arXiv research

Solves critical LYZ equation in Kähler geometry.

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

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

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

Solves Dirichlet problem for Lagrangian phase equation with critical and supercritical phase.

Estimates for special Lagrangian curvature equations in critical and convex cases.

Paper proves gradient estimates for Lagrangian mean curvature equation.

Stability of a new map derived from the equator map is analyzed.

We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb{R}^n$ with $n\geq 2$. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited state…

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-cu…

Characterizes solutions to Z-critical equations on surfaces using effective conditions.

Paper studies critical points of curvature energies in 4D.

Deep learning model solves high-dimensional PDEs using Actor-Critic approach.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.

The paper studies critical metrics on a specific type of manifold.

Trivial solution proof for heat equation on certain manifolds.

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there…

Proves stability of certain vector bundles on Kähler surfaces.

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…

Actor-critic algorithms converge to an ODE as data samples change dynamically.

The paper studies a new vacuum field equation and its solutions.

Classifies positive solutions to critical p-Laplace equation.

In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.

Rigidity theorem for critical points of Allen-Cahn equation on S³.

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 …

Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudodifferential equation $Δ^αu+Vu=0$ improves the Sobolev regularity of solutions provided the potential $V$ is integrable with the critical power $n/2α>1$.

Study semilinear equations on weighted manifolds to prove rigidity.

Neural networks trained with actor-critic algorithms converge to ODEs under weak convergence analysis.

We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.

The paper finds sign-changing solutions for a specific type of elliptic equation.

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

Extends Gauduchon's result to higher dimensions, showing balanced metrics.

Study examines wave equation decay and Strichartz estimates on conic manifolds.

Neural networks solve high-dimensional HJB PDEs with asymptotic guarantees.

Rigidity for 4D Willmore submanifolds with boundary.

Paper finds infinitely many solutions changing sign for critical fractional equations.

Study on Monge-Ampère equations with polynomial growth rates.

In this paper we derive the Euler-Lagrange equation of the functional $L_β=\int_Σ\frac{1}{\cos^βα}dμ, ~~β\neq -1$ in the class of symplectic surfaces. It is $\cos^3α{\bf{H}}=β(J(J\nabla\cosα)^\top)^\bot$, which is an elliptic equation when $β\geq 0$. We call such a surface a $β$-symplectic critical surface. We first st…

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold.In the case of constant coefficients we obtain the existence of tree …

Study finds multiple solutions for Gross-Pitaevskii equations on curved spaces.

The paper classifies solutions to semilinear equations on curved spaces.

Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.

We consider variation of energy of the light-like particle in Riemann space-time, find lagrangian, canonical momenta and forces. Equations of the critical curve are obtained by the nonzero energy integral variation in accordance with principles of the calculus of variations in mechanics. This method is shown to not lea…

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over th…