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 n 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.
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 p p -Laplace equation and show rigidity concerning the ambient manifold. The paper proves growth estimates for subsolutions of quasilinear equations.
problem Proving integral estimates on the minimal growth of subsolutions of quasilinear equations.
method Integral estimates and structural assumptions on the equation.
result Proves growth estimates for subsolutions of quasilinear equations.
Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.
problem Analyzing positive solutions to quasilinear elliptic equations on manifolds with bounded Ricci curvature.
method Employing Nash-Moser iteration technique to derive logarithmic gradient estimates and Liouville properties.
result Derives universal logarithmic gradient estimates for positive solutions under certain conditions.
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.
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.
Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.
problem Classifying Lie symmetry algebras for 2D quasilinear equations.
method Classification based on abelian Lie symmetry algebras of dimension and rank.
result Equations with specific symmetry algebras are linearizable.
The paper proves no positive weak solutions for a specific inequality on Riemannian manifolds.
problem Proving the nonexistence of positive weak solutions for a quasilinear inequality on Riemannian manifolds.
method Analyzing the volume growth of geodesic balls and applying Liouville-type theorems.
result The inequality has no positive weak solutions under certain volume growth conditions.
Sharp estimates derived for quasilinear equations on metric measure spaces.
problem Estimating solutions and eigenvalues of quasilinear equations on smooth metric measure spaces.
method Sharp estimates derived using comparisons with one-dimensional equations.
result Optimal lower bounds for the first Dirichlet eigenvalue of quasilinear operators.
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…
Proves regularity for quasilinear elliptic equations in metric spaces.
problem Regularity of quasilinear elliptic equations in metric measure spaces.
method Galerkin's method as an alternative to difference quotients.
result Second-order and Lipschitz regularity for a wide class of elliptic equations.
Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we fi…
Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.
problem Finding eigenvalues and eigenfunctions for quasilinear equations on Riemannian manifolds.
method Generalized Cheng--Yau gradient estimate for quasilinear equations on complete Riemannian manifolds.
result Eigenvalues give rise to unbounded eigenfunctions under certain conditions.
Estimates for solutions on manifolds under Ricci flow.
problem Gradient estimates for solutions on manifolds.
method Two-point function estimates for quasilinear parabolic equations under Ricci flow.
result Gradient estimates for solutions at any two points related to the distance between points.
The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold M M M with boundary ∂ M \partial M ∂ M . We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on M M M . We propose …
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.
We show, by modifying Borbély's example, that there are 3 3 3 -dimen\-sional Cartan-Hadamard manifolds M M M , with sectional curvatures ≤ − 1 \le -1 ≤ − 1 , such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces H s H^s H s with s > 2 + 2 − 3 2 s>2+\frac{2-\sqrt{3}}{2} s > 2 + 2 2 − 3 for general quasi-linear wave equation in R 1 + 3 {\mathbb R}^{1+3} R 1 + 3 by Klainerman and Rodnianski. Via this approach they obtained the l…
We study the following quasilinear elliptic system for all i = 1 , ⋯ , m i=1,\cdots,m i = 1 , ⋯ , m \begin{equation*} \label{} -div(Φ'(|\nabla u_i|^2) \nabla u_i) = H_i(u) \quad \text{in} \ \ \mathbb{R}^n \end{equation*} where u = ( u i ) i = 1 m : R n → R m u=(u_i)_{i=1}^m: \mathbb R^n\to \mathbb R^m u = ( u i ) i = 1 m : R n → R m and the nonlinearity H i ( u ) ∈ C 1 ( R m ) → R H_i(u) \in C^1(\mathbb R^m)\to \mathbb R H i ( u ) ∈ C 1 ( R m ) → R is a gen…
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 n n -Laplacian Liouville equation on the half-space R + n \mathbb{R}^{n}_{+} R + n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n = 2 n=2 n = 2 and p = n p=n 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.
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
The paper proves Liouville rigidity for Hessian equations, characterizing geometric conditions for constant solutions.
problem Characterizing geometric conditions for constant solutions in Hessian equations.
method Recursive geometric condition (Liouville admissibility) and anisotropic constructions.
result The Liouville-type property is characterized as a geometric property of the admissible set.
Study proves Liouville theorem for specific curvature equations with boundary conditions.
problem Proving Liouville theorem for σ k σ_k σ k -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 σ k σ_k σ k -curvature equations in R + n \mathbb{R}_{+}^{n} 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.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
Study proves global existence and decay for complex wave equations.
problem Global existence and decay for quasilinear wave equations with weak-null condition.
method Novel decoupling of higher order energy estimates, focusing on tangential components.
result Established global existence and decay for solutions with small data.
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.
We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order con…
Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation − Δ u = e u -Δu = e^u − Δ u = e u on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth. result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
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 K K . The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.
problem Proving gradient estimates for solutions of nonlinear equations on Riemannian manifolds.
method Employing Nash-Moser iteration technique to establish gradient estimates.
result Gradient estimates for solutions of nonlinear equations on Riemannian manifolds are proven.
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…
We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets o…
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.
Global existence and boundedness proved for quasilinear wave equations on Kerr black holes.
problem Global existence and boundedness for quasilinear wave equations on Kerr black holes.
method Combines linear inhomogeneous estimates on Kerr backgrounds and tailored physical space currents.
result Global existence, boundedness and decay for small data solutions to quasilinear wave equations on Kerr black holes.
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 n=2 n = 2 and solutions with pointwise decay assumption in n ≥ 3 n\ge3 n ≥ 3 . We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.
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.
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
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.