in this paper, by modifying cheng-yau$'$s technique to complete hypersurfaces in $s^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [h. li, hypersurfaces with constant scalar curvature in space forms, {em math. ann.} {305} (1996), 665--672].

A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale of the running coupling αs(μ ). It is common practice to guess a physical scale μ = Q which is of order of a typical momentum transfer Q in the process, and then vary the scale over a range Q/2 and 2Q. This procedure is clearly problematic since the resulting fixed-order pQCD...

In this paper we characterize the degenerate elliptic equations F(Du) = 0 whose subsolutions (F(Du) ≥ 0) satisfy the strong maximum principle. We introduce an easily computed function f on (0,∞) which is determined by F, and we show that the strong maximum principle holds depending on whether ∫ 0+ dy f(y) is infinite or finite. This complements our previous work characterizing when the (ordinar...

We state a maximum principle for the gradient of the minima of integral functionals I (u)G Ω [ f (∇u)Cg(u)] dx, on ūCW 1,1 0 (Ω ), just assuming that I is strictly convex. We do not require that f, g be smooth, nor that they satisfy growth conditions. As an application, we prove a Lipschitz regularity result for constrained minima.

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as Γ-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the...