A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

On a bifurcation governed by hysteresis nonlinearity

We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl--Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity.     

A convex-concave problem with a nonlinear boundary condition

In this paper we study the existence of nontrivial solutions of the problem

     

A generalized Minkowski problem with Dirichlet boundary condition

We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.

     

Global bifurcation results for some fourth-order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

This paper is devoted to the study of global bifurcation from infinity of nontrivial solutions of a nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. We prove the existence of two families of unbounded continua of nontrivial solutions to this problem, which emanate from bifurcation points in $ℝ\times \{\infty \}$ and possess oscillatory properties of eigenfunctions (and their derivatives) of the corresponding linear problem in some neighborhoods of these bifurcation points.     

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.     

A Hopf bifurcation in a free boundary problem with inhomogeneous media

We shall consider an interfacial problem arising reaction–diffusion models with inhomogeneous media. The purpose of this paper is to analyze the occurrence of Hopf bifurcation in the interfacial problem and to examine the effects of an inhomogeneous media. Conditions for existence of stationary solutions and Hopf bifurcation for a certain class of inhomogeneity are obtained analytically and numerically.     

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.     

A thermo-electrical problem with a nonlocal radiation boundary condition

A coupled problem arising in induction heating furnaces is studied. The thermal problem, which involves a change of phase, has a nonlocal radiation boundary condition. Convective heat transfer in the liquid is also included which makes necessary to compute the liquid motion. For the space discretization, we propose finite element methods which are combined with characteristics methods in the thermal and flow models to handle the convective terms. In the electromagnetic model they are coupled with boundary element methods (BEM/FEM). An iterative algorithm is introduced for the whole coupled model and numerical results for an industrial induction furnace are presented.     

A control problem governed by a second-order differential inclusion

This article studies some Bolza-type problems governed by second-order differential inclusions with two boundary conditions, where the controls are Young measures.     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

A singular bifurcation problem

A nonlinear and singular bifurcation problem is studied to illustrate to what extent the singularity given by a pole can influence the bifurcation behavior. Due to the singularity, well-known bifurcation analysis is not applicable. An approximation by regular problems yields the result: A compact branch which ends in a special “singular” solution bifurcates from each eigenvalue of the linearized problem.     

Free boundary surface bifurcation in the phase transition problem of elasticity

The paper is concerns the phase transition problem in an elastic medium lying in a one-parameter force field. The stability and bifurcation problem for the interface between the two phases is studied. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 169–200. Translated by N. A. Karazeeva.     

The free boundary problem without initial condition

We consider an analog of the problem of the impact of a viscoplastic rod on the wall under a nonlinear boundary condition. We investigate the behavior of the free boundary on a given time interval and as t????. We obtained a priori estimates of H?lder norms and proved the theorems of uniqueness and existence of the solution.     

A first-order boundary value problem with boundary condition on a countable set of points

LetE={E _n } be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem $$ - i\frac{{dy}}{{dx}} = \lambda y, - \alpha \leqslant x \leqslant \alpha , U(y) \equiv \int_{ - a}^a {y(t)} d\sigma (t) = 0$$ that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measuredσ, it is shown that the systemE does not form an unconditional basis of subspaces inL ²(?a, a) if at least one of the end points ±a is mass-free.