arXiv research

The paper estimates gradients for a weighted parabolic equation under geometric flow.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α…

The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

The article derives gradient estimations for semilinear equations on geometric flows.

The paper studies solutions to a nonlinear equation on Finsler manifolds with gradient estimates and Harnack inequalities.

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at infinity. We connect the behavior of non-negative solutions to the interplay betwe…

Study of limiting configurations for SU(1,2) Hitchin equation solutions.

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green's function vanishing at infinity. On…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy prob…

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of …

The paper studies heat kernels on weighted Riemannian manifolds with curvature bounds.

Motivated by applications to probability and mathematical finance, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half-space. We establish e…

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservatio…

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the proces…

Sharp estimates for parabolic equations on manifolds using symmetrization.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.

In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-…

New method for analyzing elliptic and parabolic equations.

Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.

Unified diffusive bounds for non-linear parabolic equations.

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

Using an algebraic Fourier transform of operators, we develop a method (F-method) to obtain explicit highest weight vectors in the branching laws by differential equations. This article gives a brief explanation of the F-method and its applications to a concrete construction of some natural equivariant operators that a…

Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Extends parabolic study to flat hyperkähler manifolds.

In this paper, we extend a technique due to Romero, Rubio and Salamanca establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of …

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exha…

Study proves long-term solutions to a specific equation on hyperKähler manifolds.