Milan Kučera

Ranges of nonlinear asymptotically linear operators

Abstract This paper deals with the solvability of the equation A ( u ) − S ( u ) = f , where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H , the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.

Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem

The implicit function theorem is applied in a nonstandard way to abstract variational inequalities depending on a (possibly infinite-dimensional) parameter. In this way, results on smooth continuation of solutions as well as of eigenvalues and eigenvectors are established under certain particular assumptions. The abstract results are applied to a linear second order elliptic eigenvalue problem with nonlocal unilateral boundary conditions (Schrodinger operator with the potential as the parameter).

Bifurcation of Solutions to Reaction-Diffusion Systems with Unilateral Conditions

Let Ω be a bounded domain in R N with a lipschitzian boundary 2202;Ω, let Γ D , Γ N , Γ U be open (in ∂Ω) disjoint subsets of ∂Ω, meas(∂Ω \ Γ D ⋃ Γ N ⋃Γ U ) = 0, f, g real differentiable functions on R ū, v, positive constants such that f(ū, v) = g(ū, v) = 0.

Reaction-diffusion systems: destabilizing effect of unilateral conditions

On considere un systeme de reaction-diffusion dans un domaine Ω⊂R n . Soit ū, v une solution stationnaire et spatialement homogene avec les conditions aux limites de Neumann. On etudie la stabilite linearisee de ū, v

Smooth bifurcation for variational inequalities based on the implicit function theorem

Abstract We present a certain analog for variational inequalities of the classical result on bifurcation from simple eigenvalues of Crandall and Rabinowitz. In other words, we describe the existence and local uniqueness of smooth families of nontrivial solutions to variational inequalities, bifurcating from a trivial solution family at certain points which could be called simple eigenvalues of the homogenized variational inequality. If the bifurcation parameter is one-dimensional, the main difference between the case of equations and the case of variational inequalities (when the cone is not a linear subspace) is the following: For equations two smooth half-branches bifurcate, for inequalities only one. The proofs are based on scaling techniques and on the implicit function theorem. The abstract results are applied to a fourth order ODE with pointwise unilateral conditions (an obstacle problem for a beam with the compression force as the bifurcation parameter).

Reaction-diffusion systems: stabilizing effect of conditions described by quasivariational inequalities

Reaction-diffusion systems are studied under the assumptions guaranteeing diffusion driven instability and arising of spatial patterns. A stabilizing influence of unilateral conditions given by quasivariational inequalities to this effect is described.

Spatial Patterns for reaction-diffusion systems with conditions described by inclusions

We consider a reaction-diffusion system of the activator-inhibitor type with boundary conditions given by inclusions. We show that there exists a bifurcation point at which stationary but spatially nonconstant solutions (spatial patterns) bifurcate from the branch of trivial solutions. This bifurcation point lies in the domain of stability of the trivial solution to the same system with Dirichlet and Neumann boundary conditions, where a bifurcation of this classical problem is excluded.

Global Bifurcation for a Reaction-Diffusion System with Inclusions

We consider a reaction-diffusion system exhibiting diffusion driven instability if supplemented by Dirichlet-Neumann boundary conditions. We impose unilateral conditions given by inclusions on this system and prove that global bifurcation of spatially nonhomogeneous stationary solutions occurs in the domain of parameters where bifurcation is excluded for the original mixed boundary value problem. Inclusions can be considered in one of the equations itself as well as in boundary conditions. The proof is based on the degree theory for multivalued mappings (jump of the degree implies bifurcation). We show how the degree for a class of multivalued maps including those corresponding to a weak formulation of our problem can be calculated.

Crandall-Rabinowitz Type Bifurcation for Non-differentiable Perturbations of Smooth Mappings

We consider abstract equations of the type \(F(\lambda ,u)=\tau G(\tau ,\lambda ,u)\), where \(\lambda \) is a bifurcation parameter and \(\tau \) is a perturbation parameter. We suppose that \(F(\lambda ,0)=G(\tau ,\lambda ,0)=0\) for all \(\lambda \) and \(\tau \), F is smooth and the unperturbed equation \(F(\lambda ,u)=0\) describes a Crandall-Rabinowitz bifurcation in \(\lambda =0\), that is, two half-branches of nontrivial solutions bifurcate from the trivial solution in \(\lambda =0\). Concerning G, we suppose only a certain Lipschitz condition; in particular, G is allowed to be non-differentiable. We show that for fixed small \(\tau \ne 0\) there exist also two half-branches of nontrivial solutions to the perturbed equation, but they bifurcate from the trivial solution in two bifurcation points, which are different, in general. Moreover, we determine the bifurcation directions of those two half-branches, and we describe, asymptotically as \(\tau \rightarrow 0\), how the bifurcation points depend on \(\tau \). Finally, we present applications to boundary value problems for quasilinear elliptic equations and for reaction-diffusion systems, both with small non-differentiable terms.

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

AbstractVariational inequalities