A capillarity problem for compressible liquids

Abstract

We study the existence and regularity of solutions to the capillarity problem for compressible liquids in a tube. We introduce an appropriate space of functions of bounded variation, in which the energy functional introduced recently by Robert Finn can be defined. We prove existence of a locally Lipschitz minimizer in this class.

Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry

“…, in B, u = 0, on ∂B, The equation in (1.1), where a, b are positive constants, has been introduced for modeling capillarity phenomena for compressible fluids, or for describing the geometry of the human cornea, when they are respectively supplemented with non-homogeneous conormal boundary conditions [13,14,4,15,3], or with homogeneous Dirichlet boundary conditions [20,21,22,9,24,25,23,26,10,11]. We refer to these papers for the derivation of the models, further discussions on the subject, and an additional bibliography.…”

A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis

“…The interior regularity of these bounded variation minimizers is obtained by combining a delicate approximation scheme with a "local" existence result of Serrin's type proven in [40] and with the classical gradient estimates of Ladyzhenskaya and Ural'tseva [32]. More precisely, we start from the observation, already made in [16,17,4,18,5,3], that equation (1.1) can formally be seen as the Euler equation of the functional…”

“…(−∇u,1) √ 1+|∇u| 2 is the unit upper normal to the graph of u in R N +1 . Equation (1.1) has been introduced either for modeling capillarity phenomena for compressible fluids, if supplemented with non-homogeneous conormal boundary conditions [16,17,4,18,3], or for describing the geometry of the human cornea, if supplemented with homogeneous Dirichlet boundary conditions [46,47,48,52,50,51]. We refer to these papers for the derivation of the model, further discussion on the subject and an additional bibliography.…”

“…In [54, p. 480] J. Serrin also emphasized "the delicacy of the situation when any but the simplest equations are treated". When applying these ideas to the homogeneous Dirichlet problem for (1.1), they yield its solvability assuming a smallness condition on the coefficient b and an appropriate version of the Serrin's mean convexity condition on ∂Ω: see, respectively, assumptions (2) and (3) in [40]. In [6,Remark 1] it was stated, yet without an explicit proof, that using the methods of [5] the mean convexity assumption might be suitably relaxed, allowing boundary points with negative mean curvature, at the expense however of requiring some smallness conditions both on the coefficients of the equation and on the size of the domain.…”

“…Next we prove the interior regularity of v. This exploits an argument, which was introduced in [20] and used, e.g., in [21,22,3] for the study of capillarity problems. The procedure can be summarized as follows.…”