Direction and stability of bifurcation branches for variational inequalities

Abstract

We consider a class of variational inequalities with a multidimensional bifurcation parameter under assumptions guaranteeing the existence of smooth families of nontrivial solutions bifurcating from the set of trivial solutions. The direction of bifurcation is shown in a neighborhood of bifurcation points of a certain type. In the case of potential operators, also the stability and instability of bifurcating solutions and of the trivial solution is described in the sense of minima of the potential. In particular, an exchange of stability is observed.

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

“…Moreover, bifurcating solutions form a smooth branch if N 1 , N 2 are smooth, see [19]. In this case also the bifurcation direction can be described, see [8].…”

“…The price for the use of the variational approach is that we consider only some particular situations, and that we get no information concerning the character of the set of bifurcating solutions. However, in a particular case of situations like in Examples 5.2, 5.3, our method combined with the results [8], [19] can give a bifurcation for the case of general n 1 , n 2 (Remark 3.2), and even the direction of bifurcation can be described.…”

“…R e m a r k 5.4. Example 5.2 and the second case of Example 5.3 fit into the theory developed in [8], [19]. It follows that if the critical point (d 1 , d 2 ) obtained from Theorem 3.1 or 3.2 used for the particular case n = 0 is such that the nontrivial solution of the problem (2.12), (5.8) or (2.12), (5.9) is unique up to a positive multiple and satisfies certain activity conditions then a smooth branch of nontrivial solutions of the problem (5.11) σ 1 (s)∆u + b 11 u + b 12 v + n 1 (u, v) = 0,…”

Optimization‐based stability analysis of structures under unilateral constraints

“…Then problem (35) is embedded into (14) with m := |J|, on which Algorithm 2.12 works. Consequently, subproblem (22) of Algorithm 2.12 is explicitly obtained as…”

Direction and stability of bifurcating solutions for a Signorini problem

“…To our best knowledge, up to now no analogs of the principle of exchange of stability for variational inequalities are known, with the exception of some special cases (for example obstacle problems with finitely many obstacles, see [5,6]). …”