Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions

2020 ◽  
Author(s):  
Jian Wang ◽  
Bing Yao Wu
Keyword(s):  
Gradient Estimates ◽  
Monotonicity Conditions ◽  
Local Monotonicity

scholarly journals Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: 1 < p ≤ 3n−2 2n−1

2020 ◽  
pp. 2050056
Author(s):  
Le Cong Nhan ◽  
Le Xuan Truong
Keyword(s):  
Global Regularity ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Singular Case ◽  
Regularity Estimates ◽  
Monotonicity Conditions ◽  
Vector Valued Function ◽  
Quasilinear Elliptic ◽  
Vector Valued

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form [Formula: see text] where [Formula: see text] is a finite signed Radon measure in [Formula: see text], [Formula: see text] is a bounded domain such that its complement [Formula: see text] is uniformly [Formula: see text]-thick and [Formula: see text] is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case [Formula: see text]. Our result extends the earlier results [19,22] to the strongly singular case [Formula: see text] and a recent result [18] by considering rough conditions on the domain [Formula: see text] and the nonlinearity [Formula: see text].

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

scholarly journals Gradient estimates for semilinear elliptic systems and other related results

2015 ◽  
Vol 145 (6) ◽  
pp. 1313-1330 ◽  
Author(s):  
Panayotis Smyrnelis
Keyword(s):  
Liouville Theorem ◽  
Elliptic Systems ◽  
Gradient Estimates ◽  
Alternative Form ◽  
Strong Monotonicity ◽  
Energy Tensor ◽  
Planar Domains ◽  
Stress Energy ◽  
General Phase ◽  
Double Well Potential

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

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