Periodic Solutions to Klein–Gordon Systems with Linear Couplings

In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

How to smooth a crinkled map of space–time: Uhlenbeck compactness for L ∞ connections and optimal regularity for general relativistic shock waves by the Reintjes–Temple equations

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem( Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

The 3D transient semiconductor equations with gradient-dependent and interfacial recombination

We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.