Regularity of capillary surfaces over domains with corners

Leon M. Simon

Vol. 88 (1980), No. 2, 363–377

DOI: 10.2140/pjm.1980.88.363

Abstract

Using the usual mathematical model (capillary surface equation with contact angle boundary condition) we discuss regularity of the equilibrium free surface of a fluid in a cylindrical container in case the container cross-section has corners.

It is shown that good regularity holds at a corner if the “corner angle” 𝜃 satisfies 0 < 𝜃 < π and 𝜃 + 2β > π, where 0 < β ≤ π∕2 is the contact angle between the fluid surface and the container wall.

It is known that no regularity holds in case 𝜃 + 2β < π, hence only the borderline case 𝜃 + 2β = π remains open.

Mathematical Subject Classification 2000

Primary: 49H05, 49H05

Secondary: 35R35, 58E12, 49F99

Milestones

Received: 26 March 1979

Published: 1 June 1980