The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…
In this paper, we study a nonlocal elliptic problem with the fractional Laplacian on Rn. We show that the problem has infinite positive solutions in Cτ(Rn)⋂Hlocα(Rn). Moreover each of these solutions tends to some positive constant limit at infinity. We extend Lin's result to the nonlocal problem on …
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The boundary conditions in this theory are taken as elements of the C^*-algebra generated by p…
We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in RN, N≥2, all of th…
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
We study elliptic theory on manifolds with boundary represented as a covering space. Firstly, we consider boundary value problems, where the boundary conditions are allowed to mix the values of functions in the fibers of the covering. We show that elliptic elements define Fredholm operators and prove an index formula. …
We are concerned with hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…
MetaNOR learns common nonlocal kernels for efficient metamaterial modeling.
problem Efficiently modeling wave propagation in new metamaterials.
method Meta-learns a common nonlocal kernel from existing tasks and transfers this knowledge to new tasks with minimal data.
result Substantial improvements in sampling efficiency for new metamaterials.
Nonlocal neural networks have been proposed and shown to be effective in several computer vision tasks, where the nonlocal operations can directly capture long-range dependencies in the feature space. In this paper, we study the nature of diffusion and damping effect of nonlocal networks by doing spectrum analysis on t…
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
NKN deep neural network learns governing equations and classifies images.
problem Learning governing equations and classifying images with deep neural networks.
method Nonlocal kernel network (NKN) that is resolution independent, deep, and handles various tasks.
result NKN outperforms baseline methods in learning governing equations and image classification tasks.
We prove existence, uniqueness, and regularity of viscosity solutions to the stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. We give sufficient conditions on the coefficients of the operator to obtain Hölder and Lipschitz con…
New method learns kernels in nonlocal operators robustly.
problem Learning kernels in nonlocal operators is ill-posed.
method Nonparametric regression with Tikhonov regularization.
result Robust estimator of kernel yields homogenized model.
Develops a nonlocal PINN framework using PDDO for better solution of PDEs with sharp gradients.
problem Dealing with sharp gradients in solutions of PDEs using traditional PINN approaches.
method Integrates long-range interactions (nonlocality) into PINN using Peridynamic Differential Operator (PDDO).
result Nonlocal PINN approach improves solution accuracy and parameter inference for problems with sharp gradients.
Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential…
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.
Sharp decay found for solutions of a specific equation in Lie groups.
problem Asymptotic decay of solutions to a Yamabe type equation.
method Analysis of a specific pseudodifferential operator in a homogeneous Lie group.
result Established sharp asymptotic decay of positive solutions.
Nonlocal Bayesian modeling for continuous spatio-temporal dynamics
problem Handling irregular time points, sparse observations, and nonlocal interactions in spatio-temporal forecasting
method Hierarchical Bayesian framework with coordinate-based spatial basis expansion and continuous-time ODE
result Strong forecasting and uncertainty calibration
This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …
The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.
problem Stochastic completeness on complete Riemannian manifolds.
method Proves nonlocal characterizations and provides several new conditions equivalent to stochastic completeness.
result Stochastic completeness is equivalent to genuinely nonlocal conditions, including the zero-mean identity and uniqueness of solutions to fractional equations.
Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formu…
We consider the class of measurable functions defined in all of Rn that give rise to a nonlocal minimal graph over a ball of Rn. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known…
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold correspondi…
The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear dif…
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the s…
Investigates convexity of minimizers under mass constraint using nonlocal perimeter and potential.
problem Convexity of minimizers under mass constraint.
method Nonlocal free energy with nonlocal perimeter and convex potential.
result Quantitative stability theorem for nonlocal free energy assuming symmetry on the potential.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…
Introduces a new 2C extension of the heavenly equation.
problem Solving the general heavenly equation and its extensions.
method Infinite hierarchy of nonlocal symmetries and recursion operator.
result Solutions correspond to 4D hyper-para-Hermitian metrics.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
Paper connects Bäcklund transformations to nonlocal pseudosymmetries.
problem Finding Bäcklund transformations for differential equations.
method Factorization with nonlocal pseudosymmetries to determine Bäcklund transformations as C-morphisms. result Bäcklund transformations are determined by nonlocal pseudosymmetries' invariants.
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing (s,1)-capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of (s,1)-capacity zero. The paper studies nonlocal isoperimetric problems in hyperbolic space and finds unique minimizers for small volumes.
problem Nonlocal isoperimetric problem in hyperbolic space.
method Investigates minimization of a functional with perimeter and a nonlocal term derived from the negative power of distance.
result Geodesic balls are unique minimizers for small volumes in hyperbolic space.
The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.
problem Proving the nonlocal version of the Alexandrov Theorem for sets with smooth boundaries.
method Formulated a necessary and sufficient condition for the theorem to hold, used a specific formula for the tangential derivative of the nonlocal mean curvature, and applied the method of moving planes.
result The only set with smooth boundary and constant nonlocal mean curvature is an Euclidean ball.
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
New Witten rigidity theorems for elliptic genus in various dimensions.
problem Proving rigidity theorems for elliptic genus in different dimensions.
method Combining Liu's and Han-Yu's methods to prove Witten rigidity theorems for elliptic genus in even and odd dimensions.
result Several new Witten rigidity theorems for elliptic genus in even and odd dimensions have been established.
An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …
In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
A new data-adaptive prior stabilizes kernel learning in operators.
problem Learning kernels in operators from data is ill-posed due to nonlocal dependence.
method Introduces a data-adaptive prior to stabilize the Bayesian posterior mean.
result The data-adaptive prior achieves a stable posterior with small noise limits.
The Darboux-Egoroff system of PDEs with any number n≥3 of independent variables plays an essential role in the problems of describing n-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and des…