We prove that a certain class of elliptic free boundary problems, which includes the Prandtl–Batchelor problem from fluid dynamics as special case, has two distinct nontrivial solutions for large values parameter. The first solution is global minimizer energy. energy functional nondifferentiable, so standard variational arguments cannot be used directly to obtain second solution. our limit moun...

by using the trigonometric uniform splines of order 3 with a real tension factor, a numericalmethod is developed for solving a linear second order boundary value problems (2vbp) withdirichlet, neumann and cauchy types boundary conditions. the moment at the knots isapproximated by central finite-difference method. the order of convergence of the methodand the theory is illustrated by solving tes...

We prove that if the given compact set K is convex then a minimizer of the functional I(v) = ∫ BR |∇v|dx + Per({v > 0}), 1 < p < ∞, over the set {v ∈ H 0 (BR)| v ≡ 1 on K ⊂ BR} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. N...

An abstract class of two-dimensional geometry-driven free boundary problems is considered in which the evolution of the Cauchy transform of the time-evolving domain satisfies a partial differential equation of conservation type. This idea generalizes an association between free boundary problems and the inverse gravitational problem. It is shown that this class of free boundary problems admits ...

Free Boundary Problems (FBP) motivated several studies in the past due to their relevance in applications [1 − 4]. From the mathematical point of view FBP are initial/ boundary value problems with a moving boundary [5]. The motion of the boundary is unknown (free boundary) and has to be determined together with the solution of the given partial differential equation. As a consequence the soluti...