arXiv research

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …

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 existence and uniqueness of weak solutions for specific equations.

In this paper we investigate relations between solutions to the minimal surface equation in Euclidean $3$-space $\mathbb{E}^3$, the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb{L}^3$ and the Born-Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and i…

The paper explores solutions to the distributional Bellman equation in reinforcement learning.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

Complete solutions found for Toda equations on non-compact surfaces.

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

Researchers found explicit solutions to a complex equation in advanced geometry.

The paper constructs solutions to a critical Dirac equation on spheres.

Study solves HJB equations for time-inconsistent control problems.

The paper proves constant rank theorems for special Lagrangian equations.

Abstract: Survey on quadratic Hessian equations, their properties, and open problems.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

Study improves understanding of solutions to complex equations in geometry.

Proves existence and uniqueness of solutions for a nonlinear equation on Hilbert manifold.

Stable solutions to Yang-Mills-Higgs equations on spheres and tori identified.

The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…

Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.

Study on solutions near isolated singularities in 6D Yamabe equation.

Solutions of tt*-equation from SU(2)_k fusion algebra.

Study geometric singular solutions of generalized Monge-Ampère equations.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

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

Study on positive solutions of Yamabe-type equation on spheres.

Unique solutions found for wave-like decaying null infinity equations.

Paper proves no nontrivial solutions to certain elliptic equations on graphs.

New flow solves LYZ equation on Kähler manifolds.

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

Three theorems about arbitrage bubbles in financial equations.

Convex solutions to a specific equation are smooth when the phase is smooth enough.

A universal rule-based self-learning approach using deep reinforcement learning (DRL) is proposed for the first time to solve nonlinear ordinary differential equations and partial differential equations. The solver consists of a deep neural network-structured actor that outputs candidate solutions, and a critic derived…

Classifies self-similar solutions for heat equations with positive speed.

Classification of ground state solutions to critical Dirac equation on spheres.

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…

Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.

Paper proves existence of knot solutions for specific equations.

Trivial solution proof for heat equation on certain manifolds.

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

Solutions to a differential equation link to contact structures.

New proof for convex solutions of Monge-Ampère equation.

Study on solutions of Yamabe-type equations on projective spaces.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Stable blowup solutions found for supercritical Yang-Mills equations.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.

The paper proves conditions for solutions of $J$-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.

Study bounds derivatives of solutions to a specific equation on domains.