A free boundary problem for capillary surfaces

We prove existence and regularity of capillary surfaces between two parallel hyperplanes in ℝⁿ. By geometrical arguments we also derive some properties of the minimizers of the functional related to the problem.     

A problem with free boundary

One considers the Dirichlet problem for the equation u=(u), where is the Heaviside function. Under special assumptions one constructs the solution of this problem with convex and smooth level surfaces and, in particular, with a regular free surface, which coincides with the set of level zero. One proves the solvability in the small of the problem in the neighborhood of the constructed regular solution under perturbations of the boundary condition and a smooth boundary of the domain .Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 72–83, 1986.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

A boundary value problem for Hermitian harmonic maps and applications

We study the existence and uniqueness problems for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.

     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.     

On annihilators of harmonic vector fields

Let Ω⊂ℝ^N be a smooth bounded domain. We characterize smooth vector fields g on ∂Ω which annihilate all harmonic vector fields f in Ω continuous up to ∂Ω, with respect to the pairing (dσ denotes the hypersurface measure on ∂Ω). In addition, we extend these results to differential forms with harmonic vector fields being replaced by harmonic fields, i.e., forms f satisfying df=0, δf=0. A smooth form g on ∂Ω is an annihilator if and only if suitable extensions, u and v, into Ω of its normal and tangential components on ∂Ω, satisfy the generalized Cauchy-Riemann system du=δv, δu=0, dv=0 in Ω. Finally, we prove that the described smooth annihilators are weak^* dense among all annihilators. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 90–108.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR³ \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C ¹ with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.     

A free boundary problem with moving source

A free boundary problem with a moving source is considered. The existence of a critical velocity, above which a fault line occurs, is known experimentally in welding. One possible explanation has been the formation of a teardrop-shaped weld pool with a sharp vertex at the tail end. It is proved that such a singularity cannot occur within a reasonable two-dimensional model. The model includes physical effects due to discontinuous diffusivities, convection terms, and deviations in the interfacial temperature due to terms such as curvature and linear dynamics.     

A free boundary problem of type-I superconductivity

1.IntroductionSuperconductorsofTypeIarematerialswhicharecapableofchangingfromthephaseofbeingnormalconductorstoaphasewherethereisnoresistancetothemotionoffreeelections.InnormalconductorphasethenormalizedMaxwellequations(neglectingdisplacementcurrents)aretogetherwithOhm'slawj~acEwhereuistheelectricconductivity.InasuperconductingphaseOhm'slawisnolongervalidandMaxwell'sequationsaresupplemelltedbytheGinzburg-Landaufieldequationsll].Underisothermalconditions,thechangeofphasefromsuperconductingto…     

A free boundary problem of biological interest

During fertilization of certain echinoderms, a long actin-filled tube is extended by the sperm towards the interior of the egg. This yields a parabolic free boundary problem, which differs from the classical one-phase Stefan problem by the presence of a convective term in the partial differential equation, and because the equilibrium interface condition θ(s(t),t) = 0 is here replaced by a kinetic law s′(t) = vθ(s(t),t). This problem is set in variational form and the existence of a solution is proved by means of a Faedo–Galerkin approximation procedure.     

A remark on a boundary value problem for force-free fields

Summary For the equation rotv–v=0 with not necessarily constant a boundary value problem of Neumann's type is solved by reducing it to an equivalent integral equation.

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Zusammenfassung Für die Gleichung rotv–v=0 mit nicht notwendig konstantem wird ein Neumannsches Randwertproblem durch Zuruckführung auf eine äquivalente Integralgleichung gelöst.