Bounds of Eigenfunction for Some Diffusion Operators

In this paper, we consider the $L^{\infty}$ bounds of the eigenfunctions for some diffusion operators. In case of classical Laplace eigenvalue problem $\Delta u = -\lambda u$ with Dirichlet boundary condition, we obtain the polynomial bound which is different from the result derived by heat kernel estimates. We then study the nonlocal dispersal eigenvalue problem $\int_{\Omega} J(x-y) u(y) \, dy - u(x) = -\lambda u(x)$ with bounded domain $\Omega$ and obtain an exponential bound by means of Fourier transform and nonlocal estimates.