Related papers: Hessian and gradient estimates for three dimension…

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases…

In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We…

We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon$ and thus imply analogous estimates for…

In this paper we apply various first and second derivative estimates and barrier constructions from our treatment of oblique boundary value problems for augmented Hessian equations, to the case of Dirichlet boundary conditions. As a result…

This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet…

We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.

On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. For three-dimensional phase space the concept of vector hamiltonian and vector lagrangian is entered.

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations

In this paper we derive Lieb-Thirring estimates for eigenvalues of Dirichlet Laplacians below the threshold of the essential spectrum on asymptotically Archimedean spiral-shaped regions.

In this paper, we study the interior gradient estimates for admissible solutions to prescribed curvature equations in hyperbolic space.

We develop optimization methods which offer new trade-offs between the number of gradient and Hessian computations needed to compute the critical point of a non-convex function. We provide a method that for any twice-differentiable $f\colon…

We derive Hessian estimates for convex solutions to quadratic Hessian equation by a compactness argument.

We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that…

We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally…

The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and…

Gradient descent is commonly used to find minima in rough landscapes, particularly in recent machine learning applications. However, a theoretical understanding of why good solutions are found remains elusive, especially in strongly…

We present a systematic derivation of the algorithms required for computing the gradient and the action of the Hessian of an arbitrary misfit function for large-scale parameter estimation problems involving linear time-dependent PDEs with…

A notion of internal Lagrangian for a system of differential equations is introduced. A spectral sequence related to internal Lagrangians is obtained. A connection between internal Lagrangians and presymplectic structures is investigated.…

Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $L^p$. Derivation…