Related papers: Gradient estimates for oblique derivative problems…
We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields.…
This work establishes a general stochastic maximum principle for partially observed optimal control of semi-linear stochastic partial differential equations in a nonconvex control domain. The state evolves in a Hilbert space driven by a…
Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic…
The paper proposes a new algorithm for solving global univariate optimization problems. The algorithm does not require convexity of the target function. For a broad variety of target functions after performing (if necessary) several…
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate…
In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $ M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the…
We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calder\'{o}n-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique…
Let $(M^{n},g,e^{-\phi}dv)$ be a weighted Riemannian manifold evolving by geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$. In this paper, we obtain a series of space-time gradient…
We provide quantitative gradient bounds for solutions to certain parabolic equations with unbalanced polynomial growth and non-smooth coefficients.
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in…
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a…
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can…
In this paper, we investigate boundary estimates for the Dirichlet problem for a class of fully nonlinear elliptic equations with general boundary conditions, including nonzero boundary conditions. Given specific structural conditions on…
Large deviation estimates for the following linear parabolic equation are studied: \[ \frac{\partial u}{\partial t}=\tr\Big(a(x)D^2u\Big) + b(x)\cdot D u + \int_{\R^N} \Big\{(u(x+y)-u(x)-(D u(x)\cdot y)\ind{|y|<1}(y)\Big\}\d\mu(y), \] where…
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
We study a class of nonlinear nonparametric inverse problems. Specifically, we propose a nonparametric estimator of the dynamics of a monotonically increasing trajectory defined on a finite time interval. Under suitable regularity…
We give full boundary extensions to two fundamental estimates in the theory of elliptic PDE, the weak Harnack inequality and the quantitative strong maximum principle, for uniformly elliptic equations in non-divergence form.