14 research outputs found
A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide
We will study the spectral problem related to the Laplace operator in a
singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with
contains periodic arrangement of inclusions. On the boundary of the waveguide
we consider both Neumann and Dirichlet conditions. We will prove that provided
the diameter of the inclusion is small enough in the spectrum of Laplacian
opens spectral gaps, i.e. frequencies that does not propagate through the
waveguide. The existence of the band gaps will verified using the asymptotic
analysis of elliptic operators.Comment: 26 pages, 6 figure
Spectral gaps for water waves above a corrugated bottom
In this paper, the essential spectrum of the linear problem on water waves in a water layer and in a channel with a gently corrugated bottom is studied. We show that, under a certain geometric condition, the essential spectrum has spectral gaps. In other words, there exist intervals in the positive real semi-axis that are free of the spectrum but have their endpoints in it. The position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell
Second-order -regularity in nonlinear elliptic problems
A second-order regularity theory is developed for solutions to a class of
quasilinear elliptic equations in divergence form, including the -Laplace
equation, with merely square-integrable right-hand side. Our results amount to
the existence and square integrability of the weak derivatives of the nonlinear
expression of the gradient under the divergence operator. This provides a
nonlinear counterpart of the classical -coercivity theory for linear
problems, which is missing in the existing literature. Both local and global
estimates are established. The latter apply to solutions to either Dirichlet or
Neumann boundary value problems. Minimal regularity on the boundary of the
domain is required. If the domain is convex, no regularity of its boundary is
needed at all