arXiv research

New energy identity found for biharmonic maps into spheres.

Conservation law for weakly harmonic mappings in high dimensions.

Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.

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.

Solves supercritical dHYM on projective manifolds with specific conditions.

We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can…

Solutions grow for a special type of math problem on curved spaces.

Smooth Yang-Mills fields proved in supercritical dimensions.

Wave maps with noise can lead to self-similar blowup from arbitrary initial data.

Stable blowup profile identified for wave maps in all dimensions.

Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.

Two-dimensional Lagrangian mean curvature equation solved with new inequality.

Extending isometric immersions with low regularity, especially supercritical.

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

Paper proves estimates for Lagrangian flow singularities.

Paper confirms conjecture for projective manifolds in supercritical phase.

Stable blowup solutions found for supercritical Yang-Mills equations.

Paper proves gradient estimates for Lagrangian mean curvature equation.

Let $(M,g)$ and $(K,κ)$ be two Riemannian manifolds of dimensions $m$ and $k ,$ respectively. Let $ω\in C^2(N),$ $ω> 0.$ The warped product $M\times _ωK$ is the $(m+k)$-dimensional product manifold $M\times K$ furnished with metric $g+ω^2 κ.$ We prove that the supercritical problem $$-Δ_{g+ω^2 κ}u+h u=u^{ {m+2\over …

Derives Hessian estimates for Lagrangian mean curvature equation.

The paper confirms the solvability of a complex equation for a 4D manifold.

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak …

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…

Paper proves solvability condition for complex equation on special submanifolds.

Study finds infinite sign-changing solutions for a specific equation on manifolds.

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Δ^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an…

Recently, large-scale cascading failures in complex systems have garnered substantial attention. Such extreme events have been treated as an integral part of the self-organized criticality (SOC). Recent empirical work has suggested that some extreme events systematically deviate from the SOC paradigm, requiring a diffe…

Study disproves conjecture about Hermitian-Yang-Mills solutions.

Stability of singularity formation in Yang-Mills fields in higher dimensions.

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2$ we show that for any $q<\frac{ n-d_2+2 }{n - d_2…

Proves existence and uniqueness of weak solutions for specific equations.

We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, c…

Study finds existence of $Q$-curvature metrics on even-dimensional manifolds with conical singularities.

Develops topological concepts for Morrey-Sobolev bundles in high dimensions.

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d…

Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.

We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup …

Study Neumann problem for special Lagrangian type equations.

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

Let $Ω$ be an annulus. We prove that the mean field equation $-Δψ=\frac{e\sp{-βψ}}{\int\sbΩe\sp{-βψ}} $ admits a solution with zero boundary for $β\in (-16π,-8π)$. This is a supercritical case for the Moser-Trudinger inequality.

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \…

This paper presents a new interacting particle system and uses it as a spin model for financial market microstructure. The asymptotic analysis of this stochastic process exhibits a lower bound to the contemporaneous measurement of price and trading volume under the invariant measure in the `frozen' phase of the supercr…

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a larg…

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…

We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for…

The paper characterizes when numerical criteria for PDE solvability fail and provides effective criteria for existence.

Deep learning detects bifurcations in dynamical systems.