The comparison of capillary surfaces heights in case of small gravity (Q5930176)

The author considers the question: Let \(D_i\), \(D_e\) be convex plane domains such that \(D_i\subset D_e\). Assuming a vertically downward gravity field and a constant contact angle \(\gamma\) with \(0\leq\gamma <\pi/2\), does a capillary tube of section \(D_i\) lift liquid higher over its section than does a capillary tube of the same material and section \(D_e\)? The author attributes the question to the reviewer; in fact, although the reviewer made contributions to the question, it was initially raised by M. Miranda, during informal conversation. NEWLINENEWLINENEWLINESpecifically, for prescribed \(\gamma\), \(0\leq\gamma <\pi/2\), and positive constant \(k\), one considers solutions \(u_i\), \(u_e\) of the equation NEWLINE\[NEWLINE\text{div} Tu=ku, \quad Tu\equiv{\nabla u\over \sqrt{1+ |\nabla u |^2}} \tag{1}NEWLINE\]NEWLINE in the respective (smooth convex) domains \(D_i\), \(D_e\), subject to the condition (2) \(\nu\cdot Tu=\cos \gamma\) on the respective boundaries \(\Gamma_i\), \(\Gamma_e\), with \(\nu\) the unit exterior normal; one asks whether \(u_i<u_e\) in \(D_i\).NEWLINENEWLINENEWLINEThe principal contribution of the paper is to show that if solutions of the zero gravity problem (3) \(\text{div} Tu ={|\Gamma |\over|D |}\cos \gamma\) under the boundary condition (2) exist in \(D_i\), \(D_e\), and if (4) \(|\Gamma_e |/|D_e|< |\Gamma_i|/ |D_i|\), then there exists \(k_0>0\) such that whenever \(0<k<k_0\) the Miranda question has a positive answer.NEWLINENEWLINENEWLINEThe proof of the assertion follows immediately from an asymptotic representation for solutions of (1), (2) in a domain \(D\), due to \textit{D. Siegel} [in ``Variational methods for free surface interfaces'' ed. P. Concus, R. Finn. (Menlo Park, Calif., 1985), Springer-Verlag (1987), 109-113]: if a smooth solution \(z\) of (3), (2) in \(D\) exists, and if \(z\) is normalized to have average value zero, then there holds NEWLINE\[NEWLINEu={|\Gamma|\cos\gamma \over k|D|}+z +O(k)\tag{5}NEWLINE\]NEWLINE uniformly over \(D\), as \(k\to 0\). (The author was evidently unaware of the work of Siegel, and discovered the representation independently; however, the proof of (5) as outlined in the present paper seems to the reviewer to be not correct as stated, as it depends on a gradient bound for which in the reviewer's opinion there are counterexamples.)NEWLINENEWLINENEWLINEThe author obtains a sufficient condition for the inequality (4) to hold, and notes a criterion due to Giusti for existence of solutions of (3), (2). It should be noted that solutions of (3), (2) can fail to exist, even in convex analytic domains; in such cases, the asserted comparison relation for the problem (1), (2) can fail. Thus, the author's requirement of existence of these solutions cannot be discarded.NEWLINENEWLINENEWLINEThe underlying idea of the present paper has since been further developed in a joint work of the author and the reviewer [Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Preprint 87/2000; Pacific J. Math., to appear]. In this newer work, the requirements of smoothness and convexity are relaxed, and also conditions are established under which a discontinuous reversal of the comparison relations occurs.