This paper describes the ordering degree of anionic, cationic, and zwitterionic surfactants with the increase of their packing density on Ge substrate by using Fourier transform infrared-attenuated total reflection (FTIR-ATR) spectroscopy. This work shows new insights on the conformational order of sodium dodecyl sulfate (SDS), N-hexadecyl-N-N-dimethyl-3-ammonio-1-propane-sulfonate (HPS), hexadecyl-trimethylammonium bromide (CTAB), and dodecyl trimethylammonium bromide (DTAB). DFT and semiempirical calculations are also performed in order to evaluate the effect of headgroup hydration and counterion. The CH₂ asymmetric and symmetric stretching bands for the SDS molecule show a shift of 1.7 and 0.9 cm⁻¹ to higher frequencies as the packing density increases, while it is observed a shift of 2.6 and 2.7 cm⁻¹ for the HPS molecule, respectively. The DTAB molecule shows a shift of 4.5 cm⁻¹ to lower frequencies for both CH₂ asymmetric and symmetric stretching bands as the packing density increases, indicating the decrease of gauche conformations and the increase of all-trans conformations over the aliphatic chain.

In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.

We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as key techniques used in the subquadratic case, which was studied in a previous publication of the authors, do not extend to the superquadratic setting. The main objective of the present paper is to address these difficulties. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton−Jacobi equation are obtained here through a novel set of techniques. These explore the parabolic nature of the problem through the nonlinear adjoint method. Well-posedness is proven by combining Lipschitz regularity for the Hamilton−Jacobi equation with polynomial estimates for solutions of the Fokker−Planck equation. Existence of classical solutions is then established under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to current understanding of superquadratic Hamilton−Jacobi equations.

In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton--Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker--Planck equation and a concavity argument for the nonlinearity.