In the product space H^n \times R; we obtain uniform a priori C^0 horizontal length estimates, uniform a priori C^1 boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C^1 estimates …
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48 results for “A priori Estimates”
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.
We obtain a priori estimates for solutions to the prescribed scalar curvature equation on . The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse functi…
Paper proves estimates for metrics with conic singularities.
problem Proving estimates for metrics with conic singularities.
method Reformulated Alexandrov's maximum principle for zero order estimates and Chen-Cheng's framework for higher order estimates.
result A priori estimates for conic singularities metrics with prescribed scalar curvature.
A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phasemath.AP
Paper proves estimates for Lagrangian flow singularities.
problem Understanding Lagrangian flow singularities.
method Interior a priori estimates and Jacobi inequality.
result Proves estimates for supercritical Lagrangian phase.
Study on Kähler metrics with curvature constraints.
problem Existence of constant weighted scalar curvature Kähler metrics.
method Establish -estimates for Kähler potentials.
result Extends prior results on classical cscK metrics.
Estimates neural network errors for classification problems.
problem Binary and multi-class classification problems.
method Rademacher complexity estimates and direct approximation theorems.
result A priori error estimates for regularized loss functionals.
Estimates for neural network risk nearly match Monte Carlo error rates.
problem Understanding the performance of two-layer neural networks.
method Established a priori estimates for the population risk of two-layer neural networks.
result The new estimates are nearly optimal and depend only on function norms, not model parameters.
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
Quadratic Hessian equationmath.AP
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
Paper studies unique interior points and estimates for generalized translating soliton problems.
problem Generalized translating soliton type problems.
method Proves uniqueness of interior critical points, derives and estimates using minimum principles.
result Derives a priori and estimates for solutions.
Optimal estimates derived for residual networks' generalization error.
problem Estimating the generalization error of residual networks.
method Derives optimal a priori estimates using a weighted path norm.
result Optimal error estimates are comparable to Monte Carlo error rates.
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.
We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.
New method for complex Monge-Ampère equations on Kähler manifolds.
problem Degenerate complex Monge-Ampère equations on complex manifolds.
method New approach relying on compactness and envelopes properties of quasi-plurisubharmonic functions.
result New and efficient proofs of fundamental results in Kähler geometry.
Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.
problem Quantifying the Hessian of log-concave eigenfunctions on convex domains.
method Analyzes log-concavity properties of the first Dirichlet eigenfunction on convex domains.
result Obtains quantitative estimates for the Hessian of .
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.
Estimates generalization error for two-layer ReLU NNs through minimum norm solutions.
problem Estimating generalization error for two-layer ReLU NNs trained by mean squared error.
method Uses minimum norm solutions and Neural Tangent Kernel (NTK) regime to derive generalization error bounds.
result Derives an a priori generalization error bound for two-layer ReLU NNs without requiring exponentially large number of neurons.
The paper studies equations on almost Hermitian manifolds with estimates and existence results.
problem Solving Monge-Ampère type equations on compact almost Hermitian manifolds.
method Derives a priori estimates and obtains existence results under admissible conditions.
result Existence of solutions under admissible conditions for Monge-Ampère type equations.
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.
We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori -estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the stud…
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.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Study smooth hypersurfaces with prescribed curvature in Minkowski space.
problem Existence of smooth spacelike hypersurfaces with prescribed curvature.
method Proving existence based on estimates.
result Existence of smooth spacelike hypersurfaces with prescribed curvature.
New method solves complex Monge-Ampère equations on hermitian manifolds.
problem Solving degenerate complex Monge-Ampère equations on hermitian manifolds.
method New approach using compactness and envelopes properties of quasi-plurisubharmonic functions.
result New relative a priori estimates and existence results for degenerate complex Monge-Ampère equations.
Let be a compact Kähler manifold and $\om$ a smooth closed form of bidegree which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight has fast growth at infinity, the corresponding functions are …
We derive a priori interior Hessian estimates for the special Lagrangian equation in dimension three.
Mean curvature flow of graphs with asymptotic Dirichlet conditions in Cartan-Hadamard manifoldsmath.DG
Estimates prove existence of curvature flow in curved spaces.
problem Mean curvature flow in curved spaces with boundary conditions.
method A priori estimates and existence proof for curvature flow.
result Existence of curvature flow with asymptotic Dirichlet conditions.
Paper studies Hessian quotient equations in warped product manifolds.
problem Analyzing Hessian quotient equations in warped product manifolds.
method Using standard degree theory and a priori estimates.
result Existence of star-shaped compact hypersurface solutions.
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…
The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.
problem Boundary estimates for fully nonlinear Yamabe equations on Riemannian manifolds.
method Deriving a priori second derivative estimates for subsolutions.
result Existence of smooth solutions with uniform estimates.
Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.
problem Second boundary value problem for special Lagrangian curvature potential equation.
method Method of continuity with a-priori estimate.
result Existence and uniqueness of smooth uniformly convex solutions.
We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.
We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the Kähler form on a Kähler manifold. A \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of Kähler metri…
Solves Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
method Derived estimates and gradient estimates for solutions.
result Solved Dirichlet problem with admissible subsolutions in some cases.
We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angl…
Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show th…
We derive a priori estimates for the -plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.
Solves a complex equation on manifolds with boundary.
problem Solving a Dirichlet problem for -Hessian equations on complex manifolds.
method Second order a priori estimate and blow-up argument.
result Control over all necessary norms of the solution.
Paper proves solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
problem Existence of solutions for deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds.
method Derive a priori estimates under the existence of an admissible -subsolution; prove existence of solutions under the condition of existence of a supersolution.
result Proves existence of solutions for the deformed Hermitian-Yang-Mills equation.
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
Paper constructs non-symmetric collapsing spacetimes without symmetries.
problem Forming non-symmetric collapsing spacetimes in vacuum.
method Modified Christodoulou's a priori estimates and gluing construction.
result Past geodesic completeness and asymptotic Minkowski space.
Proves interior C2 estimate for a specific equation in 3D.
problem Proving interior C2 estimate for a specific equation in 3D.
method Proves a priori interior C2 estimate for σ_2 = f in R3.
result Generalizes Warren-Yuan's result with a C2 estimate.
Hessian and gradient estimates for three dimensional special Lagrangian Equations with large phasemath.AP
We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
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 technique RRT improves OMP performance without knowing sparsity or noise.
problem Recovering sparse high-dimensional vectors without knowing sparsity or noise statistics.
method Residual ratio thresholding (RRT) to operate OMP without a priori knowledge.
result RRT achieves comparable performance to OMP with known statistics.
This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Ampére type eqution on the Kähler cone; secondly, we obtain a priori etimates for the abo…