相变梯度理论中的边界效应
Boundary effects in the gradient theory of phase transitions
摘要：
我们考虑在第一象限给定的平面区域中的具有尺度小参数ε的van der Waals自由能泛函和非齐次Dirichlet边界条件。选择边界数据的方式应使纯相之间的界面趋于水平，并固定在y轴上的某个点上，该点随着θ收敛到零而接近零。我们发现，存在一个临界尺度的钉扎点，这样，作为小参数的ε趋于零，从边界排斥和惩罚梯度的竞争效应发挥作用，在确定最佳形状的（适当重新缩放）接口。这一结果是通过自由能泛函的渐近发展来实现的。因此，这种分析不仅限于极小化，而且还编码波动的渐近概率。
英文摘要：
We consider the van der Waals' free energy functional, with a scaling small parameter epsilon, in the plane domain given by the first quadrant, and inhomogeneous Dirichlet boundary conditions. The boundary data are chosen in such a way that the interface between the pure phases tends to be horizontal and is pinned at some point on the y-axis which approaches zero as epsilon converges to zero. We show that there exists a critical scaling for the pinning point, such that, as the small parameter epsilon tends to zero, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
