Continuity in weak topology and extremal problems of eigenvalues of the 𝑝-Laplacian
Ping Yen Bryan Yan, Meirong Zhang · Transactions of the American Mathematical Society · 2010
We will study the dependence of eigenvalues of the one-dimensional p p -Laplacian on potentials or weights. Two results are obtained. One is the continuity of eigenvalues in potentials with respect to the weak topologies of L γ L^\gamma spaces, 1 ≤ γ ≤ ∞ 1\le \gamma \le \infty , and the other is the continuous differentiability of eigenvalues in potentials with respect to L γ L^\gamma norms. As applications, we will study some extremal problems of eigenvalues by developing some analytical methods.