System Upgrade on Tue, May 28th, 2024 at 2am (EDT)
Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.For online purchase, please visit us again. Contact us at customercare@wspc.com for any enquiries.
Results: 1 - 7of7
Save this search
Please login to be able to save your searches and receive alerts for new content matching your search criteria.
Gradient estimates of mean curvature equation with Neumann boundary condition in domains of Riemannian manifold
Preview AbstractWe study the prescribed mean curvature equation with Neumann boundary conditions in domains of Riemannian manifold. The main goal is to establish the gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.
GLOBAL REGULARITY FOR A SINGULAR EQUATION AND LOCAL H1 MINIMIZERS OF A NONDIFFERENTIABLE FUNCTIONAL
Preview AbstractWe prove optimal Hölder estimates up to the boundary for the maximal solution of a singular elliptic equation. The techniques used in this argument are applied to show that in some situations the maximal solution is a local minimizer of the corresponding functional in the topology of H1.
Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations
Preview AbstractThis paper concerns the gradient estimates for Neumann problem of a certain Monge–Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.
Lorentz–Morrey global bounds for singular quasilinear elliptic equations with measure data
Preview AbstractThe aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem:
{−div(A(x,∇u))=μinΩ,u=0on∂Ω,in Lorentz–Morrey spaces, where Ω⊂ℝn (n≥2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W1,p0(Ω) and the p-capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2≤p≤n, where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p>2−1n. Here in this study, we particularly restrict ourselves to the singular case 3n−22n−1<p≤2−1n. The results are central to generalize our technique of good-λ type bounds in the previous work [M.-P. Tran, Good-λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.Regularity for distribution-dependent SDEs driven by jump processes
Preview AbstractIn this paper, the following d-dimensional distribution-dependent stochastic differential equation driven by a pure jump process is studied:
Xξt=ξ+∫t0b(Xξs−,μs)ds+∫t0∫B0σ(Xξs−,z,μs)ˆN(dz,ds),t∈[0,1],where μs denotes the distribution of Xξs. The differentiability of the map ξ↦Xξt is investigated in the sense of L2(Ω;ℝd). By the Malliavin calculus for jump processes, the following Bismut type derivative formula is established,∇η𝔼f(Xξt)=𝔼(f(Xξt)Mξt),where f is a test function and Mξt is a random variable depending on the initial value ξ. Sharp gradient estimates in short time are also obtained.A Harnack inequality for a class of 1D nonlinear reaction–diffusion equations and applications to wave solutions
Preview AbstractIn this paper, a differential-geometric method is applied to build some Li–Yau–Hamilton-type Harnack inequalities for the positive solutions to a one spatial dimensional nonlinear reaction–diffusion equation in a plane geometry. The class of reaction–diffusion equation that is considered here contains several important equations some of which are Newel–Whitehead–Segel, Allen–Cahn and Fisher–KPP equations. The Harnack inequalities so derived are used to discuss some other important properties of positive solutions and in the characterization of positive wave solutions.
Differential Harnack estimates for conjugate heat equation under the Ricci flow
Preview AbstractWe prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “Δ−R(x,t)”, where R is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.