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…
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48 results for “boundary problems”
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
Study boundary value problems for elliptic operators on manifolds.
problem Characterize and analyze boundary conditions for first-order elliptic differential operators.
method Develops a new framework for elliptic boundary conditions, proving equivalence and regularity of solutions.
result Elliptic boundary conditions yield a Fredholm operator on compact manifolds.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
The study defines a boundary problem for Lagrangian submanifolds and proves a theorem.
problem Boundary problem for Lagrangian submanifolds.
method Introduced a boundary problem analogous to free boundary hypersurfaces and capillary hypersurfaces.
result Proved a Lagrangian version of Nitsche (or Hopf) type theorem.
Paper solves a mixed boundary value problem in space forms with umbilical boundaries.
problem Solving a partially overdetermined mixed boundary value problem in space forms.
method Generalizing previous results to domains with partial umbilical boundaries.
result A partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface.
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
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.
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.
Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.
problem Boundary behavior of the singular Yamabe problem near singular boundaries.
method Analysis of asymptotic behaviors and derivation of optimal estimates for background metrics.
result Solutions are well approximated by solutions in tangent cones at singular points.
Study identifies obstructions for solving a 4th-order boundary problem.
problem Solving a 4th-order boundary problem with specific curvature conditions.
method Derived Kazdan-Warner type identities using variational formulation and conformal variations.
result Obtained nontrivial integral obstructions to solvability.
Study of Hamiltonian boundary value problems with symplectic symmetries.
problem Bifurcations of solutions in Hamiltonian boundary value problems.
method Convenient, coordinate-free framework for separated Lagrangian boundary value problems; analysis of conformal symplectic symmetries.
result Existence of obstructions due to conformal symplectic symmetries on bifurcation behavior.
Blowing up solutions for supercritical Yamabe problems on manifolds with non umbilic boundarymath.DG
Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.
problem Solving supercritical Yamabe problem on manifolds with non-umbilic boundary.
method Building blowing-up solutions for a supercritical perturbation of the Yamabe problem.
result Constructed solutions for a supercritical perturbation of the Yamabe problem on manifolds with non-umbilic boundary.
On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundarymath.AP
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.
We study boundary value problems for the Dirac operator on Riemannian Spin manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …
Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…
Open problems on surfaces with boundary and constant mean curvature.
problem Open problems on compact constant mean curvature surfaces with boundary.
method Survey of current status of open problems.
result Collection of open problems in the theory of surfaces.
Proves well-posedness of Dirac operator problem on specific manifolds.
problem Well-posedness of Cauchy problem for Dirac operator on globally hyperbolic manifolds with timelike boundary.
method Transformed into symmetric hyperbolic system, proved existence and uniqueness of weak solutions, then used local methods to show smoothness.
result Smooth solutions for weak solutions of Dirac operator problem.
Elliptic boundary value problem for G2 structures on manifolds.
problem Solving G2 holonomy equation on manifolds with boundary.
method Setting up a suitable linear elliptic boundary value problem.
result Existence of certain G2 cobordisms between deformations of Calabi-Yau 3-folds.
Study of boundary value problems in 7, 4, and 3 dimensions with G2 and holonomy structures.
problem Boundary value problems for G2 structures in 7-manifolds with prescribed 3-forms.
method General observations and detailed study of reductions to 4 and 3 dimensions by imposing symmetry.
result 3-dimensional reduction admits a dual variational formulation using the real Monge-Ampere equation.
Surveying inverse problems on manifolds with boundaries.
problem Inverse problems for geometric structures on manifolds with boundaries.
method Analyzes linear and non-linear geometric inverse problems.
result Recent results on geodesic X-ray transforms and Riemannian metrics.
The study finds multiple solutions to a complex metric problem using bifurcation theory.
problem Generalizing the boundary Yamabe problem to conformally deform metrics.
method Bifurcation theory applied to fully nonlinear boundary value problems.
result Constructs examples of multiple non-homothetic solutions for specific metrics.
This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.
problem Solving the Dirichlet problem for degenerate elliptic equations on Riemannian manifolds with mean concave boundaries.
method The proof relies on a quantitative boundary estimate.
result Analogous results are obtained in complex variables and on certain product manifolds.
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.
Method proves ellipticity of vacuum spacetime boundary problems.
problem Proving ellipticity of boundary value problems for stationary vacuum spacetimes.
method Developed a general method using the projection formalism.
result Proved manifold theorem for moduli space of stationary vacuum spacetimes.
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.
Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
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.
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.
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.
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.
Paper connects minimal and maximal surfaces' boundary value problems.
problem Existence and uniqueness of minimal surfaces' boundary value problems.
method One-to-one correspondence between minimal and maximal surfaces' solutions.
result One-to-one correspondence between minimal and maximal surfaces' boundary value problems.
Proves rigidity for spherical cap eigenvalue problem.
problem Eigenvalue problem with mixed boundary conditions.
method Obata-type rigidity result for spherical cap.
result Proves rigidity for eigenvalue problem.
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…
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.
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.
Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.
problem Existence of sign-changing solutions to the Yamabe problem on manifolds with boundary.
method Variational approach, analysis of conformal invariants, and sharp energy estimates.
result Existence of least-energy nodal solutions when the manifold is positive and the boundary has non-negative constant mean curvature.
Proves unique extension of Bartnik boundary data for stationary vacuum spacetimes.
problem Unique extension of Bartnik boundary data for stationary vacuum spacetimes.
method Elliptic boundary value problem for stationary vacuum equations combined with Bartnik boundary conditions.
result Bartnik boundary data near standard flat data admits a unique stationary vacuum extension locally.
We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.
New symplectic boundary conditions for cohomology.
problem Defining cohomology for symplectic manifolds with boundary.
method Introducing elliptic boundary value problems and Hodge theories.
result Distinguishing Kähler structures of Kähler manifolds with boundary.
Solutions blow up for Yamabe problem on umbilic manifolds with nonzero Weyl tensor.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions
result Solutions blow up for linear perturbation of Yamabe problem
Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
We study the zeta determinant of global boundary problems of APS-type through a general theory for relative spectral invariants. In particular, we compute the zeta determinant for Dirac-Laplacian boundary problems in terms of a scattering Fredholm determinant over the boundary.
Researchers solve boundary and scattering rigidity problems for magnetic systems.
problem Recovering magnetic systems from boundary or scattering data.
method Reduced to magnetic systems and applied results from [DPSU07].
result Recovering -system up to a gauge.
Paper approximates free boundary for optimal investment stopping problems.
problem Optimal investment stopping problems with utility maximization.
method Dual control method to derive asymptotic properties and construct a global closed-form approximation.
result Global closed-form approximation of dual free boundary reduces computational cost.
Solutions grow for a special type of math problem on curved spaces.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions for a supercritical perturbation
result Existence of solutions for n>7 and non-vanishing Weyl tensor
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.