On a free boundary problem for minimal surfaces

ForC ⁴-embedded manifoldsS ³ which are differmorphic to the standard sphere in ³ the existence of non-constant minimal surfaces bounded byS and intersectingS orthogonally along their boundaries is deduced.This research was supported by the Sonderforschungsbereich 72 of the Deutsche-Forschungs-Gemeinschaft     

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41], [42] is optimal.     

On the free boundary problem of magnetohydrodynamics

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W₂^{2 + l,1 + l/2},1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles.     

On a free boundary problem for the relaxation transfer equation

Theoretical and Mathematical Physics - We study the free boundary problem with no initial conditions for a third-order relaxation transfer equation. First, we reduce the problem to a second-order...     

On one problem with free boundary for a nonlinear system

We formulate the filtration problem with free boundary as a problem with discontinuous nonlinearity for a degenerate elliptic or parabolic system. We prove that a solution of the Dirichlet problem exists in both cases. We study some qualitative properties of these solutions, e.g., the existence of “dead cores”. Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1155–1165, September, 1996.     

On a problem of viscous strip deformation with a free boundary

We consider a class of solutions to the 2D Navier-Stokes equations in a strip such that the longitudinal component of velocity is a linear function of the longitudinal coordinate while the transversal component and the pressure do not depend on this coordinate. One of the strip boundaries is free and the other boundary can be a solid wall or free too. We formulate conditions for a global solvability in time of corresponding initial boundary value problems, describe their asymptotic properties, give examples of exact solutions, study blowing up solutions in the case when the both strip boundaries are free.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

Perturbation and bifurcation in a free boundary problem

We study the equation with a discontinuous nonlinearity: ?Δu = λH(u ? 1) (H is Heaviside's unit function) in a square subject to various boundary conditions. We expect to find a curve dividing the harmonic (Δu = 0) region from the superharmonic (Δu = ?λ) region, defined by the equation u(x, y) = 1. This curve is called the free boundary since its location is determined by the solution to the problem. We use the implicit function theorem to study the effect of perturbation of the boundary conditions on known families of solutions. This justifies rigorously a formal scheme derived previously by Fleishman and Mahar. Our method also discovers bifurcations from previously known solution families.     

Subsolutions and supersolutions in a free boundary problem

We begin by giving some results of continuity with respect to the domain for the Dirichlet problem (without any assumption of regularity on the domains). Then, following an idea of A. Beurling, a technique of subsolutions and supersolutions for the so-called quadrature surface free boundary problem is presented. This technique would apply to many free boundary problems inR N,N≥2, which have overdetermined Cauchy data on the free boundary. Some applications to concrete examples are also given. This work was done while the author was at the University of Nancy I (France) supported by URA 750 CNRS and project Numath INRIA Lorraine.     

Analytic regularity of a free boundary problem

In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most n − 8.     

Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law.We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.Supported by Schweizerischer Nationalfonds     

On an over-determined problem of free boundary of a degenerate parabolic equation

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T ₀, to decide the initial value u ₀ such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H _u(T ₀) = {x, ?^N: u(x, T ₀) > 0}. In this paper, we first establish a priori estimate u _t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C ₀ and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ .     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

On a coefficient inverse problem for a parabolic equation in a domain with free boundary

The present paper deals with the inverse problem of determination of the coefficient of the first derivative of the unknown function with respect to a spatial variable for a one-dimensional parabolic equation in the domain whose boundary is determined by two unknown functions. The conditions of local existence and uniqueness of a solution to the inverse problem are established.