The authors study the existence of solution to p-Laplacian equation with nonlinear forcing term under optimal assumptions on the initial data, which are assumed to be measures. The existence of local solution is obtained.
The solvability of a linear equation and the regularity of the solution are discussed. The equation is arising in a geometric problem which is concerned with the realization of Alexandroff’s positive annul in R 3.
By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.
Let M n be a smooth, compact manifold without boundary, and F 0: M n → R n+1 a smooth immersion which is convex. The one-parameter families F(·, t): M n × [0, T) → R n+1 of hypersurfaces M t n = F(·, t)(M n) satisfy an initial value problem dt/dF(·, t) = −H k(·, t)ν(·, t), F(·, 0) = F 0(·), where H is the mean curvature and ν(·, t) is the outer unit normal at F(·, t), such that −H ν = $\overrightarrow H $ is the mean curvature vector, and k > 0 is a constant. This problem is called H k-flow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type I and Type II. It is proved that for Type I singularity, the limiting hypersurface satisfies an elliptic equation; for Type II singularity, the limiting hypersurface must be a translating soliton.
The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.
The authors achieve a general law of precise asymptotics for a new kind of complete moment convergence of i.i.d. random variables, which includes complete convergence as a special case. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. This extends and generalizes the corresponding results of Liu and Lin in 2006.
The author proves a global existence result for strong solutions to the quasilinear dissipative hyperbolic equation (1.1) below, corresponding to initial values and source terms of arbitrary size, provided that the hyperbolicity parameter ε is sufficiently small. This implies a corresponding global existence result for the reduced quasilinear parabolic equation (1.4) below.
It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz’s estimate of linear wave equation.