Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Optimal Liouville theorem for half-Euclidean space equations.
problem Optimal Liouville-type theorems for conformally invariant equations.
method Established optimal Liouville-type theorems for conformally invariant second-order elliptic equations.
result Proved an optimal Liouville-type theorem for equations in the half-Euclidean space.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n-Laplacian Liouville equation on the half-space R+n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n=2 and p=n. The paper classifies invariant operators and proves a Liouville theorem.
problem Classifying invariant operators and proving Liouville theorem.
method Classified Möbius invariant differential operators and established a Liouville type theorem.
result Established a Liouville type theorem for Möbius invariant equations.
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for σk-curvature equations in half spaces with nonlinear boundary conditions. method Established using positive constant curvature equations and variational functional approach.
result Proved Liouville theorem for positive constant σk-curvature equations in R+n and boundary conditions. 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.
Liouville theorems extended to graphs with bounded geometry.
problem Ancient solutions of subexponential growth on graphs.
method Extended Mosconi's results to graphs with bounded geometry.
result Nonnegative ancient solutions are stationary and harmonic.
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
Unified treatment of spacelike and timelike minimal surfaces via Liouville equation.
problem Investigating minimal surfaces in Lorentz-Minkowski space.
method Complex and paracomplex analysis, Möbius-type transformations, pseudo-isometries.
result Unified approach to both spacelike and timelike minimal surfaces.
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
problem Gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
method Nonlinear Φ-Bochner formula and Nash-Moser iteration technique for gradient bounds; maximum principle for parabolic case.
result Unified framework for gradient estimates and Liouville theorems for Φ-Laplacian equations.
We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blo…
Gradient bounds and Liouville theorems for quasi-linear equations on manifolds with nonnegative Ricci curvature.
problem Establishing bounds and theorems for solutions to quasi-linear elliptic equations on compact manifolds with nonnegative Ricci curvature.
method Gradient bounds, Liouville-type theorems, local splitting theorem, Harnack-type inequality, ABP estimate.
result Gradient bounds and Liouville-type theorems for solutions to quasi-linear equations on compact manifolds with nonnegative Ricci curvature.
We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.
New Liouville-type results for CR Yamabe equation in Heisenberg group.
problem Characterizing solutions to CR Yamabe equation in Heisenberg group.
method Integral estimates combined with divergence formula.
result Liouville-type results for bounded solutions in n=2 and solutions with pointwise decay assumption in n≥3. The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.
problem The constant mean curvature one equation for surfaces in hyperbolic 3 space.
method Group theoretic constructions and equivalence of quotient representations.
result The Darboux integrability of the equations shows that they admit equivalent quotient representations.
Paper finds solutions to a complex equation on surfaces with boundary conditions.
problem Existence of solutions to a super-Liouville equation on compact Riemannian surfaces with boundary.
method Introduced a weighted Dirac operator and constructed a Nehari manifold to show existence of non-trivial solutions.
result Existence of non-trivial solutions to the super-Liouville equation.
The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.
problem Gradient estimates and Liouville theorems for positive solutions to a specific nonlinear elliptic equation.
method Analyzes the nonlinear elliptic equation \( \Delta_{V}u^{m} + \mu(x)u + p(x)u^{\alpha} = 0 \) on smooth metric measure spaces with bounded Bakry-Émery curvature.
result Establishes gradient estimates and related Liouville theorems and Harnack inequalities.
For n≥2, we obtain Liouville type theorems for minimal surface equations in half space R+n with affine Dirichlet boundary value or constant Neumann boundary value.
This paper proves Liouville theorems for conformally invariant fully nonlinear equations.
problem Positive entire solutions of certain fully nonlinear equations are unique.
method Derives necessary and sufficient conditions for Liouville-type theorems.
result Enhanced understanding of solutions near isolated singularities.
We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.
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.
Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
problem Quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.
method Prove rigidity and classification results for the quasilinear Liouville equation associated with the n-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. result Under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles.
We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…
Paper proves Liouville theorem for curvature equation with boundary conditions.
problem Proving Liouville theorem for curvature equation with boundary conditions.
method Using Chern--Gauss--Bonnet formula and boundary conditions.
result Extends Wei's earlier result by removing the infinity condition.
In this note, we derive a Liouville theorem for the complex Monge-Ampère equation. Our result states that if the global solution u of the complex Monge-Ampère equation with constant right-hand side differs from a quadratic polynomial solution by $o(\abs{x}^2)$ at infinity, then u is a quadratic polynomial.
In this paper we study the Kato' inequality on locally finite graph. We also study the application of Kato inequality to Ginzburg-Landau equations on such graphs. Interesting properties of Schrodinger equation and a Liouville type theorem are also derived.
Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
Suggestions concerning the generalization of the geometric quantization to the case of nonlinear field theories are given. Results for the Liouville field theory are presented.
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we …
In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge-Ampère equation on product manifolds.
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…
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 establish Liouville type theorems for degenerate conformally invariant equations.
Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is …
Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
In this paper, we obtain a necessary and sufficient condition for L∞-uniqueness of Sturm-Liouville operator a(x)dx2d2+b(x)dxd−V on an open interval of $\rr$, which is equivalent to the L1-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator $\LL^V:=Δ…
Paper proves Liouville-type theorems for minimal graphs with capillary boundary.
problem Proves conditions for minimal graphs to be flat over half-spaces with capillary boundaries.
method Uses gradient estimates for mean curvature equation over R+n with capillary boundary condition, adapting maximum principle. result Minimal graphs are flat under specific conditions on growth or boundedness.
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.
Paper derives estimates for Hessian equations under concavity assumptions.
problem Estimating solutions to Hessian equations with concavity constraints.
method Exploiting concavity of sums of Hessian operators, deriving Pogorelov estimates.
result Entire admissible solutions in R^n with quadratic growth are quadratic polynomials.
Unified method improves gradient estimates for a nonlinear elliptic equation on Riemannian manifolds.
problem Gradient estimates for positive solutions to a nonlinear elliptic equation on Riemannian manifolds.
method Unified method using elliptic equation analysis.
result Improves gradient estimates and supplements previous results for different cases of constants.
In this paper, we will address to the following parabolic equation ut=Δfu+F(u) on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here F is a differentiable function defined in R. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equ…
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
The paper establishes gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
problem Gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
method Yau and Souplet-Zhang type gradient estimates for harmonic and heat equation solutions under Dirichlet boundary condition.
result Established gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
Classifies singular solutions to Liouville equation with constant Q-curvature metrics.
problem Singular solutions to Liouville equation with constant Q-curvature metrics.
method Classification through behavior at singular points and existence of radial solutions.
result Existence of metrics with singularities of arbitrary order.