The paper extends Huber's theorem to higher dimensions using n-Laplace equations.
problem Proving finite point conformal compactification for general dimensions.
method Using n-Laplace equations and strengthened Arsove-Huber's theorem.
result Established finite point conformal compactification theorem for manifolds.
In this paper, we obtain two rigidity results for p-Laplace type equation and n-Laplace equation with exponential nonlinearity on n-dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a …
Study critical quasilinear equations on Riemannian manifolds with curvature constraints.
problem Investigate critical quasilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature.
method Utilize a new nonlinear Kato inequality and Cheng-Yau type gradient estimates for positive solutions.
result Classify positive solutions to the critical p-Laplace equation and show rigidity concerning the ambient manifold. In this paper we study asymptotic behavior of n-superharmonic functions at isolated singularity using the Wolff potential and n-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study n-superharmonic functions we use a…
In the first part of the paper we investigate some geometric features of Moser-Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopo…
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Develops equivariant connections for Yang-Mills equations, simplifying interactions modeling.
problem Simplifying interactions modeling in Yang-Mills equations for different bundles.
method Introduces SO+(p,q)-equivariance to reduce Yang-Mills equations. result Models electroweak interaction and interactions with differential and wave equations.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.
problem Addressing Hermitian-Einstein equation for cyclic Higgs bundles.
method Introducing generalizations using subharmonic functions and proving existence, uniqueness, and convergence of heat equations.
result Existence, uniqueness, and convergence of solutions for heat equations.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
Study a modified Laplacian equation in spacetime.
problem Analyzing a perturbed Laplacian equation in spacetime.
method Examining the equation \( \Delta u + P |
abla u| = h |
abla u| \) in an initial data set.
result Identified new properties of the modified equation.
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
The paper proves estimates for vortex-type equations on compact Riemann surfaces.
problem Estimating vortex-type equations on compact Riemann surfaces.
method Proves \emph{a priori} estimates for vortex-type equations.
result Recover existing estimates for vortex bundle Monge-Ampère equation, prove existence and uniqueness for Calabi-Yang-Mills equations, and get estimates for J−vortex equation. Geometrically interprets two equations, showing their equivalence and providing solutions.
problem Equivalence and solutions of generalized Proudman-Johnson and r-Hunter-Saxton equations.
method Geometric interpretation through Finsler metrics and isometries.
result Equivalence of periodic and non-periodic cases as geodesic equations.
Auxiliary equations improve bounds in symplectic geometry.
problem Improving bounds for the Calabi-Yau equation.
method Adapted Monge-Ampère equations for symplectic geometry.
result Reduced the bound from exponential to L1. In this paper we introduce a new equation on the compact Kahler manifolds. Solution of this equation corresponds to the Calabi-Yau metric. New equation differs from the Monge--Ampere equation considered by Calabi and Yau.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution equations with one space variable. The main concept is that of connection defin…
Study vortices in Kähler-Yang-Mills equations on complex manifolds.
problem Solving coupled equations for Kähler metrics and connections.
method Dimensional reductions of Kähler-Yang-Mills equations to study vortices.
result Found solutions to Yang-Mills-Higgs equations related to vortices.
In this paper, we study the Poisson equation and heat equation in a model matrix geometry Mn. Our main results are about the Poisson equation and global behavior of the heat equation on Mn. We can show that if c0 is the initial positive definite matrix in Mn, then c(t) exists for all time and is positive …
Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.
problem Existence of solutions to coupled Kähler-Einstein and Hermitian-Yang-Mills equations.
method Moment map interpretation, Futaki invariant, Matsushima-Lichnerowicz theorem, deformation results.
result Nontrivial solutions produced under certain conditions.
Compactness theorem for 3D RS-SW equations on 3-manifolds.
problem Non-compactness of moduli space of solutions.
method Variational approach to 3D RS-SW equations.
result Proof of compactness theorem.
New proof of L∞ estimates for Monge-Ampère and Hessian equations on nef classes.
problem Estimating solutions to Monge-Ampère and Hessian equations on nef classes.
method Applying PDE approach to Kähler manifolds to nef classes.
result New proofs of estimates for Monge-Ampère and Hessian equations.
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
The paper explores solutions to the distributional Bellman equation in reinforcement learning.
problem Distributional reinforcement learning considers complete return distributions, not just expected returns.
method Study existence and uniqueness of solutions to general distributional Bellman equations, linking them to multivariate affine equations.
result Any solution to a distributional Bellman equation can be derived from a multivariate affine distributional equation.