Nonlocal biharmonic evolution equations with Dirichlet and Navier boundary conditions

Abstract

<p style='text-indent:20px;'>This paper studies a nonlocal biharmonic evolution equation with Dirichlet boundary condition that arises in image restoration. We prove the existence and uniqueness of solutions to the nonlocal problem by the variational method and show that the solutions of the nonlocal problem converge to the solution of the classical biharmonic equation with Dirichlet boundary condition if the nonlocal kernel is rescaled appropriately. The asymptotic behavior is discussed. Besides, we study the Navier problem by transforming it into a Dirichlet problem with a fixed point. The existence, uniqueness, convergence under the rescaling of the kernel, and asymptotic behavior of solutions to the Navier problem are discussed.</p>