Study on second Robin eigenvalue for Laplacian on manifolds.
problem Maximizing the second Robin eigenvalue for geodesic balls in nonpositively curved space forms.
method Comparison theorem and maximization analysis for the second Robin eigenvalue.
result Geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
The paper compares heat kernels on manifolds with Robin boundary conditions.
problem Comparing heat kernels on manifolds with different boundary conditions.
method Proving comparison theorems for heat kernels on geodesic balls and minimal submanifolds.
result Eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds.
Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.
problem Proving inequalities for eigenvalues of weighted Laplacian.
method Analyzing Robin boundary conditions on Rn and Hn. result Optimal domain for eigenvalues is a ball centered at the origin.
Paper finds eigenvalue bounds for hyperbolic space domains.
problem Finding eigenvalue bounds for Robin Laplacian in hyperbolic space.
method Lower and upper bounds derived for eigenvalues.
result Geodesic ball maximizes eigenvalue in negative boundary parameter case.
Upper bounds for second Robin eigenvalue on Riemannian surfaces.
problem Bounding the second Robin eigenvalue of Schrödinger operators on Riemannian surfaces.
method Geometric upper bound via Hersch balancing argument on capped surfaces.
result Sharp geometric restrictions for minimal surfaces in negatively curved manifolds.
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the p-Laplacian on a compact Riemannian manifold M with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of…
Paper proves inequality for p-Laplacian eigenvalues on curved spaces.
problem Eigenvalue inequalities for p-Laplacian on curved manifolds.
method Robin boundary conditions, lower Ricci bounds, positive asymptotic volume ratio.
result Bossel-Daners inequality extends to compact submanifolds.
On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues (λ_k(τ,α))_k of the magnetic Laplacian with the Robin boundary conditions. Here, τ is a positive number that defines the Robin condition and α is a real differential 1-form on M that represents the magnetic field. We e…
Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.
problem Counting negative eigenvalues of magnetic Pauli operator.
method Reduction to boundary Dirac operator, Atiyah-Patodi-Singer index theory, Benjamin-Ono equation conservation law.
result New formula on the number of eigenvalues of magnetic Neumann Laplacian in semi-classical limit.
Study on biharmonic Steklov problem on differential forms.
problem Characterize and estimate eigenvalues of biharmonic Steklov problem.
method Introduce boundary conditions, prove properties, derive inequalities.
result Characterize smallest eigenvalue and prove spectrum properties.
Upper bounds for Steklov eigenvalues on manifolds with boundary.
problem Investigating upper bounds for the spectrum of the Steklov-type operator on Riemannian manifolds with boundary.
method Extending the Fraser-Schoen estimate to higher Steklov eigenvalues, using relative conformal volume and isoperimetric ratio.
result Established bounds for the Steklov eigenvalues in terms of relative conformal volume and isoperimetric ratio.
Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dμ g M f dμ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a dif…
Detailed proofs and extensions of eigenvalue multiplicity bounds for spheres and plane domains.
problem Bounding the multiplicity of eigenvalues for Riemannian surfaces.
method Detailed proofs, combinatorial analysis of nodal domains, and Euler's inequality.
result Upper bounds on eigenvalue multiplicities extended to Robin boundary conditions.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
We consider the Laplacian with attractive Robin boundary conditions, \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \] in a class of bounded smooth domains Ω∈Rν; here n is the outward unit normal and α>0 is a constant. We show that for each j∈N and $α\to+\in…
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.
Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.
problem Comparing solutions of Poisson equations on Riemannian manifolds with Robin boundary.
method Using Schwarz rearrangement and isoperimetric inequalities.
result Extends results on Poisson equations with Ric≥(n−1)κ. Proposes a flexible tournament design combining knockout and round-robin.
problem Designing a tournament that eliminates participants linearly.
method Combines knockout and round-robin structures for flexible elimination.
result Flexible tournament design can eliminate participants linearly.
New heat dispersion laws established for smooth compact manifolds.
problem Understanding heat dispersion in smooth compact manifolds.
method Established new heat dispersion laws through Theorem 1.1 and explored them further with Propositions 3.1 and 3.2.
result New heat dispersion laws for smooth compact manifolds.
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.
Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.
problem Characterizing umbilical hypersurfaces in space forms.
method Using a Serrin-type partially overdetermined problem with inhomogeneous Robin boundary condition.
result Any contact angle θ ∈ (0, π) can be achieved, generalizing previous results.
We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …
Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.
problem Analyzing self-adjoint extensions of Dolbeault Laplacians on Riemann surfaces.
method Defined ζ-regularized determinants, introduced Robin mass, derived comparison formulas. result Explicit expressions for Robin mass in spinor bundles and scalar cases.
Study proves a sharp upper bound for the zero set area of a static manifold's potential.
problem Proving a sharp upper bound for the zero set area of a static manifold's potential.
method Proved a rigidity theorem for the Euclidean closed unit ball in R^3.
result Sharp upper bound for the area of the zero set of the potential.
Paper solves capillary Lp-Minkowski problem for p>1.
problem Finding capillary convex bodies with prescribed Lp-surface area measures.
method Reduction to Monge-Ampère equation with Robin boundary condition.
result Solved capillary Lp-Minkowski problem in smooth category for p>1.
For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-…
The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.
problem The study investigates the critical hyperbolic catenoid family and its geometric and spectral properties.
method The approach involves analyzing the critical hyperbolic catenoid family, identifying parameter-criticality, and studying the Robin spectrum.
result The paper proves that at a parameter-critical value a♯, the Robin nullity of Σa♯ is at least 3, with an additional kernel element in mode k=0. Critical spherical catenoids have Robin nullity and asymptotic radius determined.
problem Analyzing the critical spherical catenoids in hyperbolic space.
method Analytic results using Sturm-Liouville theory, Beta-function evaluation, and Laplace asymptotic analysis.
result Robin nullity and asymptotic radius of the critical spherical catenoid are determined.
We study the Obata equation with Robin boundary condition ∂ν∂f+af=0 on manifolds with boundary, where a∈R∖{0}. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}. Compared with their results, the si…
Paper solves capillary Orlicz-Minkowski problem with new inequalities.
problem Finding capillary convex bodies with prescribed Orlicz surface area measures.
method Continuity method and inequalities to solve the capillary even Orlicz-Minkowski problem.
result Volume-normalized smooth solutions and inequalities established.
New methods solve complex PDEs with mixed boundary conditions.
problem Solving inhomogeneous Robin type boundary value problems for linear PDEs.
method Odd and even Hilbert transforms.
result Non-standard solutions to various PDEs in finance, stochastic analysis, etc.
Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.
problem Computing the Morse index of the critical catenoid
method Physics-Informed Neural Network (PINN) enforces parity and eigenvalue as trainable parameters
result Returns eigenvalues within 10−6 to 10−4 of exact values The paper solves a specific Minkowski problem for capillary hypersurfaces.
problem Finding capillary convex bodies with prescribed dual curvature measures.
method Reduction to a Monge-Ampère type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution for θ∈(0,2π). Improved fourth-order compact scheme for option valuation with Robin boundary condition.
problem Lower convergence rates in numerical methods for American options.
method High-order compact scheme, Robin boundary condition, coupled nonlinear PDEs.
result Fourth-order convergence rate achieved without mesh refinement.
Given a three dimensional pseudo-Einstein CR manifold (M,T1,0M,θ), we study the existence of a contact structure conformal to θ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion …
New estimator stabilizes higher-order influence functions for stable statistical inference.
problem Numerical instability in estimating inverse population Gram matrix.
method Proposes a new stabilized higher-order estimator without sample splitting.
result Stabilized estimator exhibits more stable performance and similar statistical guarantees.
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k-th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
Let M be a complete connected Riemannian manifold with boundary $\pp M$, Q a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a C1-vector field Z on M. By using the reflecting diffusion process generated by L and its local time on the boundary, a probabilistic formula is presented for the semigro…
New estimator stabilizes higher-order influence functions for bilinear forms.
problem Stability issues in estimating bilinear forms using higher-order influence functions.
method Proposes a new stabilized higher-order estimator for a class of bilinear forms without sample splitting.
result New estimator exhibits more stable finite-sample performance compared to the empirical higher-order estimator.
The paper studies a 1D diffusion equation with nonlinear Robin boundary conditions and finds conditions for global and finite time blow-up or blow-down.
problem Investigating the behavior of solutions to a specific diffusion equation with nonlinear Robin boundary conditions.
method Analyzing the Ricci flow on a cylinder and applying it to the diffusion equation.
result Conditions for global and finite time blow-up or blow-down of solutions.
Classical learning assumes the learner is given a labeled data sample, from which it learns a model. The field of Active Learning deals with the situation where the learner begins not with a training sample, but instead with resources that it can use to obtain information to help identify the optimal model. To better u…
We study an extension of the classic stochastic multi-armed bandit problem which involves multiple plays and Markovian rewards in the rested bandits setting. In order to tackle this problem we consider an adaptive allocation rule which at each stage combines the information from the sample means of all the arms, with t…
New method reveals corners of drum shapes.
problem Determining the shape of drum corners from its sound.
method Locality principle and calculations of heat kernels.
result Corners are spectral invariants of the Laplacian.
Solves boundary Yamabe problem with minimal boundary scenario.
problem Existence of solutions to a conformal metric equation with constant scalar curvature.
method Iteration schemes and perturbation methods to construct sub-solutions and super-solutions.
result Complete solution for three cases based on the sign of the first eigenvalue of the conformal Laplacian.
Causal inference from observational data is hard due to discontinuous causal effects.
problem Causal inference from observational data is hard due to discontinuous causal effects.
method The problem is tackled by showing that many standard point estimates can be read as point summaries of multimodal distributions over the space of structural causal models.
result Many standard point estimates can be discontinuous summaries, while explicit posterior means and medians are continuous.