CONTINUITY AND CONTINUOUS DIFFERENTIABILITY OF HALF-EIGENVALUES IN POTENTIALS.

We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials $(a, b)\in ({\mathcal L}^\gamma, {\mathcal L}^\gamma)$, 1 ≤ γ ≤ ∞, where ${\mathcal L}^\gamma = L^\gamma([0, 1], {\mathbb R})$. Two results are obtained. One is the continuity of half-eigenvalues in $(a, b)\in({\mathcal L}^\gamma, w_\gamma)^2$, where wγ is the weak topology in ${\mathcal L}^\gamma$ space. The other is the continuous differentiability of half-eigenvalues in $(a, b)\in ({\mathcal L}^\gamma, \|\cdot\|_\gamma)^2$, where | ⋅ |γ is the Lγ norm of ${\mathcal L}^\gamma$. These results will be used to study extremal problems of half-eigenvalues in future work. [ABSTRACT FROM AUTHOR]