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

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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

Pierre-Étienne Druet ^1,

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

Received December 2013 Revised February 2014 Published December 2014

Abstract
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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.

Keywords: optimal control, Stefan-Gibbs-Thomson problem, singularity of mean-curvature type, first order optimality conditions., pointwise gradient state constraints.

Mathematics Subject Classification: Primary: 49K20, 80A22, 53A10, 35J2.

Citation: Pierre-Étienne Druet. Some mathematical problems related to the second order optimal shape of a crystallisation interface. Discrete and Continuous Dynamical Systems, 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), 35-56. doi: 10.1007/BF01182597.
[2]	M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238.
[3]	M. Delfour, G. Payre and J. Zolesio, Approximation of nonlinear problems associated with radiating bodies in space, SIAM J. Numer. Anal., 24 (1987), 1077-1094. doi: 10.1137/0724071.
[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), Berlin, 2008, Available in pdf-format at http://www.wias-berlin.de/preprint/1312/wias_preprints_1312.pdf.
[5]	P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type, Portugaliae Math., 69 (2012), 233-258. doi: 10.4171/PM/1916.
[6]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer Verlag. Berlin, Heidelberg, 2001.
[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), 1133-1156. doi: 10.1137/080737265.
[8]	J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes (VI), J. Analyse Mathématique, 11 (1963), 165-188, French. doi: 10.1007/BF02789983.
[9]	L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1, Dunod Paris, Paris, 1968, French.
[10]	G. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987. doi: 10.1007/978-1-4899-3614-1.
[11]	N. Ural'tseva, The solvability of the capillarity problem II, Vestnik Leningrad Univ., no 1, (1975), 143-149, Russian.
[12]	N. Ural'tseva, Estimates of the maximum moduli of gradients for solutions of capillary problems, Zapiski Nauchn. Sem. LOMI, 115 (1982), 274-284, Russian. English translation in J. Soviet Math, 28 (1985), 806-815.
[13]	A. Visintin, Models of Phase Transitions, Birkäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.
[14]	J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62. doi: 10.1007/BF01442543.

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), 35-56. doi: 10.1007/BF01182597.
[2]	M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equat. Oper. Th., 15 (1992), 227-261. doi: 10.1007/BF01204238.
[3]	M. Delfour, G. Payre and J. Zolesio, Approximation of nonlinear problems associated with radiating bodies in space, SIAM J. Numer. Anal., 24 (1987), 1077-1094. doi: 10.1137/0724071.
[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), Berlin, 2008, Available in pdf-format at http://www.wias-berlin.de/preprint/1312/wias_preprints_1312.pdf.
[5]	P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type, Portugaliae Math., 69 (2012), 233-258. doi: 10.4171/PM/1916.
[6]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer Verlag. Berlin, Heidelberg, 2001.
[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), 1133-1156. doi: 10.1137/080737265.
[8]	J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes (VI), J. Analyse Mathématique, 11 (1963), 165-188, French. doi: 10.1007/BF02789983.
[9]	L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, vol. 1, Dunod Paris, Paris, 1968, French.
[10]	G. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987. doi: 10.1007/978-1-4899-3614-1.
[11]	N. Ural'tseva, The solvability of the capillarity problem II, Vestnik Leningrad Univ., no 1, (1975), 143-149, Russian.
[12]	N. Ural'tseva, Estimates of the maximum moduli of gradients for solutions of capillary problems, Zapiski Nauchn. Sem. LOMI, 115 (1982), 274-284, Russian. English translation in J. Soviet Math, 28 (1985), 806-815.
[13]	A. Visintin, Models of Phase Transitions, Birkäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.
[14]	J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62. doi: 10.1007/BF01442543.

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