arXiv research

Deep neural nets solve complex insurance math equations.

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

The paper provides estimates for eigenvalues of elliptic differential problems.

Study shows solutions to certain equations form smooth manifolds.

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.

The paper examines ellipticity of specific equations on vector bundles.

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

Paper classifies minimal graph transformations into new families of surfaces.

We present some applications of ideas from partial differential equations and differential geometry to the study of difference equations on infinite graphs. All operators that we consider are examples of "elliptic operators" as defined by Y. Colin de Verdiere. For such operators, we discuss analogs of inequalities of C…

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…

The paper develops Morse homology for a class of elliptic partial differential equations.

In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefor…

Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.

The paper classifies solutions to a specific elliptic equation in the Heisenberg group.

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerat…

New Fueter sections solve monopole equations for 3/2-spinors.

Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine para…

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

The paper classifies invariant operators and proves a Liouville theorem.

The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approa…

The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.

Introduces a new PDE involving differential forms for Kähler geometry.

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,μ}$ multivalued solutions to the minimal surface equation, to produce $C^{1,μ}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations …

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nab…

New theorems prove uniqueness of solutions to geometric PDEs.

Solves division problem for L. Hörmander's systems.

Perimeter on manifolds leads to new symmetrization methods.

Survey of geometry developments, including complex structures on surfaces.

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_Ωφ\left( \left\vert \nabla u\right\vert \right) , \] where $Ω$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Ω=\mathcal{M}$ is a C…

Uniform estimates for complex equations on compact manifolds found.

Rust library solves complex equations on abstract simplicial complexes.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations whi…

We present a somewhat new proof to the $C^{2,α}$-aprori estimate for the uniform elliptic Monge-Ampere equations, in both the real and complex settings. Our estimates do not need to differentiate the equation, and only depends on the $C^{α^{\prime}}-$norm of the right hand side of the equation, $0<α<α^{\prime}$.

Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.

Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $\frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of M…

Develops a new non-abelian framework for Riemann surfaces and differential equations.

New method for analyzing elliptic and parabolic equations.

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary d…

Unified approach combining BSDEs and PINNs for solving PDEs.

Lie minimal surfaces are characterized by differential equations of principal curvatures.