Proves existence of classical Neumann problems for Hessian equations in uniformly convex domains.
problem Existence of solutions to Neumann problems for Hessian equations.
method Proving existence through uniformly convex domains and Alexandrov-Fenchel inequalities.
result Existence of classical Neumann problems for Hessian equations in uniformly convex domains.
Neumann networks solve inverse problems using truncated Neumann series.
problem Solving ill-posed linear inverse problems in imaging.
method Data-driven approach inspired by Neumann series, truncating the series to solve the problem directly.
result Trained Neumann networks outperform traditional methods and deep learning approaches.
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 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.
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.
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.
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.
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.
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.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
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.
Study proves inequalities for eigenvalues of symmetric domains in space forms.
problem Eigenvalue problem for Laplacian on symmetric domains in space forms.
method Symmetry assumptions on domain, proof of inequalities for eigenvalues.
result Proves Szegö-Weinberger type inequalities for first n positive Neumann eigenvalues.
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.
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.
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 on Neumann eigenvalues controlled by domain isoperimetric ratio.
problem Control the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue.
method Combination of analytical and numerical results, related to Yau's conjecture.
result Neumann eigenvalues are controlled by the isoperimetric ratio of the domain.
Study on learnability of Schatten--von Neumann operators in learning theory.
problem Learnability of Schatten--von Neumann operators in infinite-dimensional settings.
method Adapted representer theorem to convert infinite-dimensional optimization to convex finite-dimensional problem.
result Schatten--von Neumann operators are probably approximately correct (PAC)-learnable via practical convex program for any p<∞. 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.
New geometric conditions ensure compactness of ∂ˉ-Neumann problem.
problem Compactness of ∂ˉ-Neumann operator on specific domains. method Introduced new geometric conditions for a class of domains, proving compactness equivalence to boundary properties.
result Compactness of ∂ˉ-Neumann operator equivalent to boundary lack of analytic varieties. Dirichlet-Neumann duality for Riemannian submersions
problem Spectral geometry of Riemannian submersions
method Summation formula for reciprocal of basic Dirichlet eigenvalues
result Supersymmetric duality between basic Dirichlet and Neumann spectra
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.
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.
Solves a problem posed by Jones linking Hecke groups to subfactor indices.
problem Relation between Hecke groups and subfactor indices in von Neumann algebras.
method Using cluster C*-algebras.
result Solved the problem posed by V. F. R. Jones.
Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.
problem Determining Riemannian metrics from boundary measurements.
method Higher linearization method, integral identities, energy rigidity.
result Metrics on the target manifold are equal if the target is analytic.
Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
problem Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.
method Constructs a metric on a compact manifold to demonstrate the nonexistence of Courant-type bounds.
result Provides a negative answer to the existence of Courant-type nodal domain bounds.
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. Study spectral properties of modified Dirichlet-to-Neumann map on differential forms.
problem Spectral properties of modified Dirichlet-to-Neumann map on differential forms.
method Investigation of self-adjointness and purely discrete spectrum of the operator Λ on coclosed forms.
result Hersch-Payne-Schiffer type inequality relating eigenvalues of Λ to eigenvalues of Hodge Laplacian on the boundary.
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.
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…
A new invariant from smooth 4-manifolds using von Neumann algebras.
problem Constructing a von Neumann algebra from smooth 4-manifolds.
method Geometric construction of von Neumann algebra from smooth structure, preserving unitary equivalence under diffeomorphisms.
result A new invariant of smooth 4-manifolds, the cosmological constant, can be estimated topologically.
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open se…
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.
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.
The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition …
We establish sharp regularity and Fredholm theorems for the \bar{\partial}_b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is n…
Paper generalizes kernel mean embedding to von Neumann-algebra-valued measures.
problem Analyzing complex multivariate distributions and quantum mechanics.
method Generalizes kernel mean embedding to von Neumann-algebra-valued measures in reproducing kernel Hilbert modules.
result Injectivity and universality of the generalized KME are confirmed.
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.
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.
Proven isoperimetric inequality for Witten-Laplacian eigenvalues.
problem Proving isoperimetric inequality for lower order nonzero Neumann eigenvalues of Witten-Laplacian.
method Analytical proof using Euclidean and hyperbolic spaces.
result Strengthens Szegő-Weinberger inequality and covers Xia-Wang's progress.
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.
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.
Researchers solve a formally determined inverse problem in Lorentzian geometry.
problem Determining a Lorentzian metric from boundary measurements of the Dirichlet-to-Neumann map.
method New method using distorted plane wave solutions and geometric, topological, and unique continuation arguments.
result A globally hyperbolic metric agreeing with the Minkowski metric outside a compact set and having the same Dirichlet-to-Neumann map must be the Minkowski metric up to diffeomorphism.
New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
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.
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.
New solutions concentrate on a segment of a boundary in a Neumann problem.
problem Constructing solutions to a singularly perturbed Neumann problem.
method Constructing a new class of solutions with spikes concentrating on a segment of the boundary.
result Mean curvature function acts as friction force in the continuum limit of ODE systems.
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.
problem Finding the minimum value of the first Neumann eigenvalue for convex domains.
method Proved the sharp lower bound using diameter and width.
result Sharp lower bound for the first Neumann eigenvalue established.
Direct proof of Neumann isoperimetric inequality for convex domains.
problem Neumann isoperimetric inequality on convex domains.
method Direct proof using Riemannian manifold properties.
result New proof of Neumann isoperimetric inequality.