The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods using bolder step size selection exist. Various alternatives are investigated from both theoretical and practical...

Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation ${\Delta }_{f}u+c{u}^{-\alpha }=0$ in $M$, where $\alpha$, $c$ are two real constants and $\alpha >0$, $f$ is a smooth real valued function on $M$ and ${\Delta }_f=\Delta -\nabla f\nabla$. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty$-Bakry-Emery Ricci tensor is bounded from below and $\left|\nabla f\right|$ is bounded from above,...

Let $X=(X_1,X_2,\cdots ,X_q)$ be a system of vector fields satisfying the Hörmander condition. We prove $L_X^{2,\lambda }$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$-X_{\alpha }^{*}\left(a_{ij}^{\alpha \beta }\left(x\right)X_{\beta }{u}^j\right)=g_i-X_{\alpha }^{*}{f}_i^{\alpha }\left(x\right)\forall i=1,2,\cdots ,N,$$ where $a_{ij}^{\alpha \beta }\left(x\right)$ are bounded functions and belong to Vanishing Mean Oscillation space.

We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p > 1 of F-1 and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface....

Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in ${ℝ}^N$.

We study, with purely analytic tools, existence, uniqueness and gradient estimates of the solutions to the Neumann problems associated with a second order elliptic operator with unbounded coefficients in spaces of continuous functions in an unbounded open set Ω in ${ℝ}^N$.

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation $$u_t={\Delta }_{V}u+aulogu+bu$$ where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0,\infty )$. We suppose that the Bakry-Émery curvature and the $N$-dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.

This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathscr{H}$$$\left\{\begin{array}{c}{u}^{}\hfill \end{array}\right.$$

We introduce a new transport distance between probability measures on ${ℝ}^d$ that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is...

In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second...

The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.

Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.

Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $${\int }_{\mathrm{\Omega }}f\left(Du\right)dx,$$ dove $f$ soddisfa l'ipotesi di crescita $${\left|\xi \right|}^p-c_1\le f\left(\xi \right)\le c\left(1+{\left|\xi \right|}^q\right),$$ con $1<p<q\le 2$. L'integrando $f$ è $C^2$ e $DDf$ ha crescita $p-2$ dal basso e $q-2$ dall'alto.

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.