Some mathematical problems related to the 2nd order optimal shape of a crystallization interface
Please always quote using this URN:urn:nbn:de:0296-matheon-11470
- We consider the problem to optimize 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 minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. 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 surface derivatives, and for the surface temperature gradient.
Author: | Pierre-Etienne Druet |
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URN: | urn:nbn:de:0296-matheon-11470 |
Referee: | Fredi Tröltzsch |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2012/07/06 |
Release Date: | 2012/07/06 |
Tag: | |
Institute: | Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) |
MSC-Classification: | 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Kxx Optimality conditions / 49K20 Problems involving partial differential equations |
53-XX DIFFERENTIAL GEOMETRY (For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx) / 53Axx Classical differential geometry / 53A10 Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] | |
80-XX CLASSICAL THERMODYNAMICS, HEAT TRANSFER (For thermodynamics of solids, see 74A15) / 80Axx Thermodynamics and heat transfer / 80A22 Stefan problems, phase changes, etc. [See also 74Nxx] | |
Preprint Number: | 968 |