The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
Enhances gradient estimates for Hermitian Monge-Ampère equations.
problem Improving estimates for Hermitian Monge-Ampère equations.
method Improves gradient estimates using Evans-Krylov and third derivatives estimates.
result Enhanced estimates for second and third order derivatives.
Paper establishes L∞ estimates for complex Monge-Ampere and Hessian equations.
problem Estimating solutions to complex Monge-Ampere and Hessian equations.
method Uses PDE techniques similar to Phong et al to prove L∞ and Hölder estimates. result Establishes L∞ estimates for both complex Monge-Ampere and Hessian equations. Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
New estimate for complex Monge-Ampère equations improves previous results.
problem Improving estimates for complex Monge-Ampère equations.
method Using the ABP maximum principle to prove a new gradient estimate.
result Proves a new gradient estimate for complex Monge-Ampère equations.
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
The paper estimates curvature for a specific type of equations.
problem Estimating curvature for Hessian type equations.
method Establishing curvature estimates for a class of Hessian type equations.
result Curvature estimates for Hessian type equations have been successfully established.
Global and local estimates for a curvature equation on manifolds with boundary.
problem Estimating modified σ2 curvature equation with boundary conditions. method Global and local C2-estimates established. result Global C2-estimates for the modified σ2 curvature equation. Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
Sharp L∞ estimates proved for complex Monge-Ampère equations.
problem Proving sharp L∞ estimates for complex Monge-Ampère equations. method PDE proof covering fixed and degenerating background metrics, extends to general fully non-linear equations.
result Sharp L∞ estimates proved for complex Monge-Ampère equations. Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Estimates symplectic Calabi-Yau equation using Cheng-Yau method.
problem Estimating solutions to the Calabi-Yau equation on symplectic 4-manifolds.
method Applies a Cheng-Yau type estimate in the symplectic setting.
result Proves an a priori estimate for the symplectic Calabi-Yau equation.
Estimates for special Lagrangian curvature equations in critical and convex cases.
problem Interior estimates for special Lagrangian curvature equations.
method Establishes a priori interior curvature and gradient estimates.
result Proves interior curvature and gradient estimates for special Lagrangian curvature equations.
The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of nonlinear parabolic equations.
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
The paper proves estimates for vortex-type equations on compact Riemann surfaces.
problem Estimating vortex-type equations on compact Riemann surfaces.
method Proves \emph{a priori} estimates for vortex-type equations.
result Recover existing estimates for vortex bundle Monge-Ampère equation, prove existence and uniqueness for Calabi-Yang-Mills equations, and get estimates for J−vortex equation. Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.
problem Interior curvature estimates for convex graphs.
method Analyzes convex graphs satisfying the quotient equation σn−2σn(λ)=f(X)>0. result Interior curvature estimates for convex graphs.
Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.
problem Estimating solutions to CR heat equation on complex manifolds.
method Local and global Li-Yau type gradient estimates.
result Gradient estimates and Harnack inequality for positive solutions.
New proof of L∞ estimates for Monge-Ampère and Hessian equations on nef classes.
problem Estimating solutions to Monge-Ampère and Hessian equations on nef classes.
method Applying PDE approach to Kähler manifolds to nef classes.
result New proofs of estimates for Monge-Ampère and Hessian equations.
Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.
problem Estimating gradients for porous medium equations on manifolds with negative curvature.
method Develops Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of N-weighted Ricci curvature with N<0. result Generalizes gradient estimates for porous medium equations to manifolds with negative curvature.
The note provides uniform estimates for complex Hessian equations on compact Hermitian manifolds.
problem Uniform estimates for solutions to degenerate complex Hessian equations on compact Hermitian manifolds.
method The approach relies on corresponding a priori estimates for Monge-Ampère equations.
result Extension and short alternative proof of results for complex Hessian equations.
Paper derives estimates for complex Hessian equations on Hermitian manifolds.
problem Estimating solutions to complex Hessian equations on Hermitian manifolds.
method Derives second order estimates for solutions in a specific cone.
result Establishes second order estimates for solutions in Γk+1 cone. We prove a priori interior C2,α estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…
Derives concavity inequality and estimates for k-Hessian equations.
problem Interior estimates and curvature estimates for k-Hessian equations. method Concavity inequality and semi-convexity condition.
result Interior estimates and Liouville-type result for semi-convex solutions.
Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.
problem Derives gradient estimate for a nonlinear parabolic equation on Finsler manifolds.
method Leverages a new Laplacian comparison theorem to derive a Li-Yau type gradient estimate.
result Establishes a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.
Derives gradient estimation for a specific heat equation on evolving manifolds.
problem Gradient estimation for a generalized heat equation on evolving weighted Riemannian manifolds.
method Derives gradient estimation for a specific heat equation on evolving weighted Riemannian manifolds.
result Derives a Harnack type inequality and a Liouville type theorem as applications of gradient estimation.
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
Proves Hessian estimates for special Lagrangian equation with new proofs.
problem Interior Hessian estimates for special Lagrangian equation
method Doubling proofs, higher codimension analogue of previous methods
result Higher codimension analogue of gradient estimate for minimal hypersurfaces
The paper derives subgradient estimates for a specific nonlinear subparabolic equation on pseudo-Hermitian manifolds.
problem Deriving subgradient estimates for positive solutions to a nonlinear subparabolic equation on pseudo-Hermitian manifolds.
method Using the CR sub-Laplacian comparison property, the paper derives local subgradient estimates for positive solutions to the given equation.
result The paper establishes subgradient estimates for positive solutions to the nonlinear subparabolic equation.
We establish interior C2 estimates for convex solutions of scalar curvature equation and σ2-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces (Mn,g)⊂Rn+1 with positive scalar curvature. These estimates are consequences of an interior estimate…
Study interior estimates for solutions of Poisson equation on Riemann surfaces.
problem Interior estimates for solutions of linear Poisson equation on Riemann surfaces.
method Used Zygmund space LlnL and isoperimetric inequality. result Derived interior estimates, Harnack inequalities, and global estimate.
Paper proves gradient estimates for Lagrangian mean curvature equation.
problem Proving gradient estimates for Lagrangian mean curvature equation.
method Interior gradient estimates for critical and supercritical Lagrangian mean curvature equation.
result Solves Dirichlet boundary value problem for critical and supercritical Lagrangian mean curvature equation.
Paper establishes estimates for solutions on compact manifolds.
problem Solving fully non-linear equations on compact almost Hermitian manifolds.
method Establishes a priori estimates for solutions.
result Solves complex Hessian and Monge-Ampère equations.
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
Paper solves a long-standing problem with curvature estimates.
problem Long-standing problem in n−2 curvature equation. method Global curvature estimate for the n−2 Hessian equation. result Solves a long-standing problem in n−2 curvature equation. New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
Unified estimate for complex Monge-Ampère equations on Kähler manifolds.
problem Estimating solutions to complex Monge-Ampère equations on Kähler manifolds.
method Unified approach using PDE methods and entropy bounds to construct comparison metrics.
result Improves previous results on modulus of continuity, stability, and W1,1-estimates of Green's functions. The paper modifies Vafa-Witten equations on 4-manifolds for better solution estimates.
problem Constructing a priori estimates for solutions of Vafa-Witten equations on 4-manifolds.
method Introducing perturbation terms to the Vafa-Witten equations and proving transversality.
result The singularities of solutions can be removed, and moduli spaces constructed.
Optimizes heat equation estimates on noncompact manifolds.
problem Improving gradient estimates for heat equations on noncompact manifolds.
method Localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation.
result Essentially optimal estimates significantly improve previous results.
Study bounds derivatives of solutions to a specific equation on domains.
problem Bounding second derivatives of solutions to the σk-Yamabe equation. method Proves local pointwise second derivative estimates for positive W2,p solutions. result Establishes bounds for derivatives of solutions to the σk-Yamabe equation. The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.
problem Gradient estimates for solutions to a specific nonlinear elliptic equation on smooth metric measure spaces.
method Nash-Moser iteration technique to obtain local gradient estimates.
result New local gradient estimates for positive solutions to the equation.
The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.
problem Developing sub-gradient estimates and entropy formulas for quaternionic contact geometry.
method Establishing sub-gradient estimates and entropy formulas for the quaternionic contact heat equation.
result Two Perelman-type entropy formulas and sub-gradient estimates for the quaternionic contact heat equation.
Note on gradient estimates for complex Monge-Ampere equation.
problem Gradient estimates for solutions of complex Monge-Ampere equation.
method Estimates Lp and L∞ for gradient in terms of continuity of the right-hand side. result Gradient estimates for solutions of complex Monge-Ampere equation.