June  2021, 41(6): 2907-2946. doi: 10.3934/dcds.2020391

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

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  June 2021 Early access  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.

Citation: Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391
References:
[1]

R. AboulaichD. 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.

[2]

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

[3]

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

[4]

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

[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. 

[6]

L. A. CaffarelliC. 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.

[7]

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

[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.

[9]

D. DanielliA. 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.

[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.

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

[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.

[14]

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

[15]

J. Fernandez BonderJ. 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.

[16]

J. Fernandez BonderS. 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.

[17]

J. Fernandez BonderS. 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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.
[32]

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

[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.

[34]

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

[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.

[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. 

show all references

References:
[1]

R. AboulaichD. 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.

[2]

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

[3]

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

[4]

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

[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. 

[6]

L. A. CaffarelliC. 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.

[7]

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

[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.

[9]

D. DanielliA. 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.

[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.

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

[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.

[14]

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

[15]

J. Fernandez BonderJ. 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.

[16]

J. Fernandez BonderS. 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.

[17]

J. Fernandez BonderS. 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.

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.
[32]

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

[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.

[34]

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

[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.

[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. 

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