Classifies local boundary conditions for Dirac-type operators on manifolds.
problem Determining all local smooth boundary conditions for Dirac-type operators.
method Combining general theory of boundary value problems for Dirac operators and pointwise considerations.
result Classification of local self-adjoint regular boundary conditions for Dirac spinors in dimensions 3 and 4.
Proof of local well-posedness for a specific boundary condition in general relativity.
problem Initial boundary value problem in general relativity with umbilic boundary condition.
method Wave coordinates and key observation of momentum constraint validity for umbilic boundaries.
result Local well-posedness established for the initial boundary value problem.
Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.
New boundary conditions solve Cauchy problem for Dirac operators on spacetimes.
problem Understanding non-local boundary conditions for Dirac operators on spacetimes.
method Define and analyze a class of Lorentzian boundary conditions that are local in time and non-local in spatial directions.
result Well-posed Cauchy problem for the Dirac operator is established under these conditions.
We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the …
The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.
problem Ricci flow on manifolds with boundary.
method Proving short-time existence and uniqueness of the solution, and showing boundary conditions preservation.
result The flow preserves natural boundary conditions under certain curvature conditions.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonst…
We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold.
Novel boundary conditions for Ricci flow to deform compact manifolds.
problem Deforming compact Riemannian manifolds with boundary using Ricci flow.
method Proposed boundary conditions that make first variations of functionals (Einstein-Hilbert action, lambda-functional) without boundary terms.
result Proof of short-term existence of solutions under proposed conditions.
Proves well-posedness for Einstein equations with specific boundary conditions.
problem Well-posedness of vacuum Einstein equations with twisted Dirichlet boundary conditions.
method Proves local-in-time well-posedness for the IBVP of the Einstein equations with specified conformal class and scalar densities.
result Proves well-posedness for the Einstein equations with twisted Dirichlet boundary conditions.
PINN-FEM combines PINNs and FEM for accurate Dirichlet boundary condition enforcement.
problem Challenges in enforcing Dirichlet boundary conditions in PINNs.
method Hybrid approach combining PINNs and FEM for strong boundary condition enforcement.
result PINN-FEM outperforms standard PINN models in accuracy and robustness.
Study constant mean curvature surfaces with integrable boundary conditions.
problem Understanding surfaces with constant mean curvature under specific boundary conditions.
method Used generalized Weierstrass representation to determine potentials.
result Determined potentials for surfaces satisfying integrable boundary conditions.
Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.
problem Constructing solutions to Bogomolny equations with given boundary and asymptotic conditions.
method Using generalized Nahm pole boundary condition and real symmetry breaking condition.
result Solutions analogous to instanton solutions, satisfying different asymptotic conditions.
Lower bounds for eigenvalues on manifolds with boundary conditions.
problem Eigenvalue bounds for manifolds with boundary conditions.
method Proving lower bounds for the first non-trivial eigenvalue using Cheeger-type constants.
result Results in the spirit of Cheeger's inequality for manifolds with boundary conditions.
The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard …
Paper discusses gluing formula for zeta-determinants with Robin boundary condition.
problem Computing zeta-determinants with Robin boundary condition.
method Uses BFK type gluing formula and computes differences with Dirichlet boundary condition.
result Computes zeta-determinant on a cylinder with Robin boundary condition.
Study well-poses Dirac operator problem with APS boundary conditions.
problem Well-posedness of Cauchy problem for Dirac operator on Lorentzian manifolds.
method Derived energy estimates, established uniqueness and existence of weak solutions, introduced mollifier operators.
result Well-posedness of Cauchy problem for Dirac operator with APS boundary conditions.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.
We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer bo…
New boundary conditions improve Hamiltonian analysis in GR.
problem Improving Hamiltonian analysis in GR with IBVP.
method Presented and analyzed new boundary conditions.
result New boundary conditions lead to better Hamiltonian analysis.
Heat kernel estimates on manifolds with mixed boundary conditions.
problem Estimating heat kernels on manifolds with ends and mixed boundary conditions.
method Global harmonic function construction and h-transform technique. result Two-sided heat kernel estimates for Riemannian manifolds with mixed boundary conditions.
Study Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.
problem Analyzing Dirac-Einstein equations on manifolds with boundary conditions.
method Characterizing bubbling phenomena and classifying ground state bubbles, proving an Aubin-type inequality.
result Proved an Aubin-type inequality and existence result.
We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field me…
Study perturbs APS boundary conditions for Lorentzian Dirac operators.
problem Maintaining Fredholmness of Dirac operators under perturbations of APS boundary conditions.
method Develop criteria for perturbing compact pairs of projections to remain Fredholm.
result Criteria for perturbing APS boundary conditions without losing Fredholmness.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.
problem Characterizing smooth Riemannian manifolds with boundary.
method Introducing a Dynkin-type condition and proving its equivalence to a weighted manifold.
result Bi-Lipschitz equivalence and various spectral properties of manifolds with boundary.
Study on well-posedness of vacuum Einstein equations with specific boundary conditions.
problem Well-posedness of the initial boundary value problem for vacuum Einstein equations with geometric boundary conditions.
method Analysis of conformal-mean curvature boundary data, proving dense solution space and Holmgren-type uniqueness theorem.
result Linearized problem has a solution space with dense range in C∞, valid for general smooth linearized solutions. In the previous work ([14]) we introduced the well-posed boundary conditions P−,L0 and P+,L1 for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined…
Paper finds solutions to a complex equation on surfaces with boundary conditions.
problem Existence of solutions to a super-Liouville equation on compact Riemannian surfaces with boundary.
method Introduced a weighted Dirac operator and constructed a Nehari manifold to show existence of non-trivial solutions.
result Existence of non-trivial solutions to the super-Liouville equation.
The paper establishes gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
problem Gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
method Yau and Souplet-Zhang type gradient estimates for harmonic and heat equation solutions under Dirichlet boundary condition.
result Established gradient estimates for harmonic and heat equation solutions on manifolds with boundary.
The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.
problem Investigating the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundaries.
method Admissible boundary conditions are imposed to show the existence and uniqueness of strong solutions. For hyperbolic systems, the Cauchy problem is also well-posed in the Hadamard sense.
result Existence and uniqueness of strong solutions for the Cauchy problem are proven under admissible boundary conditions.
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.
In this paper, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the $\MIT$ bag boundary condition. Then we give some applications of these constructions for each Green function.…
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
Proves Hadamard states for Dirac fields on manifolds with timelike boundaries.
problem Existence of Hadamard states for Dirac fields with MIT boundary conditions.
method Introducing geometric Møller operator to implement unitary isomorphism between spaces of initial data.
result Existence of Hadamard states for Dirac fields with MIT boundary conditions.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
problem Finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.
method Determine rationality criteria for Durham conditions and analyze spectral curve properties.
result Rationality criteria are sufficient for finite-type solutions with complementary boundary conditions.
Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.
problem Local well-posedness of Schrödinger flow into S2 with natural boundary conditions. method Developed a new approximation scheme to solve the problem.
result Solved the local well-posedness problem for the Schrödinger flow into S2 with natural boundary conditions. We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition …
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
Study on elasticity with mixed boundary conditions, proving spectral asymptotics.
problem Analyzing spectral asymptotics for linear elasticity with mixed boundary conditions.
method Established two-term spectral asymptotics for linear elasticity on smooth compact manifolds.
result Verification of general formulae through explicit examples in 2D and 3D.
Proves a theorem for mechanical systems with reflections.
problem Action functionals on paths with reflections.
method Proves a Morse index theorem for action functionals on paths that can reflect.
result Action functionals on paths with reflections have a well-defined Morse index.
The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.
problem Compactness of hyperkaehler 4-manifolds with boundary.
method Analyzes sequences of hyperkaehler triples under topological and curvature conditions.
result Smooth convergence of hyperkaehler triples up to diffeomorphisms if boundary restrictions converge.
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…
Study local structure of Einstein metrics with boundary conditions.
problem Understanding the local structure of Einstein metrics with boundary constraints.
method Analysis of moduli space of compact Einstein metrics, focusing on boundary conformal metric and mean curvature.
result For three dimensions, the map from Einstein metrics to boundary data is generically a local diffeomorphism.
We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …
The paper studies curvature conditions on manifolds with boundary.
problem Curvature preservation on manifolds with smooth boundaries.
method Constructing a family of metrics that agree with given metrics on the boundary and interior.
result Deforming metrics to ones with totally geodesic boundary while preserving curvature conditions.