arXiv research

Study critical quasilinear equations on Riemannian manifolds with curvature constraints.

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

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

Sharp estimates derived for quasilinear equations on metric measure spaces.

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.

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

Proves regularity for quasilinear elliptic equations in metric spaces.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

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.

Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.

Estimates for solutions on manifolds under Ricci flow.

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of th…

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 …

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.

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…

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

Study of critical points for 4D conformally invariant curvature energies.

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smoo…

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.

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…

Study curve shortening flow on Riemann surfaces with conical singularities.

Study proves global existence and decay for complex wave equations.

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…

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

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

New method of symmetrization applied to PDEs on spheres.

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

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

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.

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…

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, \\…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

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

A new algorithm speeds up neural network derivative calculations.

Study optimal investment strategies with entropy regularization in volatile markets.

All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants is equal to 7, in sharp contrast with general Monge-Ampère equations …

We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem…

Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.

Extends Newton's minimal resistance problem to Lorentz-Minkowski space.

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…

Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.

Study shows conditions for nonexistence of solutions in Riemannian geometry.

Lectures on PDEs for creating surfaces with specific curvature.

New Kelvin transform for anisotropic elliptic problems.

Researchers create solutions for naked singularities in Einstein vacuum equations.

This is the first in a series Of papers in which we initiate the study Of very rough solutions to the initial value problem for the Einstein Vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques Of energy estimates and Sobolev in…