The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

1985/01/01 by L. Caffarelli, Luis Caffarelli, L. Nirenberg +3 · 70 citations
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.1007/bf02392544

Abstract

On etudie le probleme de Dirichlet dans un domaine borne Ω de R n a frontiere lisse ∂Ω:F(D 2 u)=ψ dans Ω, u=φ sur ∂Ω

Citations

• Calibrated geometries

Cited by

• Viscosity solutions to complex Hessian equations
• Curvature flow of complete hypersurfaces in hyperbolic space
• The Hessian equation on nonsmooth k-convex domains or in the presence of subsolutions
• Regularity of viscosity solutions of the σ_k-Yamabe-type Problem for k>n/2
• Ideal Gårding polynomials
• Strict Convexity for Solution of Liouville-Type Dirichlet Problems
• Conformal geometry, contact geometry, and the calculus of variations
• Hypersurfaces of prescribed curvature measure
• Notes on the curvature estimates for Hessian equations
• A fully nonlinear version of the Yamabe problem and a Harnack type inequality
• A constant rank theorem for partial convex solutions of partial differential equations
• Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential\n Equations
• Oblique boundary value problems for augmented Hessian equations III
• Differential geometry in the theory of Hessian operators
• Mixed Hessian inequalities on Hermitian manifolds and applications
• Prescribed k-symmetric curvature hypersurfaces in de Sitter space
• Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations
• Locally uniform ellipticity of the fractional Hessian operators
• Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds
• Polar sets for m-subharmonic functions on compact Hermitian manifolds
• The p-th dual Minkowski problem for the k-torsional rigidity corresponding to a k-Hessian equation
• A class of fully nonlinear equations on Riemannian manifolds with negative curvature
• Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations
• Nonuniqueness for a fully nonlinear, degenerate elliptic boundary value problem in conformal geometry
• Regularity results for the equation u₁₁u₂₂ = 1
• Singular Solutions to Special Lagrangian Equations with Subcritical Phases and Minimal Surface Systems
• A note on Alxesandrov type theorem for k-convex functions
• On the two symmetries in the theory of m-Hessian operators
• Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials
• On the existence of geodesic connecting Lagrangian graphs in ℂⁿ
• Second order estimates for augment Hessian equations of parabolic type on Riemannian manifolds
• Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions
• Interior Hessian estimates for sum Hessian quotient equation
• Semi-convex viscosity solutions of the special Lagrangian equation
• The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
• A general Liouville type theorem for some conformally invariant fully nonlinear equations
• Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature
• Symmetry of solutions of minimal gradient graph equations on punctured space
• Entire spacelike hypersurfaces with constant σ_n-1 curvature in Minkowski space
• Some shape functionals for the k-Hessian equation
• Complete conformal metrics of negative Ricci curvature in open manifolds
• On attractors of m-Hessian evolutions
• Complete surfaces with negative extrinsic curvature
• A special concavity property for positive Hessian quotient operators
• (1,1) forms with specified Lagrangian phase: A priori estimates and\n algebraic obstructions
• Fractional analogue of k-Hessian operators
• A Liouville theorem for the 2-Hessian equation on the Heisenberg group
• On the Dirichlet problem for general augmented Hessian equations
• The quermassintegral inequalities for starshaped domains
• Hessian Estimates for the Sigma-2 Equation with Variable Right-Hand Side Terms in Dimension 4
• A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications
• Dirichlet problem for Krylov type equation in conformal geometry
• On Harnack inequalities and singularities of admissible metrics in the Yamabe problem
• Dirichlet problem for degenerate Hessian quotient type curvature equations
• Stability and the deformed Hermitian-Yang-Mills equation
• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• A priori estimates for the complex Monge-Ampère equation after Krylov
• The k-Yamabe problem on CR manifolds
• Boundary operators associated to the σ_k-curvature
• Pointwise boundary estimates for fully nonlinear elliptic equations with nonzero Dirichlet boundary conditions
• Option pricing in the large risk aversion, small transaction cost limit
• On Neumann problems for elliptic and parabolic equations on bounded manifolds
• The Dirichlet Problem for the Minimal Surface System in Arbitrary Codimension
• Hypersurfaces of Constant Curvature in Hyperbolic Space I
• On the Lagrangian minimal surface equation and related problems
• The global curvature estimate for the n-2 Hessian equation
• Second order estimates for fully nonlinear elliptic equations with gradient terms on Hermitian manifolds
• Parabolic Flow for Generalized complex Monge-Ampère type equations
• The Geometry of m -Hyperconvex Domains. [europepmc]
• The obstacle problem for conformal metrics on compact Riemannian manifolds. [europepmc]

Related

• On the exterior Dirichlet problem for a class of fully nonlinear elliptic equations
• The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
• On the Dirichlet problem for a class of augmented Hessian equations
• Hessian equations with elementary symmetric functions
• A spectral characterization and an approximation scheme for the Hessian eigenvalue