A particle scheme based on implicit Euler time stepping and spatial sampling is proved to converge for first-order MFGs under displacement monotonicity, handling non-separable Hamiltonians and singular data for arbitrary horizons.
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13 Pith papers cite this work, alongside 6,903 external citations. Polarity classification is still indexing.
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2026 13representative citing papers
Proves a J-adapted Levi-Malcev decomposition for many 2-step solvable Lie algebras, confirming the Fino-Vezzoni conjecture for unimodular cases and characterizing SKT metrics on completely solvable ones.
Multiphase quadrature domains exist and are unique under sufficient conditions via constrained minimization of an energy functional over segregated states, with an example showing that energy minimization and partial balayage are not equivalent in the two-phase case.
NTK neural networks achieve minimax optimal adversarial regression rates in Sobolev spaces using gradient flow with early stopping, but minimum norm interpolants are vulnerable in the overfitting regime.
A local reconstruction scheme for Codazzi defects in 4D Lorentzian branches uses a lexicographic residual and CP1 Toeplitz visibility to select the S(U(3)×U(2))/Z6 form and standard one-generation SM exterior package.
Reaction-diffusion SPDEs with non-trace-class multiplicative noise are shown to be locally well-posed in critical Besov spaces of initial data, with regularization, blow-up criteria, and positivity.
Interior a priori Sobolev-Orlicz estimates for higher-order elliptic systems with VMO coefficients are proved for Young functions satisfying Δ2 ∩ ∇2.
Unique convex solutions exist for the second boundary value problem of mean curvature type equations with prescribed gradient image between uniformly convex bounded domains with smooth boundaries.
A PDE-based improvement-of-flatness technique for annuli provides an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions.
A nonconforming virtual element method is developed for the vanishing moment approximation of the Monge-Ampère equation in 2D, with optimal a priori error estimates in H2, H1 and L2 norms plus existence and uniqueness of the discrete solution.
A pseudospectral multishape method is developed to accurately approximate singular convolution operators in the nonlocal Cahn-Hilliard equation, enabling efficient high-resolution phase separation simulations.
Proves C^{1,1} regularity for a degenerate fully nonlinear equation on Hermitian manifolds with balanced metrics, yielding unique C^{1,1} solutions to the Donaldson equation.
The survey describes eigenvalue inequalities, spectral asymptotics, nodal domains, and new phenomena for the Dirichlet-to-Neumann map of the Helmholtz equation that do not appear in the Laplace case.
citing papers explorer
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Numerical analysis of first-order mean field games under displacement monotonicity
A particle scheme based on implicit Euler time stepping and spatial sampling is proved to converge for first-order MFGs under displacement monotonicity, handling non-separable Hamiltonians and singular data for arbitrary horizons.
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A Levi-type decomposition on two-step solvable Lie algebras with a complex structure
Proves a J-adapted Levi-Malcev decomposition for many 2-step solvable Lie algebras, confirming the Fino-Vezzoni conjecture for unimodular cases and characterizing SKT metrics on completely solvable ones.
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Multiphase quadrature domains (existence and uniqueness)
Multiphase quadrature domains exist and are unique under sufficient conditions via constrained minimization of an energy functional over segregated states, with an example showing that energy minimization and partial balayage are not equivalent in the two-phase case.
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Adversarial Robustness of NTK Neural Networks
NTK neural networks achieve minimax optimal adversarial regression rates in Sobolev spaces using gradient flow with early stopping, but minimum norm interpolants are vulnerable in the overfitting regime.
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Self-Reconstructing Codazzi Defects, $\mathbb{CP}^1$ Quantization, and the Minimal Standard-Model Carrier
A local reconstruction scheme for Codazzi defects in 4D Lorentzian branches uses a lexicographic residual and CP1 Toeplitz visibility to select the S(U(3)×U(2))/Z6 form and standard one-generation SM exterior package.
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An optimal local theory for reaction-diffusion equations driven by non-trace-class noise
Reaction-diffusion SPDEs with non-trace-class multiplicative noise are shown to be locally well-posed in critical Besov spaces of initial data, with regularization, blow-up criteria, and positivity.
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Interior a priori estimate for higher order elliptic systems in Orlicz spaces
Interior a priori Sobolev-Orlicz estimates for higher-order elliptic systems with VMO coefficients are proved for Young functions satisfying Δ2 ∩ ∇2.
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The mean curvature type hypersurfaces with prescribed gradient image
Unique convex solutions exist for the second boundary value problem of mean curvature type equations with prescribed gradient image between uniformly convex bounded domains with smooth boundaries.
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Improvement of flatness in annuli
A PDE-based improvement-of-flatness technique for annuli provides an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions.
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Nonconforming virtual element method for the Monge-Amp\`ere equation
A nonconforming virtual element method is developed for the vanishing moment approximation of the Monge-Ampère equation in 2D, with optimal a priori error estimates in H2, H1 and L2 norms plus existence and uniqueness of the discrete solution.
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Singularities in phase separation models: a spectral element approach for the nonlocal Cahn-Hilliard equation
A pseudospectral multishape method is developed to accurately approximate singular convolution operators in the nonlocal Cahn-Hilliard equation, enabling efficient high-resolution phase separation simulations.
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Regularity of a Geodesic equation in the space of mixed Volume Forms on Hermitian Manifolds
Proves C^{1,1} regularity for a degenerate fully nonlinear equation on Hermitian manifolds with balanced metrics, yielding unique C^{1,1} solutions to the Donaldson equation.
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Spectral properties of the Dirichlet-to-Neumann map for the Helmholtz equation
The survey describes eigenvalue inequalities, spectral asymptotics, nodal domains, and new phenomena for the Dirichlet-to-Neumann map of the Helmholtz equation that do not appear in the Laplace case.