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June  2021, 41(6): 2907-2946. 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, 2021, 41 (6) : 2907-2946. 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
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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

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