Keywords: Upper and lower solutions p-Laplacian operator Fractional differential equation Integral boundary condition Eigenvalue a b s t r a c t In this paper, we are concerned with the eigenvalue problem of a class of singular p-Lapla-cian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. The conditions for the existence of at least one positive sol...

We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a...

We consider the spectrum associated with the linear operator obtained when a Cahn–Hilliard system on R is linearized about a transition wave solution. In many cases it’s possible to show that the only non-negative eigenvalue is λ = 0, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans functio...

Abstract We examine the eigenvalue spectrum, ρ(μ), of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p → ∞, one can obtain two relatively simple coupled equations whose solution yields ρ(μ) for an arbitrary complex network. For scale-free graphs, with degree distribution exponent λ, we obta...

We analyze the use of edge finite element methods to approximate Maxwell’s equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are...

We consider the transmission eigenvalue problem corresponding to the scattering problem for anisotropic media for both the scalar Helmholtz equation and Maxwell’s equations in the case when the contrast in the scattering media occurs in two independent functions. We prove the existence of an infinite discrete set of transmission eigenvalues provided that the two contrasts are of opposite signs....

We present the covering set of the squared eigenfunctions for a degenerate 3×3 eigenvalue problem. The derivation follows the approach recently outlined by Yang and Kaup on this same equation (J. Math. Phys. 50 023504 (2009)). This eigenvalue problem is important since it serves as the spectral problem for the inverse scattering transform (IST) of the vector NLS equation, the Sasa– Satsuma equa...

A tissue interaction model for skin organ pattern formation is presented. Possible spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional perturbation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns suc...

We discuss optimal control problems for general unstructured linear differentialalgebraic equations of arbitrary index. We discuss controllability and observability in the higher order case and show that order reduction may lead to false results. We then derive necessary conditions in the case of linear-quadratic control problems and show how these lead to eigenvalue problems for for even matri...

Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. In this paper, we use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to...