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. 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. New method models dewetting of anisotropic particles using numerical techniques.
problem Modeling dewetting dynamics of particles with varying surface energies.
method Level set numerical approach with convolution kernels to handle anisotropic interfacial energies.
result Validated numerical scheme supports merging and splitting of interfaces.
Flood extent mapping plays a crucial role in disaster management and national water forecasting. Unfortunately, traditional classification methods are often hampered by the existence of noise, obstacles and heterogeneity in spectral features as well as implicit anisotropic spatial dependency across class labels. In thi…
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. Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
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.
In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…
Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
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.
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.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for λ above which minimizers touch the obstacle, regardless of obstacle shape. Study of mean curvature flow with obstacles using singular perturbation.
problem Obstacle problem associated to mean curvature flow.
method Geometric vanishing-viscosity approximation with singular perturbation.
result Generic level sets are distributional solutions of the obstacle problem.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
The paper finds local minimizers for obstacle avoidance on curved spaces.
problem Finding optimal paths on curved spaces avoiding obstacles.
method Minimizing an action functional with bi-Jacobi fields and biconjugate points.
result Local minimizers are classified into two categories with local uniqueness results.
The paper simplifies conditions for optimal paths on manifolds avoiding obstacles.
problem Finding optimal paths on manifolds avoiding obstacles.
method Study of sufficient conditions for optimality on Riemannian manifolds and Lie groups.
result New conditions for optimality are provided in terms of matrix invertibility.
We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of admissible curves in a way that an existence result can be obtained by a penalization …
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.
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a c…
Researchers reconstruct stiffness tensors from limited data in anisotropic elasticity.
problem Reconstructing stiffness tensors from partial data around one polarization.
method Using algebraic geometry and slowness surfaces, the approach leverages the algebraic geometry of families of slowness surfaces.
result For tensors in a dense open subset, a small amount of data around one polarization uniquely determines the entire slowness surface and stiffness tensor.
Geometric optics describes wave behavior near convex obstacles.
problem Wave behavior near convex obstacles.
method Geometric optics in L2 and H1 spaces. result Oscillations transport along grazing rays to any order.
In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator f has quadratic growth in the z-variable. In particular, we obtain existence, comparison, and stability results, and consider the opti…
The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same trave…
Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.
problem Anisotropic flows without global forcing terms and dual Orlicz Christoffel-Minkowski problems.
method Existence results for dual Orlicz Christoffel-Minkowski type problems via stationary solutions of anisotropic flows.
result Existence results for a class of dual Orlicz Christoffel-Minkowski type problems.
This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented …
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 consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original p…
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…
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.
Paper studies optimal control for a specific geometric problem.
problem Optimal control problem associated with the Paneitz obstacle problem.
method Existence and regularity results for optimal controls.
result Existence of optimal controls and their properties.
Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.
problem Gradient obstacle problems on compact Riemannian manifolds.
method Uniform semiconcavity estimates and fine convergence results for solutions and free boundaries.
result The elastic and λ-elastic sets of solutions converge to the cut locus and λ-cut locus of the manifold. 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 motion planning for points avoiding obstacles in a plane.
problem Avoiding collisions for multiple points in a plane with unknown obstacles.
method Algebraic and topological tools for motion planning.
result New topological complexity for planar motion planning.
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 Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerat…
We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.
We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. Optimizes clustering in Gaussian mixtures with varying covariance matrices.
problem Clustering with anisotropic Gaussian mixture models where covariance matrices vary.
method Proposes a computationally feasible hard EM type algorithm.
result Achieves optimal clustering rate with few iterations.
We formulate the theory of nearly autoparallel maps (generalizing conformal transforms) of locally anisotropic spaces and define the nearly autoparallel integration as the inverse operation to both covariant derivation and deformation of connections by nearly autoparallel maps. By using this geometric formalism we cons…
Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.
problem Determine Riemannian manifolds up to isometry using local source-to-solution maps.
method Comprehensive spectrum analysis and semigroup theory for nonlocal parabolic operators.
result Can determine Riemannian manifold up to isometry using local source-to-solution maps in a small open cylinder.
Vanilla SGD learns SIM from anisotropic data without explicit covariance estimation.
problem Learning SIM from anisotropic Gaussian inputs.
method Vanilla Stochastic Gradient Descent (SGD) trained on SIM with anisotropic input.
result Vanilla SGD adapts to anisotropic data's covariance structure.
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
Unique ancient solutions found for anisotropic curve shortening flow.
problem Finding unique solutions for anisotropic curve shortening flow.
method Constructing translating and ancient solutions under given conditions.
result Unique ancient and translating solutions found for anisotropic curve shortening flow.
In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. W…
Formula proves monotonicity for anisotropic minimal hypersurfaces.
problem Understanding anisotropic minimal hypersurfaces.
method Proved a monotonicity formula under a sign assumption on the Minkowski norm.
result Monotonicity formula for anisotropic minimal hypersurfaces.