We prove existence of a weak solution for a hybrid model for the electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature.
We demonstrate how arbitrarily sharp asymptotic expansions for the Landau constants can be derived, and we prove some related inequalities. The main tool used is Brounckerʼs continued fraction formula. We also show that the implied series expansion of the Landau constants is in fact divergent.
In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two scale asymptotic expansions as well as energy-like estimates.
We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f∈Lp[0,1] in terms of its Lp-modulus of continuity ωp(f;δ). These estimates are sharp for any rate of the decay of ωp(f;δ). Moreover, the constant coefficients in them depend on parameters in an optimal way.
We obtain sharp estimates of the Hardy-Vitali type p-variation of a function of two variables in terms of its mixed modulus of continuity in Lp([0,1]2).
We study a system of parabolic equations consisting of a double nonlinear parabolic equation of Forchheimer type coupled with a semilinear parabolic equation. The system describes a fluid-like driven system for active-passive pedestrian dynamics. The structure of the nonlinearity of the coupling allows us to prove the uniqueness of solutions. We provide also the stability estimate of solutions with respect to selected parameters.