半直线上带固定边界不同边界条件的多相Stefan问题
On a multi-phase Stefan problem in the half-line with different boundary conditions at the fixed boundary
AI总结：
该研究针对半直线上带不同边界条件的多相Stefan问题，通过变分方法建立解的存在唯一性，确定了Dirichlet-to-Neumann映射的性质，并推广到Robin边界情形，还研究了负连接系数的不适定问题。
AI中文摘要：
我们研究半直线x>0上热方程的多相Stefan问题的自相似解，该问题具有常数初始数据，且在固定边界x=0处带有Dirichlet、Neumann或Robin边界条件。在Dirichlet边界条件的情形下，我们证明了用于确定自由边界的非线性代数系统是梯度系统，且对应的势函数是显式写出的严格凸且强制的函数。因此，该势函数存在唯一的极小点，该点的坐标确定了自由边界并给出了所需的解。在Neumann边界条件的情形下，研究因以下事实而变得复杂：解经历的相变数量（称为其类型）无法直接从边界数据确定。对于每个固定类型n，用于确定自由边界的系统同样是梯度系统，且对应的势函数被证明在某个更宽的非物理域内是严格凸且强制的。基于这些性质，我们证明了Dirichlet-to-Neumann映射是严格递增的连续函数。这使得我们能够建立Stefan-Neumann问题解的存在性与唯一性，并确定该解的类型。该技术随后被进一步应用于具有正连接系数的Stefan-Robin问题。在最后一节中，我们还使用前几节开发的变分方法研究了具有负连接系数的某些特定不适定Stefan-Robin问题。
英文摘要：
We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the half-line $x>0$ with a constant initial data and with Dirichlet, Neumann or Robin boundary condition at the fixed boundary $x=0$. In the case of Dirichlet boundary condition we prove that a nonlinear algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, coordinates of this point determine free boundaries and provide the desired solution. In the case of Neumann boundary condition the study is complicated by the fact that number of phase transitions undergone by a solution (called its type) cannot be directly determined from the boundary data. For each fixed type $n$ the system for determination of the free boundaries is again gradient and the corresponding potential is proved to be strictly convex and coercive, but in some wider non-physical domain. On the base of these properties it is proved that the Dirichlet-to-Neumann map is a strictly increasing continuous function. This allows to establish existence and uniqueness of a solution to Stefan-Neumann problem, and to specify the type of this solution. The same technique is further applied to Stefan-Robin problem with a positive connection coefficient. In the last section we also study some particular ill-posed Stefan-Robin problem with a negative connection coefficient, using again the variational approach developed in the previous sections.
