In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing result by M. Bellassoued and J. Le Rousseau (2018) arguments Di Cristo, E. Francini, C.-L. Lin, S. Vessella, J.-N. Wang (2017), present an elementary method to under optimal regularity assumption on coefficients.

Photonic crystal fibres are novel optical devices that can be designed to guide light of a particular frequency. This requires two phenomena to occur. First, the frequency of light must be in a gap of the spectrum of the fibre’s cladding (usually a periodic arrangement of air holes), so that the cladding acts as a barrier to that frequency of light. Second, the perturbation or defect in the mid...

We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the basis of a priori estimates obtained, maximal regularity strong solvability in Sobolev spaces are proved.

We evaluate the virial coefficients Bk for k ≤ 10 for hard spheres in dimensions D = 2, · · · , 8. Virial coefficients with k even are found to be negative when D ≥ 5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D ≥ 5. Further analysis provides evidence that negative virial coefficients will be seen for some k > 10 ...

We design and analyze a coupling of discontinuous Galerkin finite element method with boundary to solve the Helmholtz equation variable coefficients in three dimensions. The is realized mortar that related an impedance trace on smooth interface. obtained has block structure nonsingular subblocks. prove quasi-optimality $$h$$ - $$p$$ -versions scheme, under threshold condition approximability pr...