arXiv research

Concave elliptic operators yield concave functions on cohomology.

Established concavity principle for curved spaces.

This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.

New method for analyzing elliptic and parabolic equations.

Study concavity of solutions to elliptic equations under conformal deformations.

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is o…

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δφ(u)$, $φ$ being an ar…

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

Study estimates eigenvalues for concave Hessian operators on convex domains.

The Anomaly flow is shown to converge on toric fibrations with the Fu-Yau ansatz, for both positive and negative values of the slope parameter $α'$. This implies both results of Fu and Yau on the existence of solutions for Hull-Strominger systems, which they proved using different methods depending on the sign of $α'$.…

By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Schördinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the f…

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we…

Derives concavity inequality and estimates for $k$-Hessian equations.

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2 μ}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $μ$ is a real parame…

Paper derives estimates for Hessian equations under concavity assumptions.

Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.

The paper extends risk measures to two-step approximations and studies log-concave distributions.

Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.

The paper proves the concavity of $p$-entropy power and applies it to functional inequalities.

The paper examines ellipticity of specific equations on vector bundles.

Note on advancements in nonlinear elliptic equations' regularity theory.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

New solutions found for elliptic sinh-Gordon and sine-Gordon equations.

Derives estimates for geometric elliptic equations on complex manifolds.

Study shows solutions to certain equations form smooth manifolds.

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

Study fully nonlinear elliptic equations on complex manifolds.

Deep neural nets solve complex insurance math equations.

Simple connection between Harnack inequalities and concavity of arrival time functions.

The paper proves the concavity of entropy power for diffusion equations and applies it to new inequalities.

Solves open problems for fully nonlinear elliptic equations on manifolds.

Solves Gursky-Streets equations for $σ_k$ Yamabe problem in dimensions $n \geq 2k$.

Establishes a concavity property for positive Hessian quotient operators.

Estimates for complex Hessian equations on Hermitian manifolds.

Study improves understanding of solutions to complex equations in geometry.

We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in different areas of geometry and analysis.

Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.

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

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

Solves a specific Dirichlet problem on Riemannian manifolds.

The paper concerns singular solutions of nonlinear elliptic equations.

New method solves elliptic equations on manifolds without grids.

Mathematical formulas for elliptic curve integrals solve anomaly equations.

Uniqueness found for elliptic equations with drift on manifolds.

Proves regularity for quasilinear elliptic equations in metric spaces.