arXiv research

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties…

Study of singular solutions to a fourth order system in a ball with a singularity.

Study on solutions to conformally invariant fourth order equations, classifying their properties.

Classifies solutions to critical sixth order equations with a singularity.

X-TFC solves parametric DEs with neural networks and physics constraints.

Optimal liquidation strategy reduces risk and improves performance.

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

Unique ancient solutions found for anisotropic curve shortening flow.

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

Paper classifies ancient solutions to 3D Ricci flow.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Study higher-dimensional Ricci flow solutions, proving uniqueness.

Ancient solutions of Ricci flow with Type I growth are classified.

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…

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

New ancient solutions found for curvature flow in 2D.

Let $n\ge 3$ and $m=\frac{n-2}{n+2}$. We construct $5$-parameters, $4$-parameters, $3$-parameters ancient solutions of the equation $v_t=(v^m)_{xx}+v-v^m$, $v>0$, in $\mathbb{R}\times (-\infty,T)$ for some $T\in\mathbb{R}$. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient so…

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

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.

Generic level sets in mean curvature flow are BV solutions.

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

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three inter…

Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.

Ancient solutions to mean curvature flow have unique shapes.

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…

Ozawa solution describes surface deformation from Davey-Stewartson II equation.

Necessary and sufficient conditions are provided for a class of warped product manifolds with non-vanishing flux to be supersymmetric solutions of 11D supergravity. Many noncompact, but complete solutions can be obtained in this manner, including the multi-membrane solution initially found by Duff and Stelle. In a diff…

In an ordinary feature selection procedure, a set of important features is obtained by solving an optimization problem such as the Lasso regression problem, and we expect that the obtained features explain the data well. In this study, instead of the single optimal solution, we consider finding a set of diverse yet nea…

Paper finds singular solutions for a specific physics problem on a sphere.

Ancient pancakes solve mean curvature flow problem.

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

Study properties of solutions with singularities in the negative cone.

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable…

We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,φ)$.

Ancient solutions on a strip are constant if polynomial, and have finite-dimensional space for slower growth.

We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of P.Daskalopoulos, R.Hamilton and N.Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solut…

Researchers find explicit solutions to complex Monge-Ampère equation.

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

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 …

We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.

We classify generalised supersymmetric fluxbranes in type II string theory obtained as Kaluza-Klein reductions of the Minkowski space vacuum of eleven-dimensional supergravity. We obtain two families of smooth solutions which contains all the known solutions, new solutions called nullbranes, and solutions interpolating…

This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The approximate solutions are constructed employing the Contraction Mapping Theorem and…

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

Survey on Kähler-Ricci flow solutions.