The paper classifies solutions to semilinear equations on curved spaces.
problem Classifying solutions to semilinear equations on manifolds with nonnegative Ricci curvature.
method Proving classification results for subcritical and critical semilinear elliptic equations.
result Strong rigidity results for nontrivial solutions in the critical case.
Trivial solution proof for heat equation on certain manifolds.
problem Proving trivial solutions for semilinear heat equations on specific manifolds.
method Analyzing pointwise monotonicity and boundedness over time.
result Trivial solutions exist only for certain values of p.
The paper classifies solutions to a specific elliptic equation in the Heisenberg group.
problem Classifying positive solutions to a critical semilinear elliptic equation in the Heisenberg group.
method Proof based on Jerison-Lee's differential identity and pointwise/integral estimates.
result The solutions are the Jerison-Lee's bubbles in the Heisenberg group.
Study semilinear equations on weighted manifolds to prove rigidity.
problem Prove rigidity of weighted manifolds via classification of semilinear equations.
method Classify positive solutions at the Sobolev-critical exponent, proving rigidity and weight triviality.
result Existence of positive solutions implies rigidity and weight triviality under certain curvature conditions.
Solves a 30-year-old problem on singular solutions for Yamabe equations.
problem Constructing singular solutions to semilinear equations without phase-plane analysis.
method Careful gluing in weighted L∞ spaces, exploiting semilinearity and stability of linearized operator. result Provides an alternative construction for the Yamabe problem with maximal singular set.
In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold M2m. Such objects satisfy the elliptic system weakly [J,ΔmJ]=0. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…
Paper proves convex domains have one maximum for semi-stable solutions.
problem Analyzing critical points of semi-stable solutions on convex domains.
method Relating critical points to an auxiliary function and using topological degree.
result Positive, semi-stable solutions have exactly one non-degenerate critical point.
The article derives gradient estimations for semilinear equations on geometric flows.
problem Gradient estimation for semilinear equations on geometric flows.
method Derives both Hamilton and Souplet-Zhang type gradient estimations.
result Gradient estimations for semilinear equations on geometric flows.
Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.
problem Determining an unknown function in a semilinear elliptic equation on complex manifolds.
method Analyzing higher order linearizations and interactions of Gaussian quasimode solutions.
result An unknown smooth function can be uniquely determined from the Dirichlet-to-Neumann map.
Derives matrix Harnack inequalities for semilinear heat equations on manifolds.
problem Bounding solutions of semilinear heat equations on manifolds with geometric constraints.
method Applies Li-Yau estimates to derive Harnack inequalities for positive solutions.
result Derives matrix Harnack inequalities for positive solutions of semilinear heat equations.
The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.
problem Analyzing semilinear elliptic equations and inequalities on pseudo-Hermitian manifolds.
method Using a generalized Jerison-Lee's formula and volume estimates.
result Established Liouville theorems and nonexistence results for specific equations and inequalities.
Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.
problem Proving a Liouville theorem for a generalized elliptic equation on H-type groups.
method Proof based on an a priori integral estimate and a generalized differential identity.
result Obtained a Liouville type theorem for the semilinear subcritical elliptic equation on H-type groups.
Derives Li & Yau estimates for heat equations on manifolds.
problem Analyzing positive solutions of semilinear heat equations on manifolds.
method Adapts Li & Yau estimates to derive new inequalities.
result Derives Harnack inequality and discusses monotonicity, convexity, decay estimates.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Sharp conditions found for solving heat equation on Riemannian manifolds.
problem Solving semilinear heat equation on Riemannian manifolds.
method Sharp conditions derived for local-in-time solvability.
result Sharp conditions on solvability given for complete and connected manifolds.
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
Paper proves no nontrivial solutions to certain elliptic equations on graphs.
problem Proving nonexistence of solutions to semilinear elliptic equations on metric graphs.
method Constructed a modified distance function and introduced test functions to show nonexistence under volume growth conditions.
result No nontrivial solutions exist for the equations under suitable conditions.
Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.
problem Analyzing positive solutions of semilinear Dirichlet problems on Riemannian manifolds with Ricci lower bound.
method Sharp pointwise gradient comparison method, derived from admissibility and structural conditions on f.
result Explicit isoperimetric-type inequality and quantitative hot-spot localization estimate.
We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.
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.
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
Study proves energy critical heat equation solutions are Type I blowups for n ≥ 7.
problem Analyzing blowup behavior of energy critical nonlinear heat equations.
method Reverse inner-outer gluing mechanism and bubbling behavior analysis.
result Proves all blowups are of Type I for n ≥ 7.
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean …
Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs
problem Nonexistence of solutions to semilinear parabolic and hyperbolic inequalities on metric graphs
method Construction of a new pseudo-metric and space-time test functions
result All solutions must be identically zero
We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry…
We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there …
New approach simplifies proof of wave equations on black holes.
problem Global existence and decay for semilinear wave equations on extremal Reissner-Nordström black holes.
method Develops a new approach based on weaker estimates, avoiding near-horizon sharp estimates.
result Simpler and more streamlined proof without requiring near-horizon sharp estimates.
Paper introduces a new method to solve complex PDEs efficiently.
problem Solving high-dimensional semilinear PDEs and BSDEs.
method Decomposes PDEs into linear and nonlinear parts, uses Deep BSDE solver with control variate method.
result Errors of the new method are much smaller than those of the original Deep BSDE solver.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.
Assume that f(s)=F′(s) where F is a double-well potential. Under certain conditions on the Lipschitz constant of f on [−1,1], we prove that arbitrary bounded global solutions of the semilinear equation Δu=f(u) on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…
We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…
Study on ground states of semilinear elliptic equations with various potential wells.
problem Characterizing ground states of semilinear elliptic equations with arbitrary potential wells.
method Analyzing solutions in convex domains and manifolds with non-negative Ricci curvature, using Morse theory and min-max methods.
result Ground states are mountain-pass type with Morse index 1 in convex domains and manifolds with non-negative Ricci curvature.
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…
Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional subm…
In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the γk functions on the space of its hermitian metrics.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.
The goal of this paper is to clarify when a semilinear stochastic partial differential equation driven by Lévy processes admits an affine realization. Our results are accompanied by several examples arising in natural sciences and economics.
Let u1 and u2 be two different positive smooth solutions of the equation Δu+n(n−2)un−2n+2=0 in Rn(n≥3). By a result of Gidas, Ni and Nirenberg, u1 and u2 are radially symmetric above the points ξ1 and ξ2, respectively. Let u be a positive C2-function on Rn su…
Paper solves portfolio problem using improved stochastic methods.
problem Finite horizon consumption-investment problem under stochastic factor framework.
method Proves existence of classical solution for semilinear equation using gradient estimates.
result Proves existence of classical solution and provides all necessary estimates.
Study asymptotically almost periodic solutions on real hyperbolic manifolds.
problem Existence and asymptotic behavior of solutions to parabolic equations.
method Dispersion and smoothing estimates, fixed point argument.
result Existence and uniqueness of asymptotically almost periodic solutions.
New method solves complex curvature equations.
problem Solving semilinear scalar curvature equations.
method Mixed convex integration method.
result New proof of scalar curvature result.
In this paper, we construct a Rabinowitz-Floer type homology for a class of non-linear problems having a \emph{starshaped} potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model…
We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…