Solves a specific Dirichlet problem on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère type equations on Hermitian manifolds.
method Solves the Dirichlet problem for Monge-Ampère type equations for (n−1)-plurisubharmonic functions on Hermitian manifolds. result Solves a specific Dirichlet problem on Hermitian manifolds.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.
problem Existence and uniqueness of solutions to the Dirichlet problem.
method Formulated using subharmonic functions; generalizes Hitchin's equation for diagonal harmonic metrics on cyclic Higgs bundles.
result Existence and uniqueness of solutions to the Dirichlet problem.
Study solves complex Hessian equation on Hermitian manifolds.
problem Solving Hessian equations on Hermitian manifolds with mixed structure.
method Derive a priori estimates and solve Dirichlet problem under conditions.
result Solvability of the Dirichlet problem for mixed Hessian equations.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
problem Solving Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.
method Uses interior C2 estimate. result Result is sharp, showing existence of singular solutions in subcritical phase.
Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.
problem Existence and uniqueness of solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Suitable settings to prove existence and uniqueness of solutions.
result Existence and uniqueness of solutions to the class of Hessian quotient equations.
Solves Dirichlet problem for elliptic equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds.
method Establishing a quantitative boundary estimate under a subsolution assumption.
result Derives solvability and regularity of the Dirichlet problem.
Paper solves curvature equations in Minkowski space for non-convex domains.
problem Solving curvature equations in non-convex domains of Minkowski space.
method Existence theorem proved via \emph{a priori} estimates and Serrin-type condition.
result Existence of solutions for curvature equations in non-convex domains.
Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.
problem Solving the Dirichlet problem for the complex Monge-Ampère equation on Hermitian manifolds with boundary.
method Weak quasi-plurisubharmonic solutions and optimal subsolution theorems for bounded and Hölder continuous quasi-plurisubharmonic functions.
result Proves continuity of solutions for measures well dominated by capacity, including Lp densities and moderate measures. In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we he…
Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère equation for (n−1)-PSH functions. method Deriving a quantitative boundary estimate under (n−1)-PSH subsolutions assumption. result Quantitative boundary estimate confirmed for specific manifolds.
Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
problem Proving radial symmetry of solutions to nonlinear equations in space forms.
method Establishing Rellich-Pohožaev type identities for Hessian quotient and k-Hessian equations.
result Radial symmetry of solutions for Hessian quotient and k-Hessian equations in space forms.
This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.
problem Solving the Dirichlet problem for degenerate elliptic equations on Riemannian manifolds with mean concave boundaries.
method The proof relies on a quantitative boundary estimate.
result Analogous results are obtained in complex variables and on certain product manifolds.
Study smooth hypersurfaces with prescribed curvature in Minkowski space.
problem Existence of smooth spacelike hypersurfaces with prescribed curvature.
method Proving existence based on C2 estimates. result Existence of smooth spacelike hypersurfaces with prescribed curvature.
Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
problem Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space
method Fully non-linear elliptic equation
result Solves if datas are strictly convex
We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori C2 estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradients of the solutions. In some special cases, we obtain the gradient estim…
Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.
problem Solving the Dirichlet problem for mean curvature equations in different spacetimes.
method Compares techniques and conditions for solvability in Euclidean and Lorentz-Minkowski spaces.
result Lorentz-Minkowski spacelike condition allows dropping mean convexity hypothesis.
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
Neural networks solve the Dirichlet problem for Monge-Ampère equations.
problem Solving the Dirichlet problem for the Monge-Ampère equation.
method Using deep input convex neural networks to find the unique convex solution.
result Deep input convex neural networks can solve the Monge-Ampère Dirichlet problem.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.
problem Understanding the behavior of solutions to the Allen-Cahn equation on manifolds with boundary.
method Analyzing the asymptotic behavior of Dirichlet minimizers, relating Neumann data to boundary geometry, and using invertibility of the linearized Allen-Cahn operator.
result Computed expansions of the solution to high order and established a projection theorem about Allen-Cahn solutions near minimal surfaces.
Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.
problem Solving Dirichlet problem for complex Monge-Ampère equation.
method Properties of subsolutions for fully nonlinear elliptic equations.
result Existence of C2-smooth strictly J-plurisubharmonic subsolution. We prove the existence of classical solutions to the Dirichlet problem for the α-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving the Dirichlet problem when the boundary data is formed by two copies of a convex func…
The paper establishes gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
problem Gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
method Yau and Souplet-Zhang type gradient estimates for harmonic and heat equation solutions under Dirichlet boundary condition.
result Established gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
Investigates regularity of solutions to complex Hessian equation.
problem Regularity of solutions to complex Hessian equation.
method Analyzes solutions to Dirichlet problem with specific density condition.
result Establishes conditions for regularity of solutions.
Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural conditions.
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniquenes…
Wave equation map reveals manifold's structure.
problem Reconstructing Lorentzian manifold from wave equation map.
method Analyzing Schwartz kernel and boundary light observation set.
result Full Lorentzian structure can be recovered under geometric assumptions.
We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan-Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation of the author. The paper is organised as follows. First we introduce Cartan-Hadamard mani…
The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…
Let $Ω\subset\r^n$ be a bounded mean convex domain. If α<0, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in Ω for the α-singular minimal surface equation with arbitrary continuous boundary data.
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…
Paper solves a complex equation for smooth domains.
problem Investigates a specific type of parabolic equation in complex domains.
method Uses J-functional to prove solution convergence.
result Proves the convergence of solutions to the equation.
Stokes equations help uniquely identify manifold metrics from boundary data.
problem Determining Riemannian metric from boundary Cauchy data.
method Proving uniqueness of metric from Stokes equations Cauchy data.
result Partial derivatives of all orders of the metric on the boundary are uniquely determined.
The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
New equations reveal viscosity from boundary measurements.
problem Determine viscosity from boundary measurements for incompressible fluids.
method Equivalent new system of elliptic equations, Dirichlet-to-Neumann map analysis.
result Dirichlet-to-Neumann map uniquely determines viscosity and its derivatives on the boundary.
Study fully nonlinear elliptic equations on complex manifolds.
problem Solving fully nonlinear elliptic equations on complex manifolds.
method Derive C2,α-estimate and prove existence theorems for solutions and Dirichlet problems. result Existence theorems for solutions on closed Hermitian manifolds with unbounded conditions.
The paper solves the Dirichlet problem for minimal surfaces on unbounded helicoidal domains.
problem Solving the Dirichlet problem for minimal surfaces on unbounded helicoidal domains.
method Analyzes G-invariant solutions in C2,α domains with helicoidal projections. result Existence of G-invariant solutions with controlled gradient at infinity. We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or…
We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show …
Proves well-posedness for Einstein equations with specific boundary conditions.
problem Well-posedness of vacuum Einstein equations with twisted Dirichlet boundary conditions.
method Proves local-in-time well-posedness for the IBVP of the Einstein equations with specified conformal class and scalar densities.
result Proves well-posedness for the Einstein equations with twisted Dirichlet boundary conditions.
The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.
problem Proving existence and interior estimates for fully nonlinear equations on Hermitian manifolds.
method Derives interior estimates and establishes the existence of smooth solutions for the Dirichlet problem and equations on closed manifolds.
result Derives interior estimates and establishes the existence of smooth solutions for the Dirichlet problem and equations on closed manifolds.
New neural network approach solves Poisson equations efficiently.
problem Approximating solutions to Poisson equations with Dirichlet boundary conditions.
method Using shallow ReLUα-networks to solve Laplace operator equations. result Neural networks can approximate solutions to the Laplace operator with Dirichlet boundary conditions efficiently.
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting…