arXiv research

The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold $M$ with boundary $\partial M$. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on $M$. We propose …

Sharp estimates derived for quasilinear equations on metric measure spaces.

Classifies Lie symmetry algebras for 2D quasilinear equations, linking symmetry to linearity.

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

Study curve shortening flow on Riemann surfaces with conical singularities.

A new algorithm speeds up neural network derivative calculations.

Prove rigidity and classification results for quasilinear Liouville equation on manifolds with nonnegative Ricci curvature.

Global existence and decay for quasilinear wave equations on various spacetimes, including Kerr black holes.

The paper studies geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.

Proves regularity for quasilinear elliptic equations in metric spaces.

New Kelvin transform for anisotropic elliptic problems.

We show, by modifying Borbély's example, that there are $3$-dimen\-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $\le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.

Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.

Estimates for solutions on manifolds under Ricci flow.

Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we fi…

The paper proves growth estimates for subsolutions of quasilinear equations.

This study optimizes crypto-market trading conditions without assuming convexity.

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order con…

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets o…

Study on dynamic curves with elastic energy and spontaneous curvature.

Geometric approach links hydrodynamic integrability to compatible nets.

New method of symmetrization applied to PDEs on spheres.

Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the l…

Smoothness of graphs evolving by fractional mean curvature is proven.

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

The paper studies eigenvalue problems on manifolds and recovers known inequalities.

Global existence and boundedness proved for quasilinear wave equations on Kerr black holes.

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of t…

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criteri…

Study proves global existence and decay for complex wave equations.

We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

Suppose that $G=(V, E)$ is a connected locally finite graph with the vertex set $V$ and the edge set $E$. Let $Ω\subset V$ be a bounded domain. Consider the following quasilinear elliptic equation on graph $G$ $$ \left \{ \begin{array}{lcr} -Δ_{p}u= λK(x)|u|^{p-2}u+f(x,u), \ \ x\inΩ^{\circ}, u=0, \ \ x\in\partial Ω, \\…

The paper proves no positive weak solutions for a specific inequality on Riemannian manifolds.

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

Survey uses Milnor fibrations to classify first integrals of differential systems.

In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} Δu$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the induced metric to establish two-sided bounds on the singular values of Du, which shows tha…

In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…

We show short-time existence for curves driven by curve diffusion flow with a prescribed contact angle $α\in (0, π)$: The evolving curve has free boundary points, which are supported on a line and it satisfies a no-flux condition. The initial data are suitable curves of class $W_2^γ$ with $γ\in (\tfrac{3}{2}, 2]$. For …

We establish area bounds for two-dimensional immersions in R^3 and R^n. Namely, for μ-stable immersions in R^3 (R^n), for graphs in $\mathbb R^3$ which solve quasilinear equations in divergence form, and for graphs which are critical for Fermat-type variational problems in R^n.

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some o…

Study on Kyle-Back model with risk aversion and non-Gaussian beliefs.

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson-Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develo…

The paper solves a complex financial optimization problem using a novel mathematical technique.

New concept of effective isometries for compliant shells.

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrov…