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We consider the Robin boundary value problem div (A ∇ u) = div f + F in Ω , a C 1 domain, with (A ∇ u - f) · n + α u = g on Γ , where the matrix A belongs to V M O (R 3) , and discover the uniform estimates on ‖ u ‖ W 1 , p (Ω) , with 1 < p < ∞ , independent of α . At the difference with the case p = 2 , which is simpler, we call here the weak reverse Hölder inequality. This estimates show that the solution of the Robin problem converges strongly to the solution of the Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter α tends to ∞ (resp. 0). [ABSTRACT FROM AUTHOR]

We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805–826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies. [ABSTRACT FROM AUTHOR]

We study the existence of local and global mild solutions of the fractional-order differential equations in an arbitrary Banach space by using the semigroup theory and the Schauder fixed-point theorem. We also give some examples to illustrate the applications of abstract results. [ABSTRACT FROM AUTHOR]

We consider a periodically heterogeneous and perforated medium filling an open domain Ω of ℝN. Assuming that the size of the periodicity of the structure and of the holes is O(ε), we study the asymptotic behavior, as ε → 0, of the solution of an elliptic boundary value problem with strongly oscillating coefficients posed in Ωε (Ωε being Ω minus the holes) with a Neumann condition on the boundary of the holes. We use Bloch wave decomposition to introduce an approximation of the solution in the energy norm which can be computed from the homogenized solution and the first Bloch eigenfunction. We first consider the case where Ωis ℝN and then localize the problem for a bounded domain Ω, considering a homogeneous Dirichlet condition on the boundary of Ω. [ABSTRACT FROM AUTHOR]