Radial limits of capillary surfaces at corners

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##### Citation

Entekhabi, Mozhgan (Nora); Lancaster, Kirk E. Radial limits of capillary surfaces at corners. Pacific Journal of Mathematics, vol. 288:no. 1:pp 55â€“67

##### Abstract

Consider a solution f is an element of C-2(Omega) of a prescribed mean curvature equation div del f/root 1+vertical bar del f vertical bar(2) = 2H (x, f) in Omega subset of R-2, where Omega is a domain whose boundary has a corner at O = (0; 0) epsilon partial derivative Omega and the angular measure of this corner is 2 alpha, for some alpha epsilon(0,pi). Suppose sup(x epsilon Omega) vertical bar f(x)vertical bar and sup(x epsilon Omega) vertical bar H (x; f(x))vertical bar are both finite. If alpha > pi/2, then the (nontangential) radial limits of f at O, namely Rf(theta) = lim(r down arrow 0) (r cos theta, r sin theta) were recently proven by the authors to exist, independent of the boundary behavior of f onand to have a specific type of behavior. Suppose partial derivative Omega 4; 2 , the contact angle gamma(.) / that the graph of f makes with one side of @ has a limit (denoted gamma(2)) at O and pi - 2 alpha < gamma 2 <2 alpha. We prove that the (nontangential) radial limits of f at O exist and the radial limits have a specific type of behavior, independent of the boundary behavior of f on the other side of partial derivative Omega. We also discuss the case 2 0; 2 and the displayed inequalities do not hold.