differential Equations, 1 (1993), 71-86.  doi: 10.1007/BF02163265.  Google Scholar

[19]

P. Harjulehto and P. Hasto, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.  Google Scholar

[20]

A. Henrot and M. Pierre, Shape Variation and Optimization. A Geometrical Analysis, EMS Tracts in Mathematics, vol. 28, European Mathematical Society, Zürich, 2018. doi: 10.4171/178.  Google Scholar

[21]

O. Kováčik and J. Rákosník, On spaces ${L}^{p(x)}$ and ${W}^{k, p(x)}$, Czechoslovak Math. J, 41 (1991), 592-618.   Google Scholar

[22]

C. Lederman, An optimization problem in elasticity, Differential Integral Equations, 8 (1995), 2025-2044.   Google Scholar

[23]

C. Lederman, A free boundary problem with a volume penalization, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), 249-300.   Google Scholar

[24]

C. Lederman and N. Wolanski, Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 27 (1998), 253-288.   Google Scholar

[25]

C. Lederman and N. Wolanski, An inhomogeneous singular perturbation problem for the $p(x)$-Laplacian, Nonlinear Anal., 138 (2016), 300-325.  doi: 10.1016/j.na.2015.09.026.  Google Scholar

[26]

C. Lederman and N. Wolanski, Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian, Interfaces Free Bound., 19 (2017), 201-241.  doi: 10.4171/IFB/381.  Google Scholar

[27]

C. Lederman and N. Wolanski, Inhomogeneous minimization problems for the $p(x)$-Laplacian, J. Math. Anal. Appl., 475 (2019), 423-463.  doi: 10.1016/j.jmaa.2019.02.049.  Google Scholar

[28]

S. Martinez, An optimization problem with volume constraint in Orlicz spaces, J. Math. Anal. Appl., 340 (2008), 1407-1421.  doi: 10.1016/j.jmaa.2007.09.061.  Google Scholar

[29]

S. Martinez and N. Wolanski, A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman, SIAM J. Math. Anal., 41 (2009), 318-359.  doi: 10.1137/070703740.  Google Scholar

[30]

K. Oliveira and E. Teixeira, An optimization problem with free boundary governed by a degenerate quasilinear operator, Differential Integral Equations, 19 (2006), 1061-1080.   Google Scholar

[31] V. D. Radulescu and D. D. Repovs, Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18601.  Google Scholar
[32]

M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[33]

E. Teixeira, The nonlinear optimization problem in heat conduction, Calc. Var. Partial Differential Equations, 24 (2005), 21-46.  doi: 10.1007/s00526-004-0313-6.  Google Scholar

[34]

E. Teixeira, Optimal design problems in rough inhomogeneous media. Existence theory, Amer. J. Math., 132 (2010), 1445-1492.   Google Scholar

[35]

N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar

[36]

N. Wolanski, Local bounds, Harnack inequality and Hölder continuity for divergence type elliptic equations with non-standard growth, Rev. Un. Mat. Argentina, 56 (2015), 73-105.   Google Scholar

[1]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[2]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[3]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028
[4]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[5]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[6]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[7]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[8]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[9]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037
[10]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[11]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[12]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[13]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401
[14]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[15]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[16]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
[17]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[18]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[19]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[20]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (19)
  • HTML views (98)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences