The second periodic eigenvalue and the Alikakos-Fusco conjecture
The author considers the periodic eigenvalue problem \(-u''+qu=\lambda u\), \(u(0)= u(2\pi)\), \(u'(0)= u'(2\pi)\), where \(q\in L^1(0,2\pi)\) and \(\int_0^{2\pi} q(x)dx=0\). It is proved that the second eigenvalue \(\lambda_1\) satisfies \(\lambda_1\leq 1\) and \(\lambda_1=1\) if and only if \(q=0\), for the following potentials: (i) \(q(x)= a_1\cos x+\sum_{n\geq 2}(a_n\cos nx+b_n\sin nx)\) and \(a_2\geq 0\) or \(a_1=0\); (ii) \(q(x)= \varepsilon Q(x)\), where \(Q\) is fixed and \(|\varepsilon|\) is small enough or sufficiently large. In particular, the one-dimensional Alikakos-Fusco conjecture is verified for these potentials.
one-dimensional Alikakos-Fusco conjecture
scientific article; zbMATH DE number 946931