arXiv research

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

Note on advancements in nonlinear elliptic equations' regularity theory.

Study solves inverse problems for equations with fractional nonlinearities.

The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…

This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in [6] does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of…

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

Study solves equations on tori for Calabi-Yau problems.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…

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

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

Solves open problems for fully nonlinear elliptic equations on manifolds.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

Paper establishes estimates for nonlinear equations on compact manifolds.

Study improves understanding of solutions to complex equations in geometry.

A gauge-invariant form of the nonlinear Hodge equations is studied.

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.

Solves geometric problems using fully nonlinear equations and Morse theory.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

The paper concerns singular solutions of nonlinear elliptic equations.

The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.

We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a no…

The paper derives subgradient estimates for a specific nonlinear subparabolic equation on pseudo-Hermitian manifolds.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…

Study on maximum principles for nonlinear equations on Riemannian manifolds.

Solves nonlinear problems on metric structures through eigenvalue counting.

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

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 $f$-…

This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models …

Study on a nonlinear elliptic equation on compact Hermitian manifolds.

The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the…

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…

We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the …

Introduces a new PDE involving differential forms for Kähler geometry.

Estimates for complex equations on manifolds derived from a conjecture.

We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.

Study fully nonlinear elliptic equations on complex manifolds.

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…

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

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.

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie alge…

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.