Study classifies solutions to a triharmonic Lane-Emden equation.
problem Classifying solutions to a specific triharmonic Lane-Emden equation.
method Derive monotonicity formula, classify solutions (positive or sign-changing, radial or not).
result New monotonicity formula for triharmonic maps as a byproduct.
Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.
problem Existence and non-existence of positive solutions for the Lane-Emden equation on Riemannian models.
method Analysis of the subcritical Lane-Emden equation on various Riemannian manifolds with polynomial volume growth.
result Subcritical regime divides into three ranges with distinct existence and non-existence phenomena.
We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation −Δgu=∣u∣p−1u in a class of Riemannian models (M,g) of dimension n≥3 which includes the classical hyperbolic space Hn as well as manifolds with sectional curvatures unbounded below. Sign properties…
The study examines stability of triharmonic hypersurfaces in space forms.
problem Stability of triharmonic hypersurfaces in space forms.
method Derivation of general stability statements, focus on specific cases of constant mean curvature in Euclidean and hyperbolic spaces, and analysis of small proper triharmonic hyperspheres and Clifford tori.
result Triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations, while in hyperbolic space they are stable.
Study on triharmonic curves in f-Kenmotsu manifolds.
problem Characterizing triharmonic curves in f-Kenmotsu manifolds.
method Investigation of necessary and sufficient conditions for Frenet curves, slant, and Legendre curves to be triharmonic. Proof of specific properties of triharmonic Frenet curves.
result Triharmonic Frenet curves with constant curvature are Frenet helices in three dimensional f-Kenmotsu manifolds.
Study on triharmonic curves in 3D spaces, proving their existence and classification.
problem Characterizing triharmonic curves in 3D homogeneous spaces.
method Analyzing curves with constant curvature in Riemannian manifolds, focusing on Frenet helices and space forms.
result Classification of triharmonic Frenet helices in space forms and Bianchi-Cartan-Vranceanu spaces.
Study on triharmonic curves in Sol space with constant curvature and torsion.
problem Characterizing triharmonic curves in the Sol space.
method Complete classification of proper triharmonic curves with constant geodesic curvature and torsion.
result Triharmonic curves form a constant angle with a Killing field of constant length.
The paper studies triharmonic hypersurfaces in space forms and proves their properties.
problem Characterizing triharmonic hypersurfaces in different space forms.
method Analyzing hypersurfaces in spheres, hyperbolic spaces, and Euclidean spaces using CMC (constant mean curvature) and triharmonic properties.
result Properties of triharmonic hypersurfaces in Euclidean space, including minimality.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. The paper proves properties of triharmonic CMC hypersurfaces with specific curvature conditions.
problem Characterizing triharmonic CMC hypersurfaces with distinct principal curvatures.
method Analyzing critical points of the tri-energy and applying geometric properties.
result Proves conditions for constant scalar curvature and minimality of hypersurfaces.
Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.
problem Existence and qualitative properties of radial solutions on Cartan-Hadamard manifolds.
method Analytical and asymptotic analysis of radial solutions, focusing on critical and supercritical exponents.
result Existence of one-parameter family of radial solutions for critical or supercritical exponents, with different dimensions of existence regions based on stochastic completeness.
Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are fin…
Derives stress-energy tensor for polyharmonic maps.
problem Characterizing polyharmonic maps between Riemannian manifolds.
method Derives stress-energy tensor and uses it to characterize polyharmonic maps.
result Characterizes polyharmonic maps, focusing on triharmonic maps.
The paper proves properties of triharmonic CMC hypersurfaces with limited curvature types.
problem Characterizing triharmonic CMC hypersurfaces with specific curvature constraints.
method Analyzing critical points of the triharmonic energy and applying geometric inequalities.
result Proves constant scalar curvature for triharmonic CMC hypersurfaces with at most 3 distinct principal curvatures.
Paper answers part of a conjecture about submersions from 3D spaces.
problem Generalized Chen's conjecture for triharmonic Riemannian submersions.
method Analyzes submersions from 3D space forms into surfaces.
result Triharmonic submersions are harmonic.
Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where 0<s≤2, p>1, m≥1, u=(ui)i=1m and ui:Rn→R. The qualitative behavior of solutions of…
Study classifies polyharmonic helices in various space forms.
problem Classifying polyharmonic helices in different space forms.
method Derived classification results for polyharmonic helices in space forms.
result Polyharmonic helices of arbitrary order in space forms of negative curvature are geodesics.
Develops a framework for generating harmonic maps from a unit ball to a sphere.
problem Creating families of harmonic maps from a unit ball to a sphere.
method Based on Toth's machinery for generating eigenmaps, combined with a generalized radial projection.
result Provides theoretical background for constructing solutions to variational problems.
The paper proves existence and instability of weak r-harmonic maps.
problem Existence and stability of weak r-harmonic maps. method Construction of critical points and analysis of stability.
result Existence and instability of weak r-harmonic maps restricted to specific dimensions. Equation embeddings learn from surrounding words to represent math equations.
problem Math equations are hard to analyze due to their uniqueness.
method Equation embeddings use surrounding words to find good representations of equations.
result Equation embeddings provide better models than existing word embedding approaches.
Paper finds new equations for pseudospherical surfaces with isometric immersions.
problem Identifying equations with isometric immersions for pseudospherical surfaces.
method Provided families of second order non-linear PDEs with local isometric immersions in E^3.
result Found equations with principal curvatures depending on finite-order jets of solutions.
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.
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 studies curvature equations and their solvability.
problem Solving curvature type equations and their Dirichlet problems.
method General class of fully nonlinear curvature equations, Christoffel-Minkowski problem, degenerate equations.
result Solvability of curvature type equations and Dirichlet problems.
Bonnet surfaces' equations link to Painlevé sixth equation.
problem Understanding Bonnet surfaces and Painlevé equations.
method Moving frame equations of Bonnet surfaces and Lax pair construction.
result Moving frame equations of Bonnet surfaces lead to a Lax pair for the sixth Painlevé equation.
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.
New methods solve complex equations, proving some have solutions.
problem Solving complex equations in infinite dimensions.
method Infinite-dimensional prequantum line bundles and moment maps.
result Proves some perturbed equations have solutions for small parameters.
New periodic solutions found for a specific equation.
problem Finding solutions to a specific partial differential equation.
method Using Bäcklund transformation for the sine-Gordon equation to generate solutions.
result New periodic exact solutions of the constant astigmatism equation.
Deep reinforcement learning solves complex differential equations.
problem Solving nonlinear differential equations.
method Rule-based deep reinforcement learning approach.
result Solver captures intrinsic nature of equations with high accuracy.
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.
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.
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…
The paper generalizes the Bott-Virasoro group and derives new Euler equations.
problem Understanding the generalized Bott-Virasoro group and its dynamics.
method Generalizing the Bott-Virasoro group using connection cochain and deriving Euler equations.
result New Euler equations derived from the generalized Bott-Virasoro group.
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 stochastic equations in Banach spaces, applying to HJM equation.
problem Existence and uniqueness of solutions to stochastic evolution equations in Banach spaces.
method Proving existence and uniqueness of solutions to stochastic evolution equations in martingale-type 2 Banach spaces.
result Existence and uniqueness of solutions to the Heath-Jarrow-Morton-Musiela equation in weighted Lebesgue and Sobolev spaces.
Study finds conservation laws for a specific class of parabolic equations.
problem Existence and structure of conservation laws for evolutionary scalar second-order differential equations.
method Calculation of linearized characteristic cohomology to find conservation laws, showing dependence on second derivatives.
result Only Monge-Ampère type equations have non-trivial conservation laws.
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.
Sharp rates and symmetry for higher order conformally invariant equations near singularities.
problem Understanding solutions near isolated singularities for higher order conformally invariant equations.
method Blow-up analysis for local integral equations, Fowler solutions, Harnack inequality.
result Sharp blow-up rates and asymptotic radial symmetry of solutions near singularities.
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…