In this paper, we present an effective route for boosting the photoelectrochemical overall water splitting to produce hydrogen and oxygen by modified TiO2 nanowire arrays (NWAs) photoanode. Such modification includes the synthesis of branched TiO2 NWAs and followed by the deposition of Au nanoparticles (NPs) via the magnetron sputtering and post-annealing for the dewetting of Au films. The physical deposition of Au on the TiO2 substrate with a hierarchical structure provides a feasible strategy to produce metal NPs with uniform size and distribution by the space confinement of the branched TiO2 NWAs. Those Au NPs rendered the intensified visible light absorption in a narrow band region due to the surface plasmon resonance (SPR) effect. The amplified electromagnetic field through the SPR excitation of the Au NPs and their passivation upon the surface of TiO2 NWAs led to the efficient generation and transfer of charge carriers. The synergistic effect between branched TiO2 NWAs and Au NPs resulted in the drastic enhancement of the photoelectrochemical overall water splitting to produce hydrogen and oxygen in the stoichiometric ratio. This work provides a facile method for the preparation of hierarchical metal/semiconductor composite photoelectrode with great potential in the efficient photoelectrochemical water splitting for hydrogen energy production and other applications in solar energy conversion.

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} with fixed constant \begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}. We reduce the problem for equation \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} propagate along locally Lipschitz singular characteristics \begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document} and time \begin{document}$ t $\end{document} can extend to \begin{document}$ +\infty $\end{document}. Essentially, we use \begin{document}$ \sigma $\end{document}-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of \begin{document}$ u $\end{document} and the complement of Aubry set using the basic idea from [9].

In this paper, we investigate the non-autonomous Hamilton-Jacobi equation $$\left\{ {\begin{array}{*{20}{c}}{{\partial _t}u + H(t,x,{\partial _x}u,u) = 0,} \\{\begin{array}{*{20}{c}}{u(x,{t_0}) = \phi (x),}&{x \in M,}\end{array}}\end{array}} \right.$$ where H is 1-periodic with respect to t and M is a compact Riemannian manifold without boundary. We obtain the viscosity solution denoted by $$T_{{t_0}}^t\phi (x)$$ and show $$T_{{t_0}}^t\phi (x)$$ converges uniformly to a time-periodic viscosity solution u* (x, t) of ∂tu + H(t, x, ∂xu, u) = 0.

We study the Lasry–Lions approximation using the kernel determined by the fundamental solution with respect to a time-dependent Tonelli Lagrangian. This approximation process is also applied to the viscosity solutions of the discounted Hamilton–Jacobi equations.

In the recent works, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators (or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM (Kolmogorov-Arnold-Moser) theory.

A system is defined by a C1 function Hon a 4 dimensional manifold M provided with a symplectic structure ω.On a 2-dimensional subsurface of energy hypersurface transverse to the vector field on M,the Poincaré return mapping associated to the Hamiltonian system(M,ω,H) is area preserving.Quasi-periodic orbits exist for monotone twist map of annulus.It is well known that Poincaré's last geometric theorem asserts the existence of at least two fixed points for an area-preserving twist homeomorphism of annulus.We believe,one of them is huperbolic and the other is elliptic.And it is helpful to study the stability of dymanical system.For area-preserving standard twist map,there is an uniformly hyperbolic structure on the closure of Birkhoff maximal orbits.These Birkhoff maximal orbits coinside with the minimal orbits in the sense of Aubry-Mather theory.In a one-degree-of-freedom lagrangian system,the minimal periodic orbits are hyperbolic.But we haven't seen any strict statement about the elliptic fixed points or periodic orbits.In this paper,using variational method we study the hyperbolicity of minimal orbits and construct a class of minimax periodic orbits.We proved that the bounded minimax periodic orbits are elliptic.

J. Mather and A. Fathi defined Ma(n)ié set and Aubry set, which are the important invariant sets in positive definite Lagrangian system, in different ways.They use variational principle and Weak KAM theory respectively. In this paper we provide a proof of the equivalence between the two kinds of definitions, and generalize A. Fathi's definition. In the end of the paper, we calculate the Ma(n)é set and Aubry set for a single pendulum system.