On the uniqueness and monotonicity of solutions of free boundary problems

Generic properties of free boundary problems in plasma physics

“…For p > 1 there exists a one to one correspondence ( [8]) between variational solutions of (F) I with γ I ≤ 0 and variational solutions of (P) λ , see Appendix A for further details. Actually, concerning variational solutions, as a consequence of the results in [3] (see also [6] and references therein), the following holds true.…”

“…However, it is not trivial to come up with the maximal branch G(D 2 ) as one needs to rule out the existence of bifurcation points as well as the monotonicity of α λ , E λ for λ > 1 p Λ(Ω, 2p). In particular the uniqueness of solutions on D 2 ( [5]) was used in a crucial way in [6], which also implies that λ(D 2 , p), as defined by Theorem 1.2, satisfies λ(D 2 , p) = λ * (D 2 , p). In general the situation is much more difficult and we come up with an alternative, in a suitably defined generic sense.…”

“…The problem (F) I arises in Tokamak's plasma physics [21,27,33] and has been studied since [8,34]. We refer to [6] for a complete list of references. We will focus here on positive solutions of (F) I .…”

“…In particular Theorem A shows that for any λ < λ * * (Ω, p) there exists at least one (variational) solution of (P) λ . The first and third author have recently proved a uniqueness and monotonicity result ( [6]). For fixed t ≥ 1 we denote Λ(Ω, t) = inf…”

“…Concerning the unique branch of positive solutions, the first author and third author recently proved the following result: Theorem B. ( [6]) Let Ω ⊂ R N be a bounded domain of class C 3 , |Ω| = 1 and p ∈ [1, p N ). Then λ * (Ω, p) ≥ 1 p Λ(Ω, 2p) and the equality holds if and only if p = 1.…”