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December  2020, 10(4): 827-854. doi: 10.3934/mcrf.2020021

On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian

1. 

Dipartimento di Matematica e Applicazioni 'R.Caccioppoli', Università di Napoli Federico Ⅱ - Complesso Universitario di Monte S. Angelo, via Cintia 80126, Napoli, Italy

2. 

Department of Differential Equations, Oles Honchar Dnipro National University, Gagarin av., 72, Dnipro, 49010, Ukraine

3. 

Dipartimento di Matematica e Applicazioni 'R.Caccioppoli', Università di Napoli Federico Ⅱ - Complesso Universitario di Monte S. Angelo, via Cintia 80126, Napoli, Italy

* Corresponding author: Gabriella Zecca

Received  July 2019 Revised  December 2019 Published  December 2020 Early access  March 2020

We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principal part and $ L^1 $-control in coefficient of the low-order term. We assume that the matrix of anisotropy belongs to BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we introduce a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces.

Citation: Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control and Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021
References:
[1]

D. J. Bergman and D. Stroud, Physical properties of macroscopically inhomogeneous media, North–HollaSolid State Physics, 46 (1992), 147-269. 

[2]

M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients, J. of Diff. Equa., 245 (2008), 2038-2054.  doi: 10.1016/j.jde.2008.07.027.

[3]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, , Birkhäuser, 2005.

[4]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equa. Appl., 7 (2000), 187-199. 

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$-cost functional, SIAM Journal on Optimization, 22 (2012), 795-820.  doi: 10.1137/110834366.

[6]

V. Chiadò Piat and F. Serra Cassano, Some remarks about the density of smooth functions in weighted Sobolev spaces, J. Convex Analysis, 1 (1994), 135-142. 

[7]

M. Chicco and M. Venturino, Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain, Annali di Matematica Pura ed Applicata, 178 (2000), 325-338.  doi: 10.1007/BF02505902.

[8]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, Solvability of an optimal control problem in coefficients for ill-posed elliptic boundary value problems, Electronic Journal of Differential Equations, 2014 (2014), 1-23. 

[9]

C. D'ApiceU. De Maio and O. P. Kogut, Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems, SIAM J. Control Optim., 50 (2012), 1174-1199.  doi: 10.1137/100815761.

[10]

C. D'ApiceU. De Maio and O. P. Kogut, On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations, Adv. Differential Equations, 15 (2010), 689-720. 

[11]

P. Di Gironimo and G. Zecca, Sobolev-Zygmund solutions for nonlinear elliptic equations with growth coefficients in BMO, Submitted.

[12]

P. Drabek, A. Kufner and F. Nicolosi, Non Linear Elliptic Equations, Singular and Degenerate Cases, (Walter de Cruyter, 1997).

[13]

T. DuranteO. P. Kupenko and R. Manzo, On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian, Ricerche di Matematica, 66 (2017), 259-292.  doi: 10.1007/s11587-016-0300-1.

[14]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426.  doi: 10.1002/cpa.3160140317.

[15]

T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I.Existence result, Mathematical Control and Related Fields, 5 (2015), 73-96.  doi: 10.3934/mcrf.2015.5.73.

[16]

T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188.  doi: 10.3233/ASY-161365.

[17]

F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.  doi: 10.1007/BF02392268.

[18]

P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 2105-2133.  doi: 10.3934/dcds.2014.34.2105.

[19]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, , Series: Systems and Control, Birkhäuser Verlag, 2011. doi: 10.1007/978-0-8176-8149-4.

[20]

P. I. Kogut and G. Leugering, Matrix-valued $L^1$-optimal control in the coefficients of linear elliptic problems, Journal for Analysis and its Applications (ZAA), 32 (2013), 433-456.  doi: 10.4171/ZAA/1493.

[21]

O. P. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), 1-18.  doi: 10.1007/s00030-016-0387-9.

[22]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed non-linear elliptic dirichlet bounday value problems, Discrete and Continuous Dynamical Systems Journal, series B, (2018).

[23]

G. I. Laptev, Monotonicity conditions for a class of quasilinear differential operators depending on parameters, Math. Notes, 96 (2014), 379-390. 

[24]

S. LeftkimmiatisA. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications, IEEE Trans. on Image Process., 21 (2012), 983-995.  doi: 10.1109/TIP.2011.2168232.

[25]

O. Levy and R. V. Kohn, Duality relations for non-ohmic composites, with applications to behavior near percolation, J. Statist. Phys., 90 (1998), 159-189.  doi: 10.1023/A:1023251701546.

[26]

V. G. Maz'ya, On cetrtain integral inequalities for functions of many variables, J. Soviet Math., 1 (1973), 205-234. 

[27]

S. E. Pastukhova, Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494.  doi: 10.1070/SM2007v198n10ABEH003892.

[28]

I. Peral, Multiplicity of Solutions for the $P$-Laplacian, , Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997.

[29]

F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. I.H. Poincaré, 26 (2009), 2001-2024.  doi: 10.1016/j.anihpc.2009.04.005.

[30]

T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. 

[31]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche Mat., 63 (2014), 355-267.  doi: 10.1007/s11587-014-0202-z.

[32]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, , Birkhäuser, 2013. doi: 10.1007/978-3-0348-0513-1.

[33]

E. Saacson and H. B. Keller, Analysis of Numerical Methods, , Wiley, 1966.

[34]

M. V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Nonlinear Partial Diff. Equa. and Related Topics, 229 (2010), 211-232. 

[35]

Ch. Schneider and W. Alt, Regularization of linear-quadratic control problems with $L^1$-control cost, In: Pötzsche C., Heuberger C., Kaltenbacher B., Rendl F. (eds) System Modeling and Optimization. CSMO 2013. IFIP Advances in Information and Communication Technology, Springer, Berlin, Heidelberg, 443 (2014).

[36]

G. Stampacchia, Èquations Elliptiques du Second Ordre à Coefficients Discontinus, Les Presses de L'Universite de Montreal, 1966.

[37]

V. V. Zhikov, Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms, Functional Analysis and Its Applications, 38 (2004), 173-183.  doi: 10.1023/B:FAIA.0000042802.86050.5e.

[38]

V. V. Zhikov and S. E. Pastukhova, Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent, Mat. Sb., 199 (2008), 19–52; translation in Sb. Math., 199 (2008), 1751–1782. doi: 10.1070/SM2008v199n12ABEH003980.

[39]

G. Zecca, An optimal control problem for nonlinear elliptic equations with unbounded coefficients, Discrete and Continous Dynamical Systems, Series B, 24 (2019), 1393-1409.  doi: 10.3934/dcdsb.2019021.

show all references

References:
[1]

D. J. Bergman and D. Stroud, Physical properties of macroscopically inhomogeneous media, North–HollaSolid State Physics, 46 (1992), 147-269. 

[2]

M. Briane and J. Casado-Diaz, Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients, J. of Diff. Equa., 245 (2008), 2038-2054.  doi: 10.1016/j.jde.2008.07.027.

[3]

D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, , Birkhäuser, 2005.

[4]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equa. Appl., 7 (2000), 187-199. 

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$-cost functional, SIAM Journal on Optimization, 22 (2012), 795-820.  doi: 10.1137/110834366.

[6]

V. Chiadò Piat and F. Serra Cassano, Some remarks about the density of smooth functions in weighted Sobolev spaces, J. Convex Analysis, 1 (1994), 135-142. 

[7]

M. Chicco and M. Venturino, Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain, Annali di Matematica Pura ed Applicata, 178 (2000), 325-338.  doi: 10.1007/BF02505902.

[8]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, Solvability of an optimal control problem in coefficients for ill-posed elliptic boundary value problems, Electronic Journal of Differential Equations, 2014 (2014), 1-23. 

[9]

C. D'ApiceU. De Maio and O. P. Kogut, Optimal control problems in coefficients for degenerate equations of monotone type: shape stability and attainability problems, SIAM J. Control Optim., 50 (2012), 1174-1199.  doi: 10.1137/100815761.

[10]

C. D'ApiceU. De Maio and O. P. Kogut, On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations, Adv. Differential Equations, 15 (2010), 689-720. 

[11]

P. Di Gironimo and G. Zecca, Sobolev-Zygmund solutions for nonlinear elliptic equations with growth coefficients in BMO, Submitted.

[12]

P. Drabek, A. Kufner and F. Nicolosi, Non Linear Elliptic Equations, Singular and Degenerate Cases, (Walter de Cruyter, 1997).

[13]

T. DuranteO. P. Kupenko and R. Manzo, On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian, Ricerche di Matematica, 66 (2017), 259-292.  doi: 10.1007/s11587-016-0300-1.

[14]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426.  doi: 10.1002/cpa.3160140317.

[15]

T. Horsin and P. I. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I.Existence result, Mathematical Control and Related Fields, 5 (2015), 73-96.  doi: 10.3934/mcrf.2015.5.73.

[16]

T. Horsin and P. I. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188.  doi: 10.3233/ASY-161365.

[17]

F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.  doi: 10.1007/BF02392268.

[18]

P. I. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 2105-2133.  doi: 10.3934/dcds.2014.34.2105.

[19]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, , Series: Systems and Control, Birkhäuser Verlag, 2011. doi: 10.1007/978-0-8176-8149-4.

[20]

P. I. Kogut and G. Leugering, Matrix-valued $L^1$-optimal control in the coefficients of linear elliptic problems, Journal for Analysis and its Applications (ZAA), 32 (2013), 433-456.  doi: 10.4171/ZAA/1493.

[21]

O. P. Kupenko and R. Manzo, Approximation of an optimal control problem in coefficient for variational inequality with anisotropic p-Laplacian, Nonlinear Differential Equations and Applications (NoDEA), 23 (2016), 1-18.  doi: 10.1007/s00030-016-0387-9.

[22]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed non-linear elliptic dirichlet bounday value problems, Discrete and Continuous Dynamical Systems Journal, series B, (2018).

[23]

G. I. Laptev, Monotonicity conditions for a class of quasilinear differential operators depending on parameters, Math. Notes, 96 (2014), 379-390. 

[24]

S. LeftkimmiatisA. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications, IEEE Trans. on Image Process., 21 (2012), 983-995.  doi: 10.1109/TIP.2011.2168232.

[25]

O. Levy and R. V. Kohn, Duality relations for non-ohmic composites, with applications to behavior near percolation, J. Statist. Phys., 90 (1998), 159-189.  doi: 10.1023/A:1023251701546.

[26]

V. G. Maz'ya, On cetrtain integral inequalities for functions of many variables, J. Soviet Math., 1 (1973), 205-234. 

[27]

S. E. Pastukhova, Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems, Sbornik: Mathematics, 198 (2007), 1465-1494.  doi: 10.1070/SM2007v198n10ABEH003892.

[28]

I. Peral, Multiplicity of Solutions for the $P$-Laplacian, , Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997.

[29]

F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. I.H. Poincaré, 26 (2009), 2001-2024.  doi: 10.1016/j.anihpc.2009.04.005.

[30]

T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000. 

[31]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche Mat., 63 (2014), 355-267.  doi: 10.1007/s11587-014-0202-z.

[32]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, , Birkhäuser, 2013. doi: 10.1007/978-3-0348-0513-1.

[33]

E. Saacson and H. B. Keller, Analysis of Numerical Methods, , Wiley, 1966.

[34]

M. V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Nonlinear Partial Diff. Equa. and Related Topics, 229 (2010), 211-232. 

[35]

Ch. Schneider and W. Alt, Regularization of linear-quadratic control problems with $L^1$-control cost, In: Pötzsche C., Heuberger C., Kaltenbacher B., Rendl F. (eds) System Modeling and Optimization. CSMO 2013. IFIP Advances in Information and Communication Technology, Springer, Berlin, Heidelberg, 443 (2014).

[36]

G. Stampacchia, Èquations Elliptiques du Second Ordre à Coefficients Discontinus, Les Presses de L'Universite de Montreal, 1966.

[37]

V. V. Zhikov, Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms, Functional Analysis and Its Applications, 38 (2004), 173-183.  doi: 10.1023/B:FAIA.0000042802.86050.5e.

[38]

V. V. Zhikov and S. E. Pastukhova, Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent, Mat. Sb., 199 (2008), 19–52; translation in Sb. Math., 199 (2008), 1751–1782. doi: 10.1070/SM2008v199n12ABEH003980.

[39]

G. Zecca, An optimal control problem for nonlinear elliptic equations with unbounded coefficients, Discrete and Continous Dynamical Systems, Series B, 24 (2019), 1393-1409.  doi: 10.3934/dcdsb.2019021.

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