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We prove that the -solution of the heart equation for the minimal surface with plateau boundary condition is continuous up to boundary in all variables.
We will study the dependence of λ(a, b), half-eigenvalues of the one-dimensional p-Laplacian, on potentials , 1 ≤ γ ≤ ∞, where . Two results are obtained. One is the continuity of half-eigenvalues in , where wγ is the weak topology in space. The other is the continuous differentiability of half-eigenvalues in , where ‖ ⋅ ‖γ is the Lγ norm of . These results will be used to study extremal problems of half-eigenvalues in future work.