March  2019, 24(3): 1393-1409. doi: 10.3934/dcdsb.2019021

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte Sant'Angelo, Via Cintia - 80126 Napoli, Italy

Dedicated to the memory of Prof. V. S. Melnik

Received  October 2017 Revised  March 2018 Published  January 2019

We study an optimal control problem associated to a Dirichlet boundary value problem of the type
$ \begin{equation*} \mbox{(BVP) }\,\,\,\,\,\,\,\, {\rm{div}}\; \left[ \beta(x)\nabla u (x) + \left( A\frac {x}{|x|^2 } +g(x) \right)u(x)\right] = {\rm{div}}\; \mathcal F,\quad u\in W^{1,p}_0( \Omega ) , \end{equation*} $
$ 1<p\leqslant 2, $
where
$ \Omega $
is a bounded regular domain of
$ \mathbb{R}^N $
,
$ 0\in \Omega , $
$ \beta: \Omega \rightarrow {\mathbb R} $
is an unbounded function satisfying
$ \beta(x)\geqslant\lambda_0>0 $
a.e.,
$ A $
is a suitably small constant, and
$ g\in L^\infty( \Omega ; \mathbb{R}^N ) $
.
We consider the vector field
$ \mathcal F $
as the control and the corresponding weak solution
$ u $
to (BVP) as the state. Our aim is to find the optimal vector field
$ \mathcal F\in L^p( \Omega ) $
so that the corresponding state
$ u\in W^{1,p}_0( \Omega ) $
is close to the desired profile in
$ L^p( \Omega ) $
while the norm of
$ u $
in
$ W^{1,p}( \Omega ) $
is not too large.
We prove that, for every
$ p $
less than
$ 2 $
and suitably close to
$ 2 $
, (BVP) admits an unique weak solution and for such values of
$ p $
, we prove the existence of optimal pairs.
Citation: Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
References:
[1]

A. Alvino, Sulla disuguaglianza di Sobolev in Spazi di Lorentz, Boll. Un. Mat. It. A (5), 14 (1977), 148-156. Google Scholar

[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Google Scholar
[3]

L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Un. Mat. It. (9), 2 (2009), 285-297. Google Scholar

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009. Google Scholar

[5]

L. Boccardo, Quelques problèmes de Dirichlet avec données dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269–1272. doi: 10.1016/S0764-4442(97)82351-8. Google Scholar

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H. Brézis and L. Nirenberg, Degree theory and BMO-Part Ⅰ: Compact manifolds with boundary, Selecta Math., 1 (1995), 197-263. doi: 10.1007/BF01671566. Google Scholar

[7]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997), 599-607. Google Scholar

[8]

M. CarozzaG. Moscariello and A. Passarelli di Napoli, Nonlinear equations with growth coefficients in BMO, Houston Journal of Mathematics, 28 (2002), 917-929. Google Scholar

[9]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, On the 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

[10]

C. D'ApiceU. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inf., 25 (2008), 461-489. doi: 10.1093/imamci/dnn010. Google Scholar

[11]

U. De Maio P. Kogut and G. Zecca, On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problem with $BMO$-anisotropic $p$-Laplacian, Applicable Analysis, (2019) to appear.Google Scholar

[12]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$, Studia Mathematica, 127 (1998), 223-231. Google Scholar

[13]

F. GiannettiL. Greco and G. Moscariello, Linear elliptic equations with lower order terms, Differential and Integral Equations, 26 (2013), 623-638. Google Scholar

[14]

O. Giubé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258. doi: 10.1007/s11118-006-9011-7. Google Scholar

[15]

L. Greco and G. Moscariello, An embedding Theorem in Lorentz-Zygmund spaces, Potential Anal., 5 (1996), 581-590. doi: 10.1007/BF00275795. Google Scholar

[16]

L. GrecoG. Moscariello and T. Radice, Nondivergence elliptic equations with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 131-143. doi: 10.3934/dcdsb.2009.11.131. Google Scholar

[17]

L. GrecoG. Moscariello and G. Zecca, Regularity for solutions to nonlinear elliptic equations, Differential and Integral Equations, 26 (2013), 1105-1113. Google Scholar

[18]

L. Greco, G. Moscariello and G. Zecca, An obstacle problem for noncoercive operators, Abstract and Applied Analysis, 2015 (2015), Article ID 890289, 8pp. doi: 10.1155/2015/890289. Google Scholar

[19]

L. Greco, G. Moscariello and G. Zecca, Very weak solutions to elliptic equations with singular convection term, Journal of Mathematical Analysis and Applications, 457 (2018), 1376–1387. doi: 10.1016/j.jmaa.2017.03.025. Google Scholar

[20]

T. Horsin and P. 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

[21]

T. Horsin and P. 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

[22]

T. HorsinP. Kogut and O. Wilk, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅱ. Approximation of solutions and optimality conditions, Mathematical Control and Related Fields, 6 (2016), 595-628. doi: 10.3934/mcrf.2016017. Google Scholar

[23]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143. Google Scholar

[24]

N. 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

[25]

H. Kim and Y. Kim, On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290. doi: 10.1137/14096270X. Google Scholar

[26]

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

[27]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104. Google Scholar

[28]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34. Google Scholar

[29]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comm. Math. Univ. Carolinae, 25 (1984), 537-554. Google Scholar

[30]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed nonlinear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamical Systems, Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155. Google Scholar

[31]

J. L. Lions, Optimal Control of System Governed by Partial Differential Equations, Springer, Berlin, 1971. Google Scholar

[32]

S. Monsurrò and M. Transirico, Noncoercive elliptic equations with discontinuous coefficients in unbounded domains, Nonlinear Analysis, 163 (2017), 86-103. doi: 10.1016/j.na.2017.07.008. Google Scholar

[33]

G. Moscariello, Existence and uniquesness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444. doi: 10.1515/ACV.2011.007. Google Scholar

[34]

R. O'Neil, Integral transforms and tensor products on Orlicz spaces and $L(p, \, q)$ spaces, J. Analyse Math., 21 (1968), 1-276. Google Scholar

[35]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche di Matematica, 63 (2014), 355–367. doi: 10.1007/s11587-014-0202-z. Google Scholar

[36]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 385-387. Google Scholar

[37]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. doi: 10.5802/aif.204. Google Scholar

[38]

B. Stroffolini, Elliptic systems of PDE with BMO coefficients, Potential analysis, 15 (2001), 285-299. doi: 10.1023/A:1011290420956. Google Scholar

[39]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556. Google Scholar

[40]

G. Zecca, Existence and uniqueness for nonlinear elliptic equations with lower-order terms, Nonlinear Analysis, 75 (2012), 899-912. doi: 10.1016/j.na.2011.09.022. Google Scholar

[41]

M. Zgurovsky, V. S. Mel'nik and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Operator Inclusions and Variation Inequalities for Earth Data Processing, Advances in Mechanics and Mathematics, 24. Springer-Verlag, Berlin, 2011. Google Scholar

[42]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511. doi: 10.4213/rm9710. Google Scholar

[43]

V. V. Zhikov, Remarks on the uniqueness of the solution of the Dirichlet problem for a second-order elliptic equation with lower order terms, Funct. Anal. Appl., 38 (2004), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e. Google Scholar

show all references

References:
[1]

A. Alvino, Sulla disuguaglianza di Sobolev in Spazi di Lorentz, Boll. Un. Mat. It. A (5), 14 (1977), 148-156. Google Scholar

[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Google Scholar
[3]

L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Un. Mat. It. (9), 2 (2009), 285-297. Google Scholar

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009. Google Scholar

[5]

L. Boccardo, Quelques problèmes de Dirichlet avec données dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269–1272. doi: 10.1016/S0764-4442(97)82351-8. Google Scholar

[6]

H. Brézis and L. Nirenberg, Degree theory and BMO-Part Ⅰ: Compact manifolds with boundary, Selecta Math., 1 (1995), 197-263. doi: 10.1007/BF01671566. Google Scholar

[7]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997), 599-607. Google Scholar

[8]

M. CarozzaG. Moscariello and A. Passarelli di Napoli, Nonlinear equations with growth coefficients in BMO, Houston Journal of Mathematics, 28 (2002), 917-929. Google Scholar

[9]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, On the 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

[10]

C. D'ApiceU. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inf., 25 (2008), 461-489. doi: 10.1093/imamci/dnn010. Google Scholar

[11]

U. De Maio P. Kogut and G. Zecca, On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problem with $BMO$-anisotropic $p$-Laplacian, Applicable Analysis, (2019) to appear.Google Scholar

[12]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$, Studia Mathematica, 127 (1998), 223-231. Google Scholar

[13]

F. GiannettiL. Greco and G. Moscariello, Linear elliptic equations with lower order terms, Differential and Integral Equations, 26 (2013), 623-638. Google Scholar

[14]

O. Giubé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258. doi: 10.1007/s11118-006-9011-7. Google Scholar

[15]

L. Greco and G. Moscariello, An embedding Theorem in Lorentz-Zygmund spaces, Potential Anal., 5 (1996), 581-590. doi: 10.1007/BF00275795. Google Scholar

[16]

L. GrecoG. Moscariello and T. Radice, Nondivergence elliptic equations with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 131-143. doi: 10.3934/dcdsb.2009.11.131. Google Scholar

[17]

L. GrecoG. Moscariello and G. Zecca, Regularity for solutions to nonlinear elliptic equations, Differential and Integral Equations, 26 (2013), 1105-1113. Google Scholar

[18]

L. Greco, G. Moscariello and G. Zecca, An obstacle problem for noncoercive operators, Abstract and Applied Analysis, 2015 (2015), Article ID 890289, 8pp. doi: 10.1155/2015/890289. Google Scholar

[19]

L. Greco, G. Moscariello and G. Zecca, Very weak solutions to elliptic equations with singular convection term, Journal of Mathematical Analysis and Applications, 457 (2018), 1376–1387. doi: 10.1016/j.jmaa.2017.03.025. Google Scholar

[20]

T. Horsin and P. 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

[21]

T. Horsin and P. 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

[22]

T. HorsinP. Kogut and O. Wilk, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅱ. Approximation of solutions and optimality conditions, Mathematical Control and Related Fields, 6 (2016), 595-628. doi: 10.3934/mcrf.2016017. Google Scholar

[23]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143. Google Scholar

[24]

N. 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

[25]

H. Kim and Y. Kim, On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290. doi: 10.1137/14096270X. Google Scholar

[26]

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

[27]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104. Google Scholar

[28]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34. Google Scholar

[29]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comm. Math. Univ. Carolinae, 25 (1984), 537-554. Google Scholar

[30]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed nonlinear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamical Systems, Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155. Google Scholar

[31]

J. L. Lions, Optimal Control of System Governed by Partial Differential Equations, Springer, Berlin, 1971. Google Scholar

[32]

S. Monsurrò and M. Transirico, Noncoercive elliptic equations with discontinuous coefficients in unbounded domains, Nonlinear Analysis, 163 (2017), 86-103. doi: 10.1016/j.na.2017.07.008. Google Scholar

[33]

G. Moscariello, Existence and uniquesness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444. doi: 10.1515/ACV.2011.007. Google Scholar

[34]

R. O'Neil, Integral transforms and tensor products on Orlicz spaces and $L(p, \, q)$ spaces, J. Analyse Math., 21 (1968), 1-276. Google Scholar

[35]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche di Matematica, 63 (2014), 355–367. doi: 10.1007/s11587-014-0202-z. Google Scholar

[36]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 385-387. Google Scholar

[37]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. doi: 10.5802/aif.204. Google Scholar

[38]

B. Stroffolini, Elliptic systems of PDE with BMO coefficients, Potential analysis, 15 (2001), 285-299. doi: 10.1023/A:1011290420956. Google Scholar

[39]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556. Google Scholar

[40]

G. Zecca, Existence and uniqueness for nonlinear elliptic equations with lower-order terms, Nonlinear Analysis, 75 (2012), 899-912. doi: 10.1016/j.na.2011.09.022. Google Scholar

[41]

M. Zgurovsky, V. S. Mel'nik and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Operator Inclusions and Variation Inequalities for Earth Data Processing, Advances in Mechanics and Mathematics, 24. Springer-Verlag, Berlin, 2011. Google Scholar

[42]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511. doi: 10.4213/rm9710. Google Scholar

[43]

V. V. Zhikov, Remarks on the uniqueness of the solution of the Dirichlet problem for a second-order elliptic equation with lower order terms, Funct. Anal. Appl., 38 (2004), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e. Google Scholar

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