arXiv research

Solves second-order PDEs using quotients and differential invariants.

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^…

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimens…

Revisits geometric PDE uniqueness in Riemannian and CR geometry.

Researchers find unique metrics solving complex PDEs for constant scalar curvature.

We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf fo…

This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems about flatness introduce a different point of view in Differential Geometry. The…

In this paper we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on 1 function of 1 variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal th…

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…

For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in…

In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had "a strong reason to believe" that there must exist a well-defined class of "not genuinely nonlinear" nonlinear PDEs. In 1978 G. Boillat coined the term "completely exceptional" to denote it. In the case of $2…

Study on 4D PDEs with half-flat conformal structure leading to Monge-Ampere equations.

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.

In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m\ne -1$) of scalar flag curvature must be locally Minkowskian in dimension $n\ge 3$. We characterize by some PDEs a Kropina metric ($…

Study proves unique compactification of hyperbolic space.

Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30…

We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation.

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic gro…

The paper introduces toric separable geometries and finds new extremal metrics.

The paper quantifies how scalar curvature changes under $C^0$ convergence in 3D.

Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.

We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Ree…

Protocol diagnoses neural HJB-PIDE solvers for Lévy jumps, revealing a missing factor in their importance-proposal density.

In \cite{LZ2} it is proved that for certain class of perturbations of the hyperbolic equation $u_t=f(u) u_x$, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations posse…

In this paper we prove a version of Lie-Bäcklund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the plane treated by E.Cartan in 1910. Many examples are provided.

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

The paper develops a new approach to solve vector-valued PDEs on manifolds with minimal regularity.

Classifies scalar second-order PDEs with low-dimensional symmetry groups.

D2SRM solves complex PDEs using deep learning.

The article proves the existence of a smooth hypersurface with constant scalar curvature and a specified boundary in hyperbolic space.

Graph Neural Simulators improve data efficiency for PDE surrogates.

The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…

New method converts video of dye plumes into PDEs for better understanding.

In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of …

Paper studies metrics with constant Q-curvature near singular points.

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using …

We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, follow…

Sharp Sobolev theory for scalar elliptic equations on minimal regular manifolds.

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captur…

This work introduces a new loss function to improve the efficiency of optimization-based PDE solvers.

Geometric approach to moment maps in complex geometry.

We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restri…

Paper studies solutions to a specific equation in conformal geometry with singular sets.

DeepONet learns operators for PDEs with varying parameters and initial conditions.

Study shows no hidden symmetries in specific spacetime metrics.

The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.