June  2015, 35(6): 2443-2463. doi: 10.3934/dcds.2015.35.2443

Some mathematical problems related to the second order optimal shape of a crystallisation interface

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received  December 2013 Revised  February 2014 Published  December 2014

We consider the problem to optimise the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimisation principle of the free energy while the temperature is solving the heat equation with radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallisation, we can expect that the interface has a global representation as a graph. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second derivatives of the surface and for the surface temperature gradient.
Citation: Pierre-Étienne Druet. Some mathematical problems related to the second order optimal shape of a crystallisation interface. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2443-2463. doi: 10.3934/dcds.2015.35.2443
References:
[1]

E. Casas and L. A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state,, Appl. Math. Optim., 27 (1993), 35.  doi: 10.1007/BF01182597.  Google Scholar

[2]

M. Dauge, Neumann and mixed problems on curvilinear polyhedra,, Integr. Equat. Oper. Th., 15 (1992), 227.  doi: 10.1007/BF01204238.  Google Scholar

[3]

M. Delfour, G. Payre and J. Zolesio, Approximation of nonlinear problems associated with radiating bodies in space,, SIAM J. Numer. Anal., 24 (1987), 1077.  doi: 10.1137/0724071.  Google Scholar

[4]

W. Dreyer, F. Duderstadt, S. Eichler and M. Naldzhieva, On unwanted nucleation phenomena at the wall of a VGF chamber,, Preprint 1312 of the Weierstass-Institute for Applied Analysis and Stochastics (WIAS), (1312).   Google Scholar

[5]

P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type,, Portugaliae Math., 69 (2012), 233.  doi: 10.4171/PM/1916.  Google Scholar

[6]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer Verlag. Berlin, (2001).   Google Scholar

[7]

M. Hintermüller and K. Kunisch, PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative,, SIAM J. Optim., 20 (2009), 1133.  doi: 10.1137/080737265.  Google Scholar

[8]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes (VI),, J. Analyse Mathématique, 11 (1963), 165.  doi: 10.1007/BF02789983.  Google Scholar

[9]

L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1,, Dunod Paris, (1968).   Google Scholar

[10]

G. Troianiello, Elliptic Differential Equations and Obstacle Problems,, Plenum Press, (1987).  doi: 10.1007/978-1-4899-3614-1.  Google Scholar

[11]

N. Ural'tseva, The solvability of the capillarity problem II,, Vestnik Leningrad Univ., (1975), 143.   Google Scholar

[12]

N. Ural'tseva, Estimates of the maximum moduli of gradients for solutions of capillary problems,, Zapiski Nauchn. Sem. LOMI, 115 (1982), 274.   Google Scholar

[13]

A. Visintin, Models of Phase Transitions,, Birkäuser, (1996).  doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[14]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces,, Appl. Math. Optim., 5 (1979), 49.  doi: 10.1007/BF01442543.  Google Scholar

show all references

References:
[1]

E. Casas and L. A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state,, Appl. Math. Optim., 27 (1993), 35.  doi: 10.1007/BF01182597.  Google Scholar

[2]

M. Dauge, Neumann and mixed problems on curvilinear polyhedra,, Integr. Equat. Oper. Th., 15 (1992), 227.  doi: 10.1007/BF01204238.  Google Scholar

[3]

M. Delfour, G. Payre and J. Zolesio, Approximation of nonlinear problems associated with radiating bodies in space,, SIAM J. Numer. Anal., 24 (1987), 1077.  doi: 10.1137/0724071.  Google Scholar

[4]

W. Dreyer, F. Duderstadt, S. Eichler and M. Naldzhieva, On unwanted nucleation phenomena at the wall of a VGF chamber,, Preprint 1312 of the Weierstass-Institute for Applied Analysis and Stochastics (WIAS), (1312).   Google Scholar

[5]

P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type,, Portugaliae Math., 69 (2012), 233.  doi: 10.4171/PM/1916.  Google Scholar

[6]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer Verlag. Berlin, (2001).   Google Scholar

[7]

M. Hintermüller and K. Kunisch, PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative,, SIAM J. Optim., 20 (2009), 1133.  doi: 10.1137/080737265.  Google Scholar

[8]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes (VI),, J. Analyse Mathématique, 11 (1963), 165.  doi: 10.1007/BF02789983.  Google Scholar

[9]

L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1,, Dunod Paris, (1968).   Google Scholar

[10]

G. Troianiello, Elliptic Differential Equations and Obstacle Problems,, Plenum Press, (1987).  doi: 10.1007/978-1-4899-3614-1.  Google Scholar

[11]

N. Ural'tseva, The solvability of the capillarity problem II,, Vestnik Leningrad Univ., (1975), 143.   Google Scholar

[12]

N. Ural'tseva, Estimates of the maximum moduli of gradients for solutions of capillary problems,, Zapiski Nauchn. Sem. LOMI, 115 (1982), 274.   Google Scholar

[13]

A. Visintin, Models of Phase Transitions,, Birkäuser, (1996).  doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[14]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces,, Appl. Math. Optim., 5 (1979), 49.  doi: 10.1007/BF01442543.  Google Scholar

[1]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

[2]

J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431

[3]

Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657

[4]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[5]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[6]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001

[7]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022

[8]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341

[9]

Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010

[10]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[11]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[12]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235

[13]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231

[14]

Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024

[15]

Andrei V. Dmitruk, Alexander M. Kaganovich. Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 523-545. doi: 10.3934/dcds.2011.29.523

[16]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505

[17]

Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213

[18]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096

[19]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[20]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]