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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  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 & 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.   Google Scholar [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.  Google Scholar [3] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, , Birkhäuser, 2005. Google Scholar [4] P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equa. Appl., 7 (2000), 187-199.   Google Scholar [5] E. Casas, R. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] C. D'Apice, U. De Maio, P. 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.   Google Scholar [9] C. D'Apice, U. 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.  Google Scholar [10] C. D'Apice, U. 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.   Google Scholar [11] P. Di Gironimo and G. Zecca, Sobolev-Zygmund solutions for nonlinear elliptic equations with growth coefficients in BMO, Submitted. Google Scholar [12] P. Drabek, A. Kufner and F. Nicolosi, Non Linear Elliptic Equations, Singular and Degenerate Cases, (Walter de Cruyter, 1997). Google Scholar [13] T. Durante, O. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [23] G. I. Laptev, Monotonicity conditions for a class of quasilinear differential operators depending on parameters, Math. Notes, 96 (2014), 379-390.   Google Scholar [24] S. Leftkimmiatis, A. 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.  Google Scholar [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.  Google Scholar [26] V. G. Maz'ya, On cetrtain integral inequalities for functions of many variables, J. Soviet Math., 1 (1973), 205-234.   Google Scholar [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.  Google Scholar [28] I. Peral, Multiplicity of Solutions for the $P$-Laplacian, , Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997. Google Scholar [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.  Google Scholar [30] T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000.   Google Scholar [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.  Google Scholar [32] T. Roubíček, Nonlinear Partial Differential Equations with Applications, , Birkhäuser, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [33] E. Saacson and H. B. Keller, Analysis of Numerical Methods, , Wiley, 1966.  Google Scholar [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.   Google Scholar [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). Google Scholar [36] G. Stampacchia, Èquations Elliptiques du Second Ordre à Coefficients Discontinus, Les Presses de L'Universite de Montreal, 1966.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.   Google Scholar [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.  Google Scholar [3] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, , Birkhäuser, 2005. Google Scholar [4] P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Diff. Equa. Appl., 7 (2000), 187-199.   Google Scholar [5] E. Casas, R. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [8] C. D'Apice, U. De Maio, P. 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.   Google Scholar [9] C. D'Apice, U. 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.  Google Scholar [10] C. D'Apice, U. 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.   Google Scholar [11] P. Di Gironimo and G. Zecca, Sobolev-Zygmund solutions for nonlinear elliptic equations with growth coefficients in BMO, Submitted. Google Scholar [12] P. Drabek, A. Kufner and F. Nicolosi, Non Linear Elliptic Equations, Singular and Degenerate Cases, (Walter de Cruyter, 1997). Google Scholar [13] T. Durante, O. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [23] G. I. Laptev, Monotonicity conditions for a class of quasilinear differential operators depending on parameters, Math. Notes, 96 (2014), 379-390.   Google Scholar [24] S. Leftkimmiatis, A. 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.  Google Scholar [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.  Google Scholar [26] V. G. Maz'ya, On cetrtain integral inequalities for functions of many variables, J. Soviet Math., 1 (1973), 205-234.   Google Scholar [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.  Google Scholar [28] I. Peral, Multiplicity of Solutions for the $P$-Laplacian, , Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997. Google Scholar [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.  Google Scholar [30] T. Radice, Regularity result for nondivergence elliptic equations with unbounded coefficients, Diff. Integral Equa., 23 (2010), 989-1000.   Google Scholar [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.  Google Scholar [32] T. Roubíček, Nonlinear Partial Differential Equations with Applications, , Birkhäuser, 2013. doi: 10.1007/978-3-0348-0513-1.  Google Scholar [33] E. Saacson and H. B. Keller, Analysis of Numerical Methods, , Wiley, 1966.  Google Scholar [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.   Google Scholar [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). Google Scholar [36] G. Stampacchia, Èquations Elliptiques du Second Ordre à Coefficients Discontinus, Les Presses de L'Universite de Montreal, 1966.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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