Eigenvalue inequalities for the Laplacian with mixed boundary conditions

An Isoperimetric Inequality for the Second Eigenvalue of the Laplacian with Robin Boundary Conditions

We prove that the second eigenvalue of the Laplacian with Robin boundary conditions is minimised amongst all bounded Lipschitz domains of fixed volume by the domain consisting of the disjoint union of two balls of equal volume.

THE FIRST EIGENVALUE OF p-LAPLACIAN SYSTEMS WITH NONLINEAR BOUNDARY CONDITIONS

Two generalized Lyapunov-type inequalities for a fractional p-Laplacian equation with fractional boundary conditions

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.

On Domain Monotonicity for the Principal Eigenvalue of the Laplacian with a Mixed Dirichlet-neumann Boundary Condition

Let Ω ⊂ Rd be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator − 1 2 ∆ in Ω − Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ0(Ω, A) > 0 denote its principal eigenvalue. We discuss the question of monotonicity of λ0(Ω, A) in its dependence on the domain Ω. The main point of thi...

Mixed Finite Element Approximation of the Vector Laplacian with Dirichlet Boundary Conditions

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. How...