n-Laplacians on Metric Graphs and Almost Periodic Functions: I

The spectra of n-Laplacian operators $$(-\Delta )^n$$ ( - Δ ) n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved using the theory of almost periodic functions. To achieve this, new results concerning roots of functions close to almost periodic functions are proved. The results obtained on almost periodic functions are of general interest outside the theory of differential operators.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

On spectral asymptotics and bifurcation for Carrier equations with odd superlinear term

In this paper, we consider the existence of eigenvalues and relative eigenfunctions for Carrier equations and present spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.

Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in [Formula: see text]-norm is given.