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.
We describe 4th order ODEs satisfying two contact invariant conditions of Bryant in terms of the Ricci tensor of a certain gl(2,R) valued connection. We also provide nonhomogeneous examples of such ODEs.
We discuss some open problems and recent progress related to the 4th order Paneitz operator and Q curvature in dimensions other than 4.
This paper continues the work of our previous paper [8], where we generalize kth-powers of the Euclidean Dirac operator D_x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D_x^3 and D_x^4 to higher spin spaces in the ca…
MATLAB toolbox pde2path solves geometric PDEs and finds bifurcations in immersed surfaces.
problem Finding bifurcations in geometric PDEs of immersed surfaces.
method Solving PDEs for surface displacement, updating surface, detecting and localizing bifurcations, and switching branches.
result Symmetry breaking bifurcations in various geometric surfaces.
New techniques solve Riccati equations on 3D manifolds, finding 4th order metric obstructions.
problem Solving Riccati-type equations with algebraic constraints on 3D Riemannian manifolds.
method Real algebraic geometry techniques, focusing on connection coefficients and Hessian equations.
result Obstruction to solving Riccati equations has order 4 in metric coefficients.
We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the…
Recently, there has been a trend to combine independent component analysis and canonical polyadic decomposition (ICA-CPD) for an enhanced robustness for the computation of CPD, and ICA-CPD could be further converted into CPD of a 5th-order partially symmetric tensor, by calculating the eigenmatrices of the 4th-order cu…
The paper addresses optimal control on Riemannian manifolds, introducing biased splines for robotic systems.
problem Optimal control on Riemannian manifolds with a mathematically natural cometric not capturing true motion cost.
method Encoding torque-based actuators into a cometric, characterizing optimal solutions via a 4th order differential equation.
result Identified a tensor as the geometric source of biasing solutions away from ordinary splines and geodesics.
Discovering the latent structure from many observed variables is an important yet challenging learning task. Existing approaches for discovering latent structures often require the unknown number of hidden states as an input. In this paper, we propose a quartet based approach which is \emph{agnostic} to this number. Th…
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
The paper studies hyperkähler structures and adapted complex structures using the Monge-Ampère equation.
problem Finding hyperkähler structures and adapted complex structures in tangent bundles.
method Analyzing the asymptotic expansion of the Monge-Ampère equation and using gauge transformations.
result Explicit computation of 4th order terms in the asymptotic expansion and equivalence to gauge transformations.
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…
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 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.
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.
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 compares 5 ODE solvers on 3 case studies, finding varying accuracy.
problem Comparing estimation accuracy of 5 ODE solvers on 3 case studies.
method Used 5 different numerical ODE solvers (Euler's, Heun's, Midpoint, Runge-Kutta 4th order, ODE45) on 3 case studies and compared their results.
result Different solvers have varying accuracy depending on the case study.
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.
We study boundary value problems for the Dirac operator on Riemannian Spinc 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 …
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.
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.
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…
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.
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.
In this paper we establish a connection between free boundary minimal surfaces in a ball in R3 and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
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.
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.
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.
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.
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.
In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds satisfying this boundary condition and we prove a Lagrangian version of Nitsche (…
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.
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…
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.
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 MP-system up to a gauge. In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we pres…
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.