Proves long-term behavior of curvature flows with Neumann boundary.
problem Neumann boundary value problem for general curvature flow.
method Long-term existence and convergence results via Neumann boundary condition.
result First result for Neumann boundary problem of non-Monge-Ampere type.
Extends Neumann boundary solvability to Riemannian manifolds.
problem Global regularity of Monge-Ampère equations with Neumann boundary conditions.
method Analysis on Riemannian manifolds, extending Euclidean results.
result Extends Neumann boundary solvability to Riemannian manifolds.
Proves an optimal Minkowski inequality in space forms.
problem Proving an optimal Minkowski inequality in space forms.
method Applying the Reilly formula to a Neumann boundary value problem.
result Optimal Minkowski type inequality in space forms.
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.
We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
Existence of YMH fields with boundary conditions proven.
problem Existence of Yang--Mills--Higgs fields with boundary conditions.
method Study of convergence and blow-up behavior of Sacks-Uhlenbeck type α-YMH fields as α→1, regularity theorem for coupled systems.
result Existence of smooth YMH fields up to the boundary under certain conditions.
The paper solves a class of nonlinear flows with boundary conditions, preserving convexity and proving long-term existence.
problem Solving a class of fully nonlinear flows with nonlinear Neumann boundary conditions.
method Analyzing the convexity and proving long-term existence of the flow.
result Preserving convexity and proving long-term existence of the flow.
New manifolds found with equal Robin and Steklov spectra.
problem Finding manifolds with equal Robin and Steklov spectra.
method Used Sunada method and torus action method to construct manifolds.
result Constructs manifolds with equal Robin and Steklov spectra under various boundary conditions.
Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.
problem Express zeta-determinant of Dirichlet-to-Neumann operator on forms.
method Expresses zeta-determinant as difference of Laplacian determinants with boundary conditions.
result Computes terms explicitly for dimensions 2 and 3.
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.
Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are fi…
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.
Eigenvalue bounds on manifolds with lower Ricci curvature.
problem Eigenvalue inequalities on Riemannian manifolds with lower Ricci curvature bounds.
method Revisit and prove classical inequalities for Dirichlet and Neumann boundary value problems.
result Eigenvalue multiplicity bounds and related open problems discussed.
Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.
problem Rigidity of weak solutions for anisotropic N-Laplacian equations with boundary conditions.
method Established a key integral inequality involving anisotropic gradient and second fundamental form, proving rigidity under natural monotonicity assumptions.
result All weak solutions to Neumann boundary problems are constant without a priori boundedness assumption.
The paper shows connections can be uniquely determined by their boundary data.
problem Determining unique connections from boundary measurements.
method Defined a Dirichlet-to-Neumann map for twisted Dirac Laplacians and showed its pseudodifferential properties.
result Equal Dirichlet-to-Neumann maps imply locally gauge equivalent connections.
The paper optimizes domains for the first nontrivial eigenvalue in Riemannian manifolds.
problem Optimizing domains for the first nontrivial eigenvalue in Riemannian manifolds.
method Analyzing locally extremal domains and using an extended shape derivative formula.
result Critical domains for the first nontrivial eigenvalue are identified.
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.
Let Ω be an open, bounded domain in the plane with connected and smooth boundary, and ω an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue μ>0. If the boundary value of ω is a nonzero constant along the boundary, denoting 0=μ1(Ω)<μ2(Ω)<=... the set of all Neumann eigen…
On any given compact (n+1)-manifold M with non-empty boundary, it is proved that the moduli space of Einstein metrics on M is a smooth, infinite dimensional Banach manifold under a mild condition on the fundamental group. Thus, the Einstein moduli space is unobstructed. The Dirichlet and Neumann boundary maps to data o…
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…
Estimates derived for solutions of Neumann problems on Riemannian manifolds.
problem Gradient and second order estimates for solutions of fully nonlinear elliptic equations on compact Riemannian manifolds.
method Derivation of gradient and second order {\em a priori} estimates.
result Existence and regularity results for solutions of Neumann problems.
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 on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.
problem Finding conditions for constant solutions to Brezis-Nirenberg type problems.
method Developed a study involving nonlinear partial differential equations on spheres and hemispheres with zero Neumann boundary condition.
result Conditions for equations to have only constant solutions.
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
Study elastic Dirichlet-to-Neumann map to uniquely determine metrics and spectral invariants.
problem Uniquely determine metrics of Riemannian manifolds from elastic Dirichlet-to-Neumann maps.
method Explicitly get matrix-valued full symbol for elastic Dirichlet-to-Neumann map, prove metric uniqueness, calculate spectral invariants.
result Elastic Dirichlet-to-Neumann map uniquely determines the metric of a real-analytic Riemannian manifold.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n-Laplacian Liouville equation on the half-space R+n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n=2 and p=n. In this paper we continue the study started in part I (posted). We consider a planar, bounded, m-connected region Ω, and let $\bordΩ$ be its boundary. Let T be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function…
The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.
problem Finding surfaces with commuting boundary Laplacian and Dirichlet-to-Neumann map.
method Investigated 2D submanifolds of R^2, classified surfaces of genus 0 or with k≥3 boundary components.
result One-parameter family of 2D submanifolds with commuting Laplacian and Dirichlet-to-Neumann map.
Study inverse boundary value problem for Monge-Ampère equation on convex domains.
problem Determine a positive source function from the Dirichlet-to-Neumann map for Monge-Ampère equation.
method Recover Hessian as Riemannian metric, prove DN map uniqueness, develop asymptotic expansions, solve nonlocal ∂-equation. result DN map uniquely determines positive source function in convex Euclidean plane domains.
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.
Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.
problem Understanding properties of Sturm-Liouville problems with zero potential.
method Developed simple criteria for assessing properties of regular Sturm-Liouville problems in terms of coefficient functions.
result Proved various properties of Sturm-Liouville problems with zero potential under Neumann boundary conditions.
Anisotropic metric on manifolds uniquely determined by boundary data.
problem Determining Riemannian metrics on compact manifolds from boundary measurements.
method Analysis of the Dirichlet-to-Neumann map for the Laplace-Beltrami operator.
result Riemannian metrics can be uniquely determined up to isometry.
Proves existence of solution to Neumann problem for Hessian equations.
problem Existence of classical solution to Neumann boundary problem for Hessian equations.
method Established a priori derivative estimates up to second order.
result Affirmative answer to a conjecture by N. Trudinger.
Study proves solutions concentrate on a capillary surface in a manifold.
problem Existence of solutions for a nonlinear Neumann boundary condition equation.
method Inspired by Pacard and Ritoré, constructs solutions concentrating to a capillary surface.
result Solutions concentrate asymptotically to a given volume nondegenerate capillary hypersurface.
Study new Ricci bounds for metric measure spaces, preserving properties under time changes.
problem Extend Ricci bounds to non-synthetic spaces and understand their behavior under time changes.
method Introduce distribution-valued lower Ricci bounds BE1(κ,∞), prove equivalence with gradient estimates, and show preservation under time changes. result Distribution-valued Ricci bounds BE1(κ,∞) are preserved under arbitrary time changes and imply sharp gradient estimates. In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…
Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.
problem Smoothness of solutions to nonlinear equations with Neumann boundary conditions on Riemannian manifolds.
method Integral refinement of Bochner's identity.
result Semilinear Calderón-Zygmund type results on Sobolev regularity.
Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.
problem Obstructing higher conformal Yang-Mills equations on conformally compact manifolds.
method Formal asymptotics, Dirichlet-to-Neumann maps, higher transverse derivative boundary operators.
result Obstructing current is the variation of a conformally invariant coefficient in the interior Yang-Mills energy expansion.
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
Study consumption-investment problem in markets with rank-based returns.
problem Consumption-investment problem in markets with rank-based returns.
method Derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem.
result Explicit solutions for unconstrained, open market constraints, and fully invested cases.
Solves Neumann problem on CR manifold boundary.
problem Neumann problem on CR manifold boundary.
method Analyzes CR Yamabe operator and contact forms.
result Solves Neumann problem and finds contact form.
This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges toward…
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
New subdomains found on cylinder and sphere with nonconstant curvatures.
problem Existence of subdomains with specific curvature properties.
method Functional analytic approach to overcome regularity issues.
result First examples of nonhomogeneous, nonisoparametric subdomains.
Study solves a mathematical problem related to elliptic Schroedinger-to-Neumann maps.
problem Solving a Cherrier-Escobar problem for elliptic Schroedinger-to-Neumann maps.
method Using algebraic topological argument of Bahri-Coron, assuming positive eigenvalue and Green function.
result Solvability of the extended problem under specified conditions.
Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.
problem Longtime existence of mean curvature flow in GRW spacetimes.
method Proved longtime existence using perpendicular Neumann boundary condition and null convergence condition.
result Metric of solution is conformal to GRW leaf's metric in asymptotic time.