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. Smooth solutions up to evolving free boundaries for degenerate equations.
problem Degenerate parabolic equations with evolving free boundaries.
method Smooth short-time existence using linear degenerate equations on a fixed domain.
result Smoothness up to the free boundary for the p-Laplacian evolution equation and α-Gauss curvature flow. Recent research connects Hörmander's old work to modern boundary Laplacian analysis.
problem How close is the Dirichlet-to-Neumann map to the boundary Laplacian?
method Investigates techniques from Hörmander's 1950s manuscript to solve modern boundary Laplacian problems.
result Obtained results for DtN maps on non-smooth boundaries, Helmholtz equation, and differential forms.
Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.
problem Modeling conic Laplacian on \(\mb P^1\) with specific boundary conditions.
method Fourier decomposition, Legendre equations, gluing map, Friedrichs spectrum, Weyl function.
result Explicit computation of eigenfunctions and \(S\)-matrix.
New method generalizes eigenvalue inequality to surfaces with boundaries.
problem Eigenvalue inequality for surfaces with boundaries.
method Generalized Rohleder's approach to differential forms, presenting Hodge-Laplacian spectrum.
result Obtained inequality for eigenvalues of Hodge-Laplacian and Dirichlet problems.
The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an e…
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.
The paper computes a residue density for a specific Laplacian on compact manifolds.
problem Computing a noncommutative residue density for a complex Laplacian.
method Explicit computation of the noncommutative residue density associated with equivariant twisted Bismut Laplacian with torsion.
result Proves equivariant twisted Kastler-Kalau-Walze type theorems with torsion on compact manifolds with boundary.
BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.
problem Comparing combinatorial and Hodge Laplacians for discrete data.
method Introducing Boundary-Induced Graph (BIG) Laplacians using DEC.
result BIG Laplacian eigenvalues converge to Hodge Laplacian for simple shapes.
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.
New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.
problem Positive scalar curvature metrics on manifolds with boundary that cannot be extended.
method Analytic techniques related to the prescribed scalar curvature problem in conformal geometry.
result Obstruction to positivity of conformal Laplacians given by a real-valued ξ-invariant.
Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.
problem Determining geometric data from boundary symbol of connection Laplacian.
method Analyze symbol of Dirichlet-to-Neumann operator associated with connection Laplacian.
result Geometric data on boundary and normal derivatives are determined by symbol.
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.
Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.
problem Investigate the first Robin eigenvalue of the p-Laplacian on Riemannian manifolds.
method Prove eigenvalue comparison theorems and establish sharp bounds.
result Sharp lower bounds for the first Robin eigenvalue of the p-Laplacian.
The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.
problem Analyzing boundary conditions and thin-shell limits for viscous operators on arbitrary smooth hypersurfaces.
method Decomposing the ambient Bochner Laplacian into intrinsic and radial pieces, proving results for stress-free and Hodge boundary conditions.
result Universal thin-shell limits for viscous operators on arbitrary smooth hypersurfaces, including stress-free and Hodge boundary conditions.
We consider a notion of conservation for the heat semigroup associated to a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitabl…
Lower bounds for eigenvalues on manifolds with boundary conditions.
problem Eigenvalue bounds for manifolds with boundary conditions.
method Proving lower bounds for the first non-trivial eigenvalue using Cheeger-type constants.
result Results in the spirit of Cheeger's inequality for manifolds with boundary conditions.
In manifold learning, algorithms based on graph Laplacians constructed from data have received considerable attention both in practical applications and theoretical analysis. In particular, the convergence of graph Laplacians obtained from sampled data to certain continuous operators has become an active research topic…
Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.
problem Approximating eigenvalues of manifolds with cylindrical boundaries.
method Using truncated graph Laplacians constructed from (ε,ρ)-proximity graphs. result Eigenvalues of truncated graph Laplacians converge to Dirichlet eigenvalues of the Laplace-Beltrami operator.
In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the…
The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary co…
The paper finds upper bounds for eigenvalues of a weighted Laplacian.
problem Finding bounds for eigenvalues of a weighted Laplacian.
method Analyzing the operator \( L_h \) with Neumann boundary conditions.
result Existence of upper bounds for eigenvalues of the weighted Laplacian.
Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
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.
Magnitude study on manifolds using fractional Laplacian.
problem Magnitude invariant of compact metric spaces via fractional Laplacian.
method Semiclassical analysis of nonlocal boundary value problem related to fractional Laplacian.
result Asymptotic expansion of magnitude in terms of curvature invariants.
We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boun…
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
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.
Study on p-Laplacian problems with critical exponent, focusing on existence of solutions.
problem Existence of least energy solutions for nonlinear p-Laplacian problems with critical exponent.
method Proving existence of solutions through critical point theory and variational methods.
result Significant difference in existence results between p-Laplacian and Laplacian cases.
A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.
problem Ambiguity in viscous operator choice for Navier-Stokes equations on Riemannian manifolds.
method Kinematic construction of strain rate from Lie-dragged vectors, excluding Hodge Laplacian due to antisymmetric part.
result Kinematic selection uniquely identifies the deformation Laplacian, resolving analytical obstructions.
We present gluing formulas for zeta regularized determinants of Dolbeault laplacians on Riemann surfaces. These are expressed in terms of determinants of associated operators on surfaces with boundary satisfying local elliptic boundary conditions. The conditions are defined using the additional structure of a framing, …
Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted p-Laplacian with 1<p<∞ on a compact Bakry-Émery manifold (Mn,g,f), without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+…
Derives integral formula for differential forms on compact spaces with applications.
problem Integral formula for differential forms on compact spaces with boundary.
method Derives a weighted Reilly type integral formula.
result Lower bounds for spectrum and eigenvalues of differential forms.
For a bounded domain Ω with a piecewise smooth boundary in an n-dimensional Euclidean space Rn, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Lap…
Uniqueness proof for Calderón's problem on real-analytic vector bundles.
problem Recovering topology and geometry from boundary measurements of vector bundles.
method Real-analyticity assumption for uniqueness proof.
result Topology and geometry of vector bundles can be recovered from boundary measurements.
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various fo…
The paper explores inequalities on weighted Riemannian manifolds with boundary.
problem Developing inequalities on weighted Riemannian manifolds with boundary.
method Using a Reilly type integral formula associated with the φ-Laplacian.
result Provided inequalities of Brascamp-Lieb type and Colesanti type.
I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued…
The paper studies the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with specific boundary conditions.
problem Analyzing the asymptotic behavior of a zeta-regularized determinant on a cuspidal end.
method Specifying and analyzing the behavior of the pseudo-Laplacian with Alvarez--Wentworth boundary conditions.
result Finding the asymptotic behavior of the zeta-regularized determinant for various parameter values.
Upper bounds found for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
problem Finding upper bounds for eigenvalues of weighted Steklov and (p,q)-Laplacian problems.
method Proving upper bounds using the weighted p-Laplace operator and (p,q)-Laplacian on submanifolds. result Reilly-type upper bounds for the first eigenvalues of Steklov and (p,q)-Laplacian problems.
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.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between…
Paper extends noncompact Llarull's theorem to manifolds with boundary.
problem Finding upper bounds for scalar curvature infimum in noncompact manifolds.
method Using deformed Dirac operators to relax boundary conditions.
result Upper bound for scalar curvature infimum in terms of Laplacian spectrum.
We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energ…
Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zeta-determinant given by Burghelea, Friedlander and Kappeler with some assumptions, we prov…