The study proves the existence of free boundary minimal disks in convex regions.
problem Proving the existence of free boundary minimal disks in convex regions.
method Based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.
result Existence of at least three embedded free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.
Proves limit of free boundary minimal hypersurfaces inherit non-trivial Jacobi fields.
problem Compactness and finiteness of free boundary minimal hypersurfaces.
method Proves one-sided Harnack inequality for minimal graphs on balls with many holes.
result Limit of almost properly embedded free boundary minimal hypersurfaces inherit non-trivial Jacobi fields.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.
problem Constructing free boundary minimal surfaces in product spaces of balls.
method Extremal eigenvalue approach involving mixed Steklov-Neumann eigenvalues.
result No absolute maximum exists for the problem in product spaces.
Proposes neural networks for solving complex free boundary problems.
problem Solving free boundary and Stefan problems with complex interfaces.
method Physics-informed neural networks for approximating solutions and boundaries.
result Successfully approximates solutions and moving boundaries in various Stefan problems.
Study finds rigid properties of boundary-free hypersurfaces in specific data sets.
problem Rigidity of free boundary hypersurfaces in initial data sets with boundary.
method Extending local splitting theorems and applying results on free boundary MOTS.
result Rigidity results for compact free boundary hypersurfaces in initial data sets with boundary.
In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type mi…
Study finds optimal retirement timing in uncertain wage scenarios.
problem Optimal retirement timing in presence of uncertain wages.
method Formulated as a free boundary problem in an incomplete market.
result Developed a method to determine optimal retirement timing.
Develops a new flow with free boundary in barrier surfaces.
problem Free-boundary problems in barrier surfaces.
method Develops Brakke flow with free boundary, proves compactness, Gaussian monotonicity, and existence.
result Proves existence of free-boundary Brakke flows.
Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
Foundations for free boundary Brakke flows established.
problem Analyzing free boundary flows through singularities
method Introducing unit-regular and cyclic free boundary flows, proving avoidance principle, and introducing inner and outer flows.
result General tools for analyzing free boundary flows through singularities.
Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.
problem Optimal weighted Minkowski inequality for umbilical hypersurfaces with free boundary.
method Generalization of Xia's result to free boundary case in space forms.
result Free boundary version of optimal weighted Minkowski inequality in space forms.
Study proves the critical catenoid is the free boundary minimal annulus.
problem Characterizing free boundary minimal surfaces.
method Holomorphic techniques from Fraser and Schoen.
result Proves the critical catenoid is the free boundary minimal annulus.
Survey on recent advances in free boundary minimal surfaces in unit ball.
problem Constructing and understanding free boundary minimal surfaces in the unit ball.
method Review of various techniques and examples, uniqueness results, Morse index estimates, eigenvalue estimates, and compactness results.
result Discussion of critical catenoid uniqueness conjecture and sharp area bounds.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Study anisotropic obstacle problem for minimal surfaces using Cahn-Hoffman transform.
problem Anisotropic obstacle problem for minimal surfaces.
method Cahn-Hoffman transform to convert to isotropic problem with generalized Robin boundary condition.
result Optimal regularity of the solution and C1,1 regularity of the free boundary. Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Optimal inequality for free boundary hypersurfaces in convex domains.
problem Proving an optimal Heintze-Karcher inequality for free boundary hypersurfaces.
method Analyzing anisotropic free boundary hypersurfaces in convex domains.
result Optimal Heintze-Karcher-type inequality achieved for anisotropic free boundary Wulff shapes.
The paper connects minimal surfaces to cones in a ball.
problem Understanding minimal surfaces with free boundaries.
method Establishing a connection between minimal surfaces and cones.
result Doubly connected minimal surfaces in a ball are catenoids.
The paper provides curvature estimates for stable free boundary minimal hypersurfaces.
problem Proving uniform curvature estimates for stable free boundary minimal hypersurfaces.
method Using the theory of minimal laminations developed by Colding and Minicozzi.
result Uniform curvature estimates for stable free boundary minimal hypersurfaces.
Study on stability of free boundary Willmore problem using new gradient inequality.
problem Stability of free boundary Willmore problem.
method New Łojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds.
result Existence and convergence of solutions for the free boundary Willmore flow.
The study constructs free boundary CMC annuli in spherical and hyperbolic balls.
problem Finding free boundary CMC annuli in spherical and hyperbolic balls.
method Constructing free boundary CMC annuli with constant mean curvature H in geodesic balls of S^3 and H^3.
result Embedded free boundary CMC annuli exist for certain mean curvatures in both spaces.
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
The study finds conditions for free boundary Hamiltonian stationary discs in complex 2-space.
problem Conditions for free boundary Hamiltonian stationary Lagrangian discs in complex 2-space.
method Established conditions for weakly conformal, branched Ω-free boundary Hamiltonian stationary Lagrangian immersions of discs. result If conditions are met, a disc is a free boundary minimal immersion.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
New solutions found for a free boundary problem using refined methods.
problem Free boundary problem with two component boundaries.
method Refined Lyapunov-Schmidt reduction method and variational arguments.
result Global minimizers of energy functional found in dimension 8.
Study of free boundary minimal Möbius bands in spherical caps.
problem Characterizing minimal surfaces with free boundary in spherical caps.
method Analyzing spectral properties and geometric constraints.
result Proves that any free boundary minimal Möbius band in spherical caps must be intrinsically rotationally symmetric.
Smooth solutions up to evolving free boundaries for degenerate equations.
problem Degenerate parabolic equations with evolving free boundaries.
method Smooth short-time existence using linear degenerate equations on a fixed domain.
result Smoothness up to the free boundary for the p-Laplacian evolution equation and α-Gauss curvature flow. The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
New minimal discs and annuli found in ellipsoids.
problem Constructing minimal surfaces in ellipsoids.
method Equivariant variational methods.
result At least three distinct embedded free boundary minimal annuli in ellipsoids.
Smooth solution found for free boundary problem in higher dimensions.
problem One phase free boundary problem in R^N
method Constructing a smooth axially symmetric solution
result Existence of a smooth solution with catenoid type free boundary
Paper solves inequalities for convex hypersurfaces with free boundary in a ball.
problem Finding inequalities for convex hypersurfaces with free boundary in a ball.
method Introduced quermassintegrals and used a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.
result Obtained new Alexandrov-Fenchel inequalities for convex free boundary hypersurfaces.
Proves smoothness of minimal surfaces near polyhedral boundaries.
problem Smoothness of free-boundary minimal surfaces near polyhedral domains.
method Allard-type regularity theorem for minimal surfaces in convex polyhedra.
result Minimal surfaces are C1,α graphical over a free-boundary plane if close to it. The paper proves infinitely many free boundary minimal hypersurfaces in compact manifolds.
problem Existence of free boundary minimal hypersurfaces in compact Riemannian manifolds.
method Adaptations of A. Song's work and Marques-Neves' resolution to Yau's conjecture, combined with Li-Zhou's regularity theorem.
result Proves the existence of infinitely many almost properly embedded free boundary minimal hypersurfaces in compact manifolds.
Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
Paper calculates index for free boundary minimal surfaces.
problem Free boundary minimal submanifolds with specific boundary conditions.
method Uses two simpler problems: fixed boundary and Dirichlet-to-Neumann map for Jacobi fields.
result Morse index of a free boundary minimal annulus is 4 if and only if it is the critical catenoid.
Study introduces a new mean curvature flow for spherical boundaries.
problem Mean curvature flow for free boundary hypersurfaces in a ball.
method Guan-Li type volume preserving mean curvature flow for star-shaped free boundary hypersurfaces.
result The flow converges to a free boundary spherical cap for star-shaped initial data.
No free boundary Möbius bands exist in a 3D ball.
problem Proving the non-existence of Möbius bands with free boundaries in a 3D ball.
method Analytical proof based on geometric properties.
result Proves the non-existence of free boundary Möbius bands in the unit three-ball.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
Study on stock trading model with uncertain market status, proving free boundaries and optimal strategies.
problem Optimal trading strategies in a stock market with uncertain market status.
method Free boundary problem, variational inequality system, degenerate operator, C^∞-smoothness.
result All four switching free boundaries are no-overlapping, monotonic, and C^∞-smooth, and their relative localities are completely determined.
The paper generalizes free boundary min-max theory to equivariant settings.
problem Existence of minimal hypersurfaces with free boundary in manifolds with boundary.
method Free boundary min-max theory generalized to equivariant settings.
result Existence of nontrivial smooth almost properly embedded G-invariant minimal hypersurfaces with free boundary. The paper develops a theory for free boundary minimal surfaces with genus at least one.
problem Finding minimal surfaces with specific genus and boundary conditions.
method Using sweepouts of surfaces of genus g≥1 and m≥1 ideal boundary components, the paper constructs a min-max theory for free boundary minimal surfaces.
result The width for the area functional can be achieved by a bubble tree limit of branched genus g free boundary minimal surfaces with nodes.
Simply connected surfaces with large constant mean curvature and free boundaries concentrate at critical points of the boundary's mean curvature.
problem Surfaces with large constant mean curvature and free boundaries.
method Proving concentration at critical points of the boundary's mean curvature.
result Simply connected H-surfaces concentrate at critical points of the boundary's mean curvature.
Proves existence of least area free boundary hypersurface in compact manifolds.
problem Existence of minimal hypersurfaces with least area in compact manifolds.
method Proves existence using volume spectrum and properties of minimal hypersurfaces.
result Existence of free boundary minimal hypersurface of least area.
Study compactness of minimal hypersurfaces with free boundaries.
problem Understanding the behavior of minimal hypersurfaces with free boundaries.
method Analyzes geometric conditions for strong convergence and discusses limit behavior.
result Establishes conditions for strong one-sheeted graphical convergence and discusses multi-sheeted convergence.
Sharp area bounds for 2D free boundary minimal surfaces in a sphere.
problem Sharp bounds for the area of minimal surfaces in a geodesic ball of the sphere.
method Extending earlier work by Brendle and Fraser-Schoen, applying to higher dimensions.
result New sharp bounds for area of free boundary minimal surfaces in higher dimensions.