Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner’s equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped ‘mathematical’ plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

中文翻译：

---

具有不同边界条件的测地线 Loewner 路径

Loewner 类方程受到沿实轴的非恒定边界条件的影响，给出了在上半复平面中生长的狭缝的测地线路径。这个问题是由拉普拉斯生长引起的，其中狭缝代表在扩散场中生长的细指。单个手指沿着由出现在 Loewner 方程中的强制函数确定的弯曲路径。该函数是通过求解常微分方程找到的，该方程的项取决于共形映射的“数学”平面中扩散场流线的曲率特性。边界条件指定沿实轴的场变量的分段常数值或放置在实轴上的偶极子的影响，揭示了狭缝生长的一系列行为。这些包括沿着实轴的区域，从那里不可能生长狭缝，路径增长到无限的区域，或路径在有限时间内向实轴弯曲的区域。还计算了沿实轴受分段常数边界条件影响的对称路径对，证明了增长到无穷大的路径向 π/5 的分叉角渐近演化。