• Previous Article
    Nomonotone spectral gradient method for sparse recovery
  • IPI Home
  • This Issue
  • Next Article
    Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map
August  2015, 9(3): 791-814. doi: 10.3934/ipi.2015.9.791

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

1. 

Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shan'xi, China, China

2. 

LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China

3. 

LSEC, NCMIS, Institute of Systems Science, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China

Received  November 2014 Revised  March 2015 Published  July 2015

This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.
Citation: Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems and Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984. doi: 10.1088/0266-5611/19/4/312.

[3]

G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comp. Appl. Math., 198 (2007), 307-320. doi: 10.1016/j.cam.2005.06.048.

[4]

S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563. doi: 10.1088/0266-5611/12/5/002.

[5]

M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336. doi: 10.1088/0266-5611/22/4/012.

[6]

F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936. doi: 10.1088/0266-5611/21/6/008.

[7]

M. Berggren, Approximation of very weak solutions to boundary value problems, SIAM J. Numer. Anal., 42 (2004), 860-877. doi: 10.1137/S0036142903382048.

[8]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp. doi: 10.1088/0266-5611/26/9/095016.

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{th}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[10]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal., 35 (2001), 595-605. doi: 10.1051/m2an:2001128.

[11]

E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611. doi: 10.1137/050626600.

[12]

A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206. doi: 10.1088/0266-5611/22/4/005.

[13]

J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, Z. Angew. Math. Mech., 81 (2001), 665-674. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.

[14]

P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978. doi: 10.1137/1.9780898719208.

[15]

A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570. doi: 10.1088/0266-5611/17/3/313.

[16]

F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538. doi: 10.1007/BF02576643.

[17]

K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819. doi: 10.1137/080735369.

[18]

H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884. doi: 10.1081/NFA-100108313.

[19]

D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), 299-314. doi: 10.1080/01630569108816430.

[20]

A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements, J. Ind. Univ. Math., 38 (1989), 527-556. doi: 10.1512/iumj.1989.38.38025.

[21]

A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation, Tr. Mosk. Mat. Obs., 52 (1991), 139-176.

[22]

H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations, J. Inverse Ill-Posed Problems, 5 (1997), 437-454. doi: 10.1515/jiip.1997.5.5.437.

[23]

W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008.

[24]

W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems, 23 (2007), 2401-2415. doi: 10.1088/0266-5611/23/6/008.

[25]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.

[26]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009.

[27]

G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994. doi: 10.1088/0266-5611/13/4/006.

[28]

V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, New York, 2006.

[29]

M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083. doi: 10.1080/0003681021000029819.

[30]

S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed problems, 3 (1995), 21-46. doi: 10.1515/jiip.1995.3.1.21.

[31]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085.

[32]

R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513.

[33]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52.

[34]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications], Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967.

[35]

R. Li and W. Liu, The C++ software library AFEPack., Available from: , (). 

[36]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[37]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972.

[38]

M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting, in Industrial Engineering and Systems Management (IESM), Proceedings of 2013 International Conference on, Rabat, Morocco, 2013, p431.

[39]

S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces, Numer. Funct. Anal. Optim., 21 (2000), 901-916. doi: 10.1080/01630560008816993.

[40]

N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995.

[41]

A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington D.C., 1977.

[42]

W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body, Inverse Problems, 17 (2001), 1957-1975. doi: 10.1088/0266-5611/17/6/325.

[43]

K. Yosida, Functional Analysis, $5^{th}$ edition, Springer-Verlag, Berlin-New York, 1978.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984. doi: 10.1088/0266-5611/19/4/312.

[3]

G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comp. Appl. Math., 198 (2007), 307-320. doi: 10.1016/j.cam.2005.06.048.

[4]

S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563. doi: 10.1088/0266-5611/12/5/002.

[5]

M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336. doi: 10.1088/0266-5611/22/4/012.

[6]

F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936. doi: 10.1088/0266-5611/21/6/008.

[7]

M. Berggren, Approximation of very weak solutions to boundary value problems, SIAM J. Numer. Anal., 42 (2004), 860-877. doi: 10.1137/S0036142903382048.

[8]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp. doi: 10.1088/0266-5611/26/9/095016.

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{th}$ edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[10]

M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal., 35 (2001), 595-605. doi: 10.1051/m2an:2001128.

[11]

E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611. doi: 10.1137/050626600.

[12]

A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206. doi: 10.1088/0266-5611/22/4/005.

[13]

J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, Z. Angew. Math. Mech., 81 (2001), 665-674. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V.

[14]

P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978. doi: 10.1137/1.9780898719208.

[15]

A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570. doi: 10.1088/0266-5611/17/3/313.

[16]

F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538. doi: 10.1007/BF02576643.

[17]

K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819. doi: 10.1137/080735369.

[18]

H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884. doi: 10.1081/NFA-100108313.

[19]

D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), 299-314. doi: 10.1080/01630569108816430.

[20]

A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements, J. Ind. Univ. Math., 38 (1989), 527-556. doi: 10.1512/iumj.1989.38.38025.

[21]

A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation, Tr. Mosk. Mat. Obs., 52 (1991), 139-176.

[22]

H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations, J. Inverse Ill-Posed Problems, 5 (1997), 437-454. doi: 10.1515/jiip.1997.5.5.437.

[23]

W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008.

[24]

W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems, 23 (2007), 2401-2415. doi: 10.1088/0266-5611/23/6/008.

[25]

M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5.

[26]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009.

[27]

G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994. doi: 10.1088/0266-5611/13/4/006.

[28]

V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, New York, 2006.

[29]

M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083. doi: 10.1080/0003681021000029819.

[30]

S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed problems, 3 (1995), 21-46. doi: 10.1515/jiip.1995.3.1.21.

[31]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085.

[32]

R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667. doi: 10.1002/cpa.3160380513.

[33]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52.

[34]

R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications], Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967.

[35]

R. Li and W. Liu, The C++ software library AFEPack., Available from: , (). 

[36]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[37]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972.

[38]

M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting, in Industrial Engineering and Systems Management (IESM), Proceedings of 2013 International Conference on, Rabat, Morocco, 2013, p431.

[39]

S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces, Numer. Funct. Anal. Optim., 21 (2000), 901-916. doi: 10.1080/01630560008816993.

[40]

N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995.

[41]

A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington D.C., 1977.

[42]

W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body, Inverse Problems, 17 (2001), 1957-1975. doi: 10.1088/0266-5611/17/6/325.

[43]

K. Yosida, Functional Analysis, $5^{th}$ edition, Springer-Verlag, Berlin-New York, 1978.

[1]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control and Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[2]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[3]

Savin Treanţă. Characterization of efficient solutions for a class of PDE-constrained vector control problems. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 93-106. doi: 10.3934/naco.2019035

[4]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014

[5]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks and Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501

[6]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[7]

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control and Related Fields, 2020, 10 (2) : 333-363. doi: 10.3934/mcrf.2019041

[8]

Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control and Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013

[9]

Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066

[10]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[11]

Juan Carlos De los Reyes, Carola-Bibiane Schönlieb. Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization. Inverse Problems and Imaging, 2013, 7 (4) : 1183-1214. doi: 10.3934/ipi.2013.7.1183

[12]

Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems and Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028

[13]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105

[14]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927

[15]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control and Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017

[16]

Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021220

[17]

Gianmarco Manzini, Annamaria Mazzia. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. Journal of Computational Dynamics, 2022, 9 (2) : 207-238. doi: 10.3934/jcd.2021020

[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[19]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[20]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (147)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]