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Advances in Mathematics, ISSN 0001-8708, 02/2018, Volume 325, pp. 1 - 33
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the... 
Integral curvature bounds | Isoperimetric Constant | [formula omitted] Hessian estimates | Hessian estimates | MATHEMATICS | REGULARITY | FLOWS | L-2 Hessian estimates | RIEMANNIAN-MANIFOLDS
Journal Article
Journal de mathématiques pures et appliquées, ISSN 0021-7824, 10/2016, Volume 106, Issue 4, pp. 744 - 767
We establish sharp W2,p regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the... 
Regularity theory | Fully nonlinear elliptic equations | A priori [formula omitted] estimates | A priori W | estimates | MATHEMATICS | A priori W-2,W-p estimates | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | SPECIAL LAGRANGIAN EQUATIONS
Journal Article
Journal de mathématiques pures et appliquées, ISSN 0021-7824, 12/2013, Volume 100, Issue 6, pp. 785 - 805
Let (X,ω) be an n-dimensional compact Kähler manifold and fix an integer m such that 1⩽m⩽n. We study degenerate complex Hessian equations of the form... 
[formula omitted]-subharmonic | Weak solution | Hessian operator | Kähler manifold | (ω, m)-subharmonic | MATHEMATICS | MATHEMATICS, APPLIED | (omega, m)-subharmonic | PLURISUBHARMONIC-FUNCTIONS | INEQUALITY | DIRICHLET PROBLEM | WEAK SOLUTIONS | Kahler manifold | MONGE-AMPERE EQUATION | Sects
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 04/2016, Volume 270, Issue 7, pp. 2691 - 2714
We establish an interior C2 estimate for k+1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity... 
[formula omitted] convex | Rigidity theorem | Interior [formula omitted] estimate | k-Hessian equations | estimate | Interior C | K-Hessian equations | K+1 convex | MATHEMATICS | Interior C-2 estimate | HYPERSURFACES | k+1 convex | EQUATION
Journal Article
Applied Mathematics Letters, ISSN 0893-9659, 04/2020, Volume 102, p. 106124
In this paper, we focus on the convergence analysis of the unique solution for a Dirichlet problem of the general k-Hessian equation in a ball. By introducing... 
Unique solution | Radial solutions | [formula omitted]-Hessian equation | Error estimate | Convergence analysis | SCHRODINGER-EQUATIONS | EXISTENCE | EIGENVALUE | MATHEMATICS, APPLIED | k-Hessian equation | NONEXISTENCE | BOUNDARY-VALUE PROBLEM | FRACTIONAL DIFFERENTIAL-EQUATION | SYSTEM MODEL | MULTIPLE POSITIVE SOLUTIONS
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 02/2015, Volume 268, Issue 3, pp. 555 - 584
We consider fourth order elliptic systems in divergence form with greatly discontinuous coefficients in a nonsmooth domain. Our domain is composed of a finite... 
Global [formula omitted] estimate | Composite Reifenberg domain | BMO space | Fourth order elliptic system | estimate | Global W | MATHEMATICS | PARABOLIC EQUATIONS | Global W-2,W-P estimate | NONSMOOTH DOMAINS
Journal Article
Automatica, ISSN 0005-1098, 10/2014, Volume 50, Issue 10, pp. 2606 - 2614
We present the first q-Gaussian smoothed functional (SF) estimator of the Hessian and the first Newton-based stochastic optimization algorithm that estimates... 
[formula omitted]-Gaussian perturbations | Hessian estimate | Two-timescale algorithms | Stochastic optimization | q-Gaussian perturbations | APPROXIMATION | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Models | Algorithms | Mathematical optimization | Analysis
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 01/2018, Volume 167, pp. 51 - 66
In this paper we study the k-convex solutions to the boundary blow-up k-Hessian problem Sk(D2u)=H(x)up for x∈Ω,u(x)→+∞ as dist(x,∂Ω)→0.Here k∈{1,2,…,N},Sk(D2u)... 
[formula omitted]-convex solution | [formula omitted]-Hessian equation | Boundary blow up | Sub-supersolution method | k-convex solution | k-Hessian equation | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | 2ND-ORDER ELLIPTIC-EQUATIONS | BEHAVIOR | DIRICHLET PROBLEM | MONGE-AMPERE EQUATION
Journal Article
Complex Variables and Elliptic Equations, ISSN 1747-6933, 10/2019, Volume 64, Issue 10, pp. 1739 - 1755
We consider the Dirichlet problem for the complex Hessian equation of order . First, we give a specific estimate on the modulus of continuity of the solution... 
Complex Hessian equation | O. Celebi | m-hyperconvex domain | m-subharmonic functions | 32W20 | 32U40 | 32U05 | 32T15 | MATHEMATICS | MONGE-AMPERE OPERATOR | HOLDER CONTINUOUS SOLUTIONS | DIRICHLET PROBLEM | Dirichlet problem | Formulas (mathematics) | Continuity (mathematics)
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 01/2017, Volume 148, pp. 106 - 125
Journal Article
Bulletin des sciences mathématiques, ISSN 0007-4497, 06/2017, Volume 141, Issue 4, pp. 267 - 311
Journal Article
Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 11/2018, Volume 509, pp. 140 - 150
Journal Article
Pattern Recognition, ISSN 0031-3203, 2005, Volume 38, Issue 8, pp. 1239 - 1260
In this paper we describe a new shape-from-shading method. We show how the parallel transport of surface normals can be used to impose curvature consistency... 
EM algorithm | formula omitted | Vector transport | Shape-from-shading | Surface curvature | surface curvature | PERCEPTION | shape-from-shading | SURFACE | vector transport | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Algorithms
Journal Article
Complex Variables and Elliptic Equations, ISSN 1747-6933, 09/2018, Volume 63, Issue 9, pp. 1271 - 1289
We prove a global weighted Lorentz and Lorentz-Morrey estimates of the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equation... 
weighted Lorentz estimates | Fully nonlinear elliptic equations | Lorentz-Morrey spaces | viscosity solutions | weight functions | 35J60 | 35D40 | Lorentz–Morrey spaces | EXISTENCE | MATHEMATICS | PARABOLIC EQUATIONS | COEFFICIENTS | CONVEXITY ASSUMPTIONS | DOMAINS | Viscosity | Dirichlet problem | Nonlinearity | Nonlinear equations | Elliptic functions | Formulas (mathematics)
Journal Article
Journal of Statistical Computation and Simulation, ISSN 0094-9655, 03/2016, Volume 86, Issue 5, pp. 922 - 941
Journal Article
Communications in Mathematical Physics, ISSN 0010-3616, 7/2018, Volume 361, Issue 1, pp. 1 - 52
Journal Article
ESAIM: Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 07/2017, Volume 23, Issue 3, pp. 827 - 850
We prove a generalization of the Li−Yau estimate for a broad class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger−Yau... 
Mathematical analysis | Formulas (mathematics) | Hyperspaces
Journal Article
Interfaces and Free Boundaries, ISSN 1463-9963, 2015, Volume 17, Issue 1, pp. 93 - 115
We consider equations with the simplest hysteresis operator at the right-hand side. Such equations describe the so-called processes "with memory" in which... 
Sub-caloric functions | Free boundary | Monotonicity formula | Quadratic growth estimates | Hysteresis | REACTION-DIFFUSION EQUATIONS | MATHEMATICS | MATHEMATICS, APPLIED | monotonicity formula | quadratic growth estimates | sub-caloric functions | SPATIALLY DISTRIBUTED HYSTERESIS | hysteresis
Journal Article