Asymptotic expansion at a corner for the capillary problem: The singular case
Consider the capillary equation \[ \text{div} Tu=\kappa u,\qquad Tu={\nabla u\over\sqrt{1+|\nabla u|^ 2}},\qquad\kappa>0, \] in a domain \(\Omega\) containing a protruding corner of opening angle \(2\alpha\), and with contact angle \(\gamma\) that is constant on \(\partial\Omega\) in a neighborhood of the vertex point \(V\), \(0\leq\gamma<\pi/2\). In earlier work, Concus and Finn demonstrated a dichotomy of behavior, showing that every solution is bounded at \(V\) if \(\alpha+\gamma\geq\pi/2\), while in the complementary (singular) case there holds \(| u-h_{-1}(\vartheta)r^{-1}|<\text{const}\), with \(h_{-1}(\vartheta)=\Bigl(\cos\vartheta-\sqrt{k^ 2-\sin^ 2\vartheta}\Bigr)(k\kappa )^{-1}\), \(k=\sin\alpha/\cos\gamma\) in polar coordinates centered at \(V\), so that all solutions are unbounded. In the present paper, the author establishes for the singular case the definitive result that for any given \(m\geq 0\) there are constant \(r_ 0\), \(A\) and functions \(h_{4l-1}(\vartheta)\), \(l=0,\ldots,m\), all independent of the particular solution considered, such that \[ \mid u(x)- \sum^ m_{l=0}h_{4l-1}(\vartheta)r^{4l-1}\mid\leq Ar^{4m+3} \] in the sector \(\Omega_{r_ 0}\). A particular consequence is that the expression \(h_{-1}(\vartheta)r^{-1}\) originally introduced by Concus and Finn is asymptotic to the solution up to \(O(r^ 3)\). The proof of the expansion is something of a tour de force, using the same comparison principle that was exploited for the original result. The functions \(h_{4l-1}(\vartheta)\) appear as solutions of linear two point boundary value problems on \((-\alpha,\alpha)\). The case \(\gamma=0\) leads to singular equations and requires special consideration.
linear two point boundary value problems
scientific article; zbMATH DE number 508854