Related papers: Gradient Estimates and Applications for Neumann Se…

In a previous paper we have presented a general formalism for computing Feynman diagrams for scalar fields in curved spacetime at any loop order using heat kernel methods. The main technique used is the expansion of the fully off-diagonal…

We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained…

For a semigroup $P_t$ generated by an elliptic operator on a smooth manifold $M$, we use straightforward martingale arguments to derive probabilistic formulae for $P_t(V(f))$, not involving derivatives of $f$, where $V$ is a vector field on…

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and…

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under…

In this preliminary study, we provide two methods for estimating the gradients of functions of real value. Both methods are built on derivative estimations that are calculated using the standard method or the Squire-Trapp method for any…

Gradient algorithms are classical in adaptive control and parameter estimation. For instantaneous quadratic cost functions they lead to a linear time-varying dynamic system that converges exponentially under persistence of excitation…

In this paper, we are concerned with the gradient estimate of the electric field due to two nearly touching dielectric inclusions, which is a central topic in the theory of composite materials. We derive accurate quantitative…

This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach,…

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact…

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and…

In this paper, we consider a class of important nonlinear elliptic equations $$\Delta u + a(x)u\log u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$…

This paper studies global a priori gradient estimates for divergence-type equations patterned over the $p$-Laplacian with first-order terms having polynomial growth with respect to the gradient, under suitable integrability assumptions on…

Stochastic-gradient sampling methods are often used to perform Bayesian inference on neural networks. It has been observed that the methods in which notions of differential geometry are included tend to have better performances, with the…

This paper addresses a class of nonsmooth and nonconvex optimization problems defined on complete Riemannian manifolds. The objective function has a composite structure, combining convex, differentiable, and lower semicontinuous terms,…

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local…

For Dirac operators, which have discrete spectra, the concept of eigenvalues gradient is given and formulae for this gradients are obtained in terms of normalized eigenfunctions. It is shown how the gradient is being used to describe…

In this paper, firstly, we study gradient estimates for positive solution of the following equation \begin{equation*} \Delta_\xi(u)-\partial_t u- q u =A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $…

For a locally Lipschitz continuous function $f:X\to\mathbb{R}$ the generalized gradient $\partial f(x)$ of Clarke is used to develop some (set-valued) gradient on a set $A\subset X$. Existence, uniqueness and some approximation are…

A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calder\'on-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish…