Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.
problem Proving finite ends and linear energy growth for solutions to the Allen-Cahn equation.
method Curvature decay estimate on level sets, indirect blow-up technique, Toda system analysis.
result Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.
Finite-gap solutions approximate jets of initial data for certain BKM systems.
problem Approximating jets of initial data for specific PDE systems.
method Using finite-reduction map to finite-gap solutions of Stäckel systems.
result Full jet-surjectivity for KdV and Kaup--Boussinesq, partial for Camassa--Holm.
Finite index solutions to Bernoulli problem are always axially symmetric.
problem Entire solutions to the Bernoulli free boundary problem with finite Morse index in 3D.
method Proof of axial symmetry for finite index solutions.
result Finite index solutions to the Bernoulli problem in 3D are axially symmetric.
Ancient solutions to curve shortening with finite total curvature created by gluing Grim Reapers.
problem Creating ancient solutions to curve shortening with finite total curvature.
method Constructing ancient solutions by gluing Grim Reapers along their asymptotes.
result Ancient solutions to curve shortening with finite total curvature can be created.
Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.
problem Characterizing ancient solutions on an infinite strip with polynomial and exponential growth.
method Analyzing parabolic equations on an infinite strip, proving properties of ancient solutions.
result Ancient solutions on the strip are constant if they grow polynomially, and have a finite-dimensional space for slower exponential growth.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Researchers study ancient and finite solutions of Laplacian coflow and modified coflow on a 7D Heisenberg group.
problem Analyzing the behavior of G2-structures under Laplacian and modified Laplacian coflow on a 7D Heisenberg group. method Examining the ancient, finite, and eternal solutions of the Laplacian coflow and modified coflow.
result Ancient solutions for Laplacian coflow and finite, ancient, or eternal solutions for modified coflow on the 7D Heisenberg group.
We present a 1-parameter family of finite action solutions to the S0(2,1) Hitchin's equations and explore some of its basic properties. For a fixed value of the parameter, the solution is smooth. We conclude by showing a multi-particle generalization of our basic solutions.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
Local existence and uniqueness of Bakry-Émery Ricci flow solutions on finite graphs.
problem Analyzing the behavior of Ricci flow on finite graphs.
method Local existence and uniqueness proof for solutions of the Bakry-Émery Ricci flow.
result Local existence and uniqueness of solutions to the Ricci flow on finite graphs.
Paper solves convertible bond valuation using finite elements with penalty method.
problem Valuation of convertible bonds under penalty TF model.
method Solves TF system of equations using P1 and P2 finite elements with penalty method.
result Numerical solutions compare favorably with finite difference method.
Estimates for VPMCF show ancient MCF solutions and finite-time behavior.
problem Volume Preserving Mean Curvature Flow (VPMCF) behavior and singularities.
method Nonlocal estimates and blowup analysis.
result Ancient solutions to MCF and finite-time behavior of VPMCF.
Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.
problem Characterizing the qualitative behavior of 4-harmonic and ES-4-harmonic maps.
method Proving triviality of finite energy solutions for both maps.
result Finite energy solutions of both 4-harmonic and ES-4-harmonic maps are trivial.
The paper solves integrable systems of PDEs, including famous equations.
problem Constructing solutions for multicomponent integrable PDEs.
method Reduction to a finite-dimensional system, using Nijenhuis geometry.
result Animations of multi-component soliton and cnoidal solutions.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
Study finite curvature solutions on surfaces with nonnegative Gauss curvature.
problem Finite total curvature solutions of Liouville equation on surfaces with nonnegative Gauss curvature.
method Analyzes asymptotic behavior of solutions on complete surfaces.
result Two extremal cases identified: Euclidean plane or flat cylinder, with specific decay conditions.
We consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numeric…
Study smooth solutions to fractional mean curvature flow, proving uniqueness and finite extinction time.
problem Understanding evolution of surfaces with fractional mean curvature.
method Established a comparison principle and evolutions equations for fractional geometric quantities.
result Proved uniqueness and finite extinction time for compact solutions.
Finite energy solutions classified for Seiberg-Witten equations on complex plane and Riemann surface.
problem Classifying solutions to Seiberg-Witten equations with finite energy.
method Established a classification theorem for solutions on X=CimesΣ with finite analytic energy. result Finite energy solutions correspond to polynomial maps from C to H0(Σ,L+,∂ˉ). Ancient solutions to Ricci flow in 3D are mostly cylinders or solitons.
problem Understanding finite-time singularities in Ricci flow on compact 3-manifolds.
method Proved every noncompact ancient κ-solution in 3D is isometric to specific models.
result Ancient κ-solutions in 3D are either shrinking cylinders or the Bryant soliton.
The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.
problem Understanding the singularities and behavior of the Kähler-Ricci flow.
method Li-Yau type and Harnack estimates for weighted Ricci potential functions.
result Finite time singularities are shown to sub-converge to ancient solutions on analytic normal varieties.
In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …
In this note, we first prove that the solution of mean curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval [0,T) …
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
problem Solving the Pohlmeyer--Lund--Regge equation and understanding Lund--Regge curve evolution.
method Finite-gap construction using hyperelliptic spectral data, Baker--Akhiezer function, and SU(2)-frame. result Explicit theta-quotient formula for PLR solutions and criteria for Lund--Regge curve evolution.
Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
problem Finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
method Determine rationality criteria for Durham conditions and analyze spectral curve properties.
result Rationality criteria are sufficient for finite-type solutions with complementary boundary conditions.
The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An i…
Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.
problem Existence and behavior of networks under anisotropic curvature flow.
method Existence, uniqueness, and regularity of maximal geometric solutions proven.
result Existence of maximal geometric solutions and behavior under finite time.
We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.
The study explores conformal planes with finite areas.
problem Geometry of conformal planes with finite areas.
method Analyzes several questions about conformal planes.
result Exploration of conformal planes with finite areas.
New study confirms some mean curvature flow solutions have bounded mean curvature.
problem Existence of mean curvature flow singularities with bounded mean curvature.
method Construction of specific solutions in RN for N≥8. result A nontrivial subset of solutions has uniformly bounded mean curvature.
Classifies self-similar solutions for heat equations with positive speed.
problem Classifying self-similar solutions for semilinear heat equations.
method Analyzes the semilinear heat equation ut=Δu+∣u∣p−1u for p>1. result Finite time blowing up solutions converge to a positive constant after rescaling.
Researchers create finite time blow-up solutions for harmonic map flow into S^2.
problem Constructing finite time blow-up solutions for harmonic map flow into S^2.
method Constructing finite time blow-up solutions using asymptotically singular scaling and reverse bubbling.
result Finite time blow-up solutions constructed precisely at given points in the domain.
For word-equations in groups, we find a logarithmic bound on non-solutions.
problem Finding the length of non-solutions to word-equations in groups.
method Analyzing finite-rank free groups and applying results to broader classes of groups.
result Logarithmic bound on non-solutions for word-equations in groups.
We use the solution set of a real ordinary differential equation which has order n which is at least 2 to construct a smooth curve C in R^n. We describe when C is a proper embedding of infinite length with finite total first curvature.
Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar cur…
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn, which imply asymptotic behavior of the solutions at i…
Study on solutions of degenerate equations on manifolds, linking behavior to geometry and decay rates.
problem Behavior of solutions to degenerate parabolic equations on manifolds with inhomogeneous density.
method Analysis of Cauchy problem on Riemannian manifolds, considering weight function as capacitary coefficient.
result Estimates of vanishing rate and finite speed of propagation in subcritical ranges, universal bounds and blow-up in supercritical ranges.
Let H=Δ+V be a Schrödinger on a complete non-compact manifold. It is known since the work of Fischer-Colbrie and Schoen that the finiteness of the negative spectrum of H implies the existence of a function φ solution of Hφ=0 outside a compact set. This has consequences for minimal surfaces and for the finitenes…
The paper analyzes the asymptotic sequential Rademacher complexity for finite function classes.
problem Understanding the complexity of finite function classes in asymptotic settings.
method Using viscosity solutions of a G-heat equation and sublinear expectation theory, the paper derives the asymptotic sequential Rademacher complexity.
result The asymptotic sequential Rademacher complexity is expressed in terms of the viscosity solution of a G-heat equation and the expected value of the largest order statistics of a multidimensional G-normal random variable.
We solve a linear equation on affine manifolds, finding finite-dimensional solutions.
problem Solving a linear equation on affine manifolds.
method Proving the space of solutions is finite-dimensional and characterizing the maximal dimension.
result The maximal dimension of solutions is achieved only on strongly projectively flat manifolds.
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.
We discuss intrinsic aspects of Krupka's approach to finite-order variational sequences. We give intrinsic isomorphisms of the quotient subsheaves of the short finite-order variational sequence with sheaves of forms on jet spaces of suitable order, obtaining a new finite-order (short exact) variational sequence which i…
The paper studies harmonic map flow and its singularities, proving global solutions and no-loss-of-topology.
problem Analyzing and resolving singularities in harmonic map flow.
method Analysis of finite-time singularities and a canonical way to flow beyond them.
result Proves global solutions and no-loss-of-topology for arbitrary maps.
Develops theory linking Schrödinger equations to manifold ends, proving finiteness.
problem Understanding the number of ends in Riemannian manifolds.
method Variant of Li-Tam theory, polynomial growth analysis, Sobolev inequality.
result Finiteness of manifold ends under scaling invariant Sobolev inequality.
The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.
problem Existence of finite-energy solutions to a singular elliptic equation on Riemannian manifolds.
method ε-regularized problems, mountain pass arguments, and limiting procedures.
result Existence of nonnegative finite-energy supersolutions under certain conditions.
We consider minimal surfaces M which are complete, embedded and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation Δu+f(u)=0inR3. Here f=−W′ with W bistable and balanced, for instance W(u)=41(1−u2)2. We assume that …
Transforms solutions of Davey-Stewartson II equation geometrically.
problem Solving the Davey-Stewartson II equation.
method Moutard transform and spinor representation of surfaces.
result Constructs examples of solutions with smooth initial data losing regularity.
The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that th…