arXiv research

Researchers found solutions to minimal surface equations in 6D.

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

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

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

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…

Alternative proof of weak solutions to mean curvature flow using minimizing movements.

New boundary functions yield infinitely many solutions for minimal surfaces.

Sharp bounds found for minimal surface solutions.

We construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimesional Euclidean space and gravitational instantons possessing two killing vectors. By Calabi's correspondence, we derive a family of explici…

Investigates polynomial solutions to minimal surface equation, proving constraints and structure theorems.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

New insights on solutions to Allen-Cahn equation with degenerate minimal hypersurfaces.

The paper evaluates biased methods for alpha-divergence minimization.

Hasse principle applied to area-minimizing submanifolds across different homology types.

New ancient and eternal solutions found for mean curvature flow from minimal surfaces.

In this paper, we study multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.

Study finds minimum growth rate for surface solutions.

Study totally real flat minimal surfaces in hyperquadric.

Study finds all possible 5D minimal supergravity solutions with nondegenerate horizons.

New method solves supercooled Stefan problem, proving minimal solutions are physical.

The study finds dense clusters of solutions in a simple neural network model, providing bounds for their existence.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.

Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.

An explicit solution found for maximizing/minimizing agreement in a 2x2 table.

Minimal surface equation results in constant solutions on RCD spaces.

ERM with $f$-divergence regularization yields unique solution.

We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,μ}$ multivalued solutions to the minimal surface equation, to produce $C^{1,μ}$ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…

Study compares nodal sets of solutions to the Allen-Cahn equation.

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the correspondi…

Study shows zero level sets of solutions to Allen-Cahn equation are minimal surfaces with zero mean curvature.

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions …

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…

Paper proves existence of solutions for complex surface diffusion equation.

Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.

Study finds weak solutions for complex map flows with optimal lifespan.

Strictly stable Allen-Cahn hypersurfaces have multiplicity one.

Given a compact Riemannian manifold $(M^n,g)$ and a fixed cohomology class, $[α^*] \in H^k(M)$, we consider the existence of a minimizer $α\in [α^*]$ of the generalized minimal surface energy $\int_M \sqrt{1+|α|^2} dV_g$. When $k = 1$, we prove the existence of unique minimizers for every cohomology class $[α^*]$. Next…

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical results for the minimal surface equation can be generalized. We also conjecture…

Study minimal surfaces in 4D, find specific tori with total curvature -8π.

In this paper we study the non-existence of solutions to the Dirichlet problem for minimal graphs of codimension $\geq 2$, including certain situations over domain $Ω$ even with non-$C^1$ boundary $\partial Ω$.

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

A construction of blowing up solutions to the modified Novikov-Veselov equation is proposed. It is based on the Moutard transformation of two-dimensional Dirac operators and its geometrical interpretation via surface geometry. An explicit example of such a solution constructed by using the Enneper minimal surface is di…

Study shows instability of specific cone solutions in high-dimensional spaces.

Rank minimization has attracted a lot of attention due to its robustness in data recovery. To overcome the computational difficulty, rank is often replaced with nuclear norm. For several rank minimization problems, such a replacement has been theoretically proven to be valid, i.e., the solution to nuclear norm minimiza…

Paper proves uniqueness of weak solutions for Plateau flow.

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at infinity. If the cone is $C^{3,α}$-regular and mean convex (but not area-minimizing…