New model predicts grain boundary migration in metals.
problem Anisotropic grain boundary migration in polycrystals.
method Level set-finite element formulation based on thermodynamics and mechanics.
result First analytical solution for anisotropic grain boundary configurations.
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 study shows that certain graphs are regular at boundary points.
problem Boundary regularity of anisotropic minimal Lipschitz graphs.
method Proves regularity for graphs with bounded anisotropic mean curvature and atomic energy condition.
result Regularity at boundary points with density bounded above by 1/2 + σ.
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. Paper proves flatness of anisotropic minimal graphs in half-spaces.
problem Anisotropic minimal graphs with free boundaries in half-spaces.
method Proves flatness using linear growth conditions.
result Anisotropic minimal graphs in half-spaces are flat if they have at most one-sided linear growth.
Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.
problem Rigidity of weak solutions for anisotropic N-Laplacian equations with boundary conditions.
method Established a key integral inequality involving anisotropic gradient and second fundamental form, proving rigidity under natural monotonicity assumptions.
result All weak solutions to Neumann boundary problems are constant without a priori boundedness assumption.
Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.
problem Anisotropic parabolic obstacle problems and Stefan problem.
method Cahn-Hoffman transform and anisotropic mean curvature flow.
result Optimal regularity of the solution and C1,α-regularity of the evolving free boundary. Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
We develop an explicit and tractable representation of a twist-grain-boundary phase of a smectic A liquid crystal. This allows us to calculate the interaction energy between grain boundaries and the relative contributions from the bending and compression deformations. We discuss the special stability of the 90 degree g…
The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
Compact hypersurfaces minimize area in convex cones with free boundary.
problem Finding compact hypersurfaces minimizing area in convex cones with free boundary.
method Minimizing an anisotropic area functional under a volume constraint.
result Compact hypersurfaces are contained in a Wulff-shape.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Paper proves anisotropic Minkowski inequality and related inequalities.
problem Proving anisotropic Minkowski inequality and related inequalities.
method Utilizes a nonlinear potential theoretic approach.
result Sharp anisotropic Minkowski inequality and related inequalities proved.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
Paper solves anisotropic capillary Minkowski problem for p ≥ 1.
problem Anisotropic capillary convex bodies and their properties.
method Introduced anisotropic capillary p-sum and computed variations of quermassintegrals. result Solved the anisotropic capillary Lp-Minkowski problem for p≥1. Large twist-angle grain boundaries in layered structures are often described by Scherk's first surface whereas small twist-angle grain boundaries are usually described in terms of an array of screw dislocations. We show that there is no essential distinction between these two descriptions and that, in particular, their…
Anisotropic metric on manifolds uniquely determined by boundary data.
problem Determining Riemannian metrics on compact manifolds from boundary measurements.
method Analysis of the Dirichlet-to-Neumann map for the Laplace-Beltrami operator.
result Riemannian metrics can be uniquely determined up to isometry.
The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
We generalize Penrose's notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Li…
Study optimizes perimeter in convex domains with anisotropic constraints.
problem Optimizing perimeter in convex domains with anisotropic constraints.
method Analytical properties, topological features, and geometric measure theory results.
result Sharp isoperimetric inequalities and existence of minimizers.
Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transf…
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…
The paper proves a Wulff inequality for minimal submanifolds with boundary in Euclidean space.
problem Proving a Wulff inequality for minimal submanifolds with boundary.
method Associating a nonnegative anisotropic weight to the boundary of minimal submanifolds and proving the inequality.
result The Wulff inequality constant is independent of the weights and depends only on m and n. We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …
This study recovers electromagnetic parameters on boundaries from impedance and admittance data.
problem Recovering anisotropic electromagnetic parameters from boundary impedance and admittance data.
method Formulated inverse boundary value problem for time-harmonic Maxwell's equations on differential 1-forms.
result Knowledge of impedance and admittance maps determines tangential entries of induced metrics at the boundary.
Given an elliptic integrand of class C3, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.
We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider n≥2, p∈(1,+∞) and Σ an n-dimensional, closed hypersurface in Rn+1, boundary of a convex, open set. We show that …
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…
New method uses broken scattering to uniquely identify Finsler manifolds.
problem Identifying Finsler manifolds from scattering data.
method Uses broken scattering relation to compare geodesics.
result Two reversible Finsler manifolds with the same broken scattering relation are isometric.
We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.
In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the…
After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $…
Given a positive function F on Sn which satisfies a convexity condition, for 1≤r≤n, we define the r-th anisotropic mean curvature function HrF for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface…
The paper studies curvature measures and volume-preserving flows on convex bodies.
problem Characterizing and understanding convex bodies through anisotropic curvature measures.
method Developed anisotropic curvature measures, used Minkowski formulas and Heintze-Karcher inequalities, and analyzed volume-preserving flows.
result Characterized Wulff shapes via anisotropic curvature measures and proved convergence of volume-preserving flows.
Paper proves existence of anisotropic dynamical horizons in gravitational collapse.
problem Existence of apparent horizons in gravitational collapse.
method Scale-critical hyperbolic method and non-perturbative elliptic techniques.
result Smooth and spacelike apparent horizons emerge from general initial data in gravitational collapse.
Study shows limits of volume-constrained sets are finite unions of Wulff shapes.
problem Analyzing the behavior of sets with degenerating ellipticity.
method Proving rigidity of L1-accumulation points of volume-constrained almost-critical sets. result Limits of volume-constrained sets are finite unions of φ-Wulff shapes. The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.
problem Proving Alexandrov-Fenchel inequalities for anisotropic mixed volumes.
method Introducing a fully nonlinear locally constrained anisotropic curvature flow.
result The flow converges smoothly and exponentially to a scaled Wulff shape.
Study reconstructs Riemannian metric from Cherenkov radiation in complex media.
problem Reconstructing internal geometry of inhomogeneous anisotropic targets.
method Mathematical model of waves in medium, including vector-valued wave operator and phase velocity.
result Riemannian metric inside a bounded region can be reconstructed from boundary measurements of Cherenkov radiation.
Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.
problem Determining an unknown function in a semilinear elliptic equation on complex manifolds.
method Analyzing higher order linearizations and interactions of Gaussian quasimode solutions.
result An unknown smooth function can be uniquely determined from the Dirichlet-to-Neumann map.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
Isoperimetric regions in scaled product manifolds are products of regions in each factor.
problem Characterizing isoperimetric regions in anisotropically scaled product manifolds.
method Analyzing regions with smooth boundaries in scaled product manifolds.
result Isoperimetric regions in scaled product manifolds are products of regions in each factor.
In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
ECN framework improves training on noisy structured labels.
problem Structured errors in fine-grained annotations lead to biased models.
method Error-Correcting Networks (ECN) framework.
result ECN improves fine-grained annotation prediction.
The paper proves a Willmore-type inequality for unbounded convex sets.
problem Proving a Willmore-type inequality for unbounded convex sets.
method Analytical proof involving hypersurfaces, contact angle conditions, and asymptotic volume ratio.
result The Willmore-type inequality holds for unbounded closed convex sets with certain conditions.
The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.
problem Evolution of spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Anisotropic inverse mean curvature flow with Neumann boundary condition.
result The flow converges to a hyperbolic plane as time tends to infinity.