October  2022, 42(10): 5017-5036. doi: 10.3934/dcds.2022084

An optimization problem in heat conduction with volume constraint and double obstacles

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

*Corresponding author: Cong Wang

Received  January 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The first author is supported by the China Scholarship Council (No. 202006040127)

We consider the optimization problem of minimizing $ \int_{\mathbb{R}^n}|\nabla u|^2\,{\mathrm{d}}x $ with double obstacles $ \phi\leq u\leq\psi $ a.e. in $ D $ and a constraint on the volume of $ \{u>0\}\setminus\overline{D} $, where $ D\subset\mathbb{R}^n $ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is $ C^{1,1} $ locally in $ D $ and Lipschitz continuous in $ \mathbb{R}^n $ and that the free boundary $ \partial\{u>0\}\setminus\overline{D} $ is smooth. Moreover, when the boundary of $ D $ has a plane portion, we show that the minimizer is $ C^{1,\frac{1}{2}} $ up to the plane portion.

Citation: Xiaoliang Li, Cong Wang. An optimization problem in heat conduction with volume constraint and double obstacles. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 5017-5036. doi: 10.3934/dcds.2022084
References:
[1]

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.

[2]

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

[3]

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

[4]

L. AmbrosioI. FonsecaP. Marcellini and L. Tartar, On a volume constrained variational problem, Arch. Ration. Mech. Anal., 149 (1999), 23-47.  doi: 10.1007/s002050050166.

[5]

H. Brézis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1973/74), 831-844.  doi: 10.1512/iumj.1974.23.23069.

[6]

L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations, 5 (1980), 427-448.  doi: 10.1080/0360530800882144.

[7]

L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402.  doi: 10.1007/BF02498216.

[8]

L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Anal. Math., 37 (1980), 285-295.  doi: 10.1007/BF02797689.

[9]

M. Chipot, Sur la régularité de la solution d'inéquations variationnelles elliptiques, C. R. Acad. Sci. Paris Sér., 288 (1979), 543-546. 

[10]

H. J. Choe, Regularity for certain degenerate elliptic double obstacle problems, J. Math. Anal. Appl., 169 (1992), 111-126.  doi: 10.1016/0022-247X(92)90106-N.

[11]

G. Dal MasoU. Mosco and M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem, Acta Math., 163 (1989), 57-107.  doi: 10.1007/BF02392733.

[12]

J. Fernández BonderS. Martínez and N. Wolanski, An optimization problem with volume constraint for a degenerate quasilinear operator, J. Differential Equations, 227 (2006), 80-101.  doi: 10.1016/j.jde.2006.03.006.

[13]

J. Fernández 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), 1612-1635.  doi: 10.1137/040613615.

[14]

A. Figalli and J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., 215 (2019), 311-366.  doi: 10.1007/s00222-018-0827-8.

[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Equatios of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2001. 
[16]

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

[17]

G. P. Leonardi and P. Tilli, On a constrained variational problem in the vector-valued case, J. Math. Pures Appl., 85 (2006), 251-268.  doi: 10.1016/j.matpur.2005.07.004.

[18]

G. M. Lieberman, Regularity of solutions to some degenerate double obstacle problems, Indiana Univ. Math. J., 40 (1991), 1009-1028.  doi: 10.1512/iumj.1991.40.40045.

[19]

F. Lin, Lectures on elliptic free boundary problems, in Lectures on the Analysis of Nonlinear Partial Differential Equations, Part 4, Morningside Lect. Math. 4, International Press, Somerville, MA, 2016,115-193.

[20]

J. Mu and W. P. Ziemer, Smooth regularity of solutions of double obstacle problem involving degenerate elliptic equations, Comm. Partial Differential Equations, 16 (1991), 821-843.  doi: 10.1080/03605309108820780.

[21]

A. Petrosyan and T. To, Optimal regularity in rooftop-like obstacle problem, Comm. Partial Differential Equations, 35 (2010), 1292-1325.  doi: 10.1080/03605302.2010.483265.

[22]

O. Savin and H. Yu, On the fine regularity of the singular set in the nonlinear obstacle problem, Nonlinear Anal., 218 (2022), Paper No. 112770. doi: 10.1016/j.na.2021.112770.

[23]

O. Savin and H. Yu, Regularity of the singular set in the fully nonlinear obstacle problem, to appear, J. Eur. Math. Soc.

[24]

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

[25]

R. Teymurazyan and J. M. Urbano, A free boundary optimization problem for the $\infty$-Laplacian, J. Differential Equations, 263 (2017), 1140-1159.  doi: 10.1016/j.jde.2017.03.010.

[26]

H. Yu, An optimization problem in heat conduction with minimal temperature constraint, interior heating and exterior insulation, Calc. Var. Partial Differential Equations, 55 (2016), Art. 130, 15 pp. doi: 10.1007/s00526-016-1071-y.

show all references

References:
[1]

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.

[2]

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

[3]

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

[4]

L. AmbrosioI. FonsecaP. Marcellini and L. Tartar, On a volume constrained variational problem, Arch. Ration. Mech. Anal., 149 (1999), 23-47.  doi: 10.1007/s002050050166.

[5]

H. Brézis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1973/74), 831-844.  doi: 10.1512/iumj.1974.23.23069.

[6]

L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations, 5 (1980), 427-448.  doi: 10.1080/0360530800882144.

[7]

L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402.  doi: 10.1007/BF02498216.

[8]

L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Anal. Math., 37 (1980), 285-295.  doi: 10.1007/BF02797689.

[9]

M. Chipot, Sur la régularité de la solution d'inéquations variationnelles elliptiques, C. R. Acad. Sci. Paris Sér., 288 (1979), 543-546. 

[10]

H. J. Choe, Regularity for certain degenerate elliptic double obstacle problems, J. Math. Anal. Appl., 169 (1992), 111-126.  doi: 10.1016/0022-247X(92)90106-N.

[11]

G. Dal MasoU. Mosco and M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem, Acta Math., 163 (1989), 57-107.  doi: 10.1007/BF02392733.

[12]

J. Fernández BonderS. Martínez and N. Wolanski, An optimization problem with volume constraint for a degenerate quasilinear operator, J. Differential Equations, 227 (2006), 80-101.  doi: 10.1016/j.jde.2006.03.006.

[13]

J. Fernández 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), 1612-1635.  doi: 10.1137/040613615.

[14]

A. Figalli and J. Serra, On the fine structure of the free boundary for the classical obstacle problem, Invent. Math., 215 (2019), 311-366.  doi: 10.1007/s00222-018-0827-8.

[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Equatios of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2001. 
[16]

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

[17]

G. P. Leonardi and P. Tilli, On a constrained variational problem in the vector-valued case, J. Math. Pures Appl., 85 (2006), 251-268.  doi: 10.1016/j.matpur.2005.07.004.

[18]

G. M. Lieberman, Regularity of solutions to some degenerate double obstacle problems, Indiana Univ. Math. J., 40 (1991), 1009-1028.  doi: 10.1512/iumj.1991.40.40045.

[19]

F. Lin, Lectures on elliptic free boundary problems, in Lectures on the Analysis of Nonlinear Partial Differential Equations, Part 4, Morningside Lect. Math. 4, International Press, Somerville, MA, 2016,115-193.

[20]

J. Mu and W. P. Ziemer, Smooth regularity of solutions of double obstacle problem involving degenerate elliptic equations, Comm. Partial Differential Equations, 16 (1991), 821-843.  doi: 10.1080/03605309108820780.

[21]

A. Petrosyan and T. To, Optimal regularity in rooftop-like obstacle problem, Comm. Partial Differential Equations, 35 (2010), 1292-1325.  doi: 10.1080/03605302.2010.483265.

[22]

O. Savin and H. Yu, On the fine regularity of the singular set in the nonlinear obstacle problem, Nonlinear Anal., 218 (2022), Paper No. 112770. doi: 10.1016/j.na.2021.112770.

[23]

O. Savin and H. Yu, Regularity of the singular set in the fully nonlinear obstacle problem, to appear, J. Eur. Math. Soc.

[24]

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

[25]

R. Teymurazyan and J. M. Urbano, A free boundary optimization problem for the $\infty$-Laplacian, J. Differential Equations, 263 (2017), 1140-1159.  doi: 10.1016/j.jde.2017.03.010.

[26]

H. Yu, An optimization problem in heat conduction with minimal temperature constraint, interior heating and exterior insulation, Calc. Var. Partial Differential Equations, 55 (2016), Art. 130, 15 pp. doi: 10.1007/s00526-016-1071-y.

[1]

Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations and Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017

[2]

Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101

[3]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[4]

Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365

[5]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[6]

Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122

[7]

Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605

[8]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049

[9]

N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549

[10]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

[11]

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres. Fractional order optimal control problems with free terminal time. Journal of Industrial and Management Optimization, 2014, 10 (2) : 363-381. doi: 10.3934/jimo.2014.10.363

[12]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

[13]

H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. Mathematical Biosciences & Engineering, 2015, 12 (4) : 739-760. doi: 10.3934/mbe.2015.12.739

[14]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control and Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[15]

Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046

[16]

H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass, Jay A. Rosenheim. Optimal design for dynamical modeling of pest populations. Mathematical Biosciences & Engineering, 2018, 15 (4) : 993-1010. doi: 10.3934/mbe.2018044

[17]

K.F.C. Yiu, K.L. Mak, K. L. Teo. Airfoil design via optimal control theory. Journal of Industrial and Management Optimization, 2005, 1 (1) : 133-148. doi: 10.3934/jimo.2005.1.133

[18]

Boris P. Belinskiy. Optimal design of an optical length of a rod with the given mass. Conference Publications, 2007, 2007 (Special) : 85-91. doi: 10.3934/proc.2007.2007.85

[19]

Xueling Zhou, Bingo Wing-Kuen Ling, Hai Huyen Dam, Kok-Lay Teo. Optimal design of window functions for filter window bank. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1119-1145. doi: 10.3934/jimo.2020014

[20]

Yannick Privat, Emmanuel Trélat. Optimal design of sensors for a damped wave equation. Conference Publications, 2015, 2015 (special) : 936-944. doi: 10.3934/proc.2015.0936

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (60)
  • HTML views (36)
  • Cited by (0)

Other articles
by authors

[Back to Top]