Weighted Steklov Eigenvalues

• Weighted Steklov eigenvalues are boundary spectral quantities defined via weighted mass conditions in harmonic and nonlinear problems, offering a framework to link geometry with spectral analysis.
• They are characterized through variational principles, Dirichlet-to-Neumann mappings, and pseudodifferential formulations, with applications in optimization over boundary densities.
• Recent work leverages homogenization, finite element verification, and nonlinear generalizations to derive rigorous spectral bounds and study extremal density issues in complex domains.

Weighted Steklov eigenvalues are boundary spectral quantities for problems in which the spectral parameter remains in a Steklov-type boundary condition while the boundary mass, the ambient measure, or the elliptic operator is weighted. In the most standard formulation one considers

$$\begin{cases} \Delta u=0 & \text{in }\Omega,\ \partial_\nu u=\lambda \rho u & \text{on }\partial\Omega, \end{cases}$$

with $\rho\in L^\infty(\partial\Omega)$, $\rho\ge 0$; in parallel, the literature also studies weighted manifold versions with drift Laplacians, Wentzell extensions, nonlinear $p$- and $(p,q)$-analogues, and homogenized limits in which oscillatory or concentrating masses produce effective weighted boundary or interior spectral problems (Girouard et al., 2014, Batista et al., 2015, Salort, 2020, Menovschikov, 10 May 2026). The subject therefore sits at the intersection of spectral geometry, boundary pseudodifferential operators, homogenization, rearrangement optimization, and verified numerical analysis.

1. Boundary-density formulations and nonlinear generalizations

The boundary-density model on a bounded Lipschitz domain $\Omega\subset \mathbb R^N$ takes the form

$\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$

with $\rho\in L^\infty(\partial\Omega)$, $\rho\ge 0$, $\rho\not\equiv 0$. Its weak formulation is

$\rho\in L^\infty(\partial\Omega)$0

and the spectrum is discrete,

$\rho\in L^\infty(\partial\Omega)$1

The traces of eigenfunctions are orthonormal in $\rho\in L^\infty(\partial\Omega)$2,

$\rho\in L^\infty(\partial\Omega)$3

and nontrivial modes satisfy weighted boundary mean zero,

$\rho\in L^\infty(\partial\Omega)$4

The variational characterization is correspondingly written on

$\rho\in L^\infty(\partial\Omega)$5

with Rayleigh quotient

$\rho\in L^\infty(\partial\Omega)$6

and weighted orthogonality constraints (Kao et al., 26 Sep 2025).

The same structural pattern appears in the survey treatment of generalized Steklov boundary conditions

$\rho\in L^\infty(\partial\Omega)$7

where $\rho\in L^\infty(\partial\Omega)$8, $\rho\in L^\infty(\partial\Omega)$9, and $\rho\ge 0$0 is interpreted as a boundary mass density for a vibrating membrane. The survey also singles out the sloshing problem as a degenerate case with

$\rho\ge 0$1

which shows that vanishing weights are intrinsic to the theory rather than exceptional (Girouard et al., 2014).

Nonlinear weighted Steklov problems replace the Laplacian by quasilinear operators while preserving boundary weighting. For the $\rho\ge 0$2-Laplacian one studies

$\rho\ge 0$3

with positive boundary weight $\rho\ge 0$4, and the eigenvalues are defined by a Krasnosel'skii genus minimax principle (Salort, 2020). A different nonlinear branch is the weighted $\rho\ge 0$5-Steklov problem

$\rho\ge 0$6

whose first nontrivial eigenvalue is the variational level

$\rho\ge 0$7

with nonlinear boundary orthogonality

$\rho\ge 0$8

removing constants (Menovschikov, 10 May 2026).

2. Variational and operator-theoretic structure

The weighted Steklov problem is naturally a generalized eigenvalue problem for an interior energy form and a boundary mass form. In the standard Hilbert setting one writes

$\rho\ge 0$9

so that

$p$0

A decisive point is that the boundary form is only positive semidefinite on $p$1: in the unweighted case its kernel is exactly $p$2, and for a weighted form with $p$3 a.e. the same identification persists, whereas if $p$4 vanishes on part of the boundary the kernel enlarges. The abstract framework allowing the second bilinear form to be only positive semidefinite was developed precisely to treat this Steklov structure (You et al., 2018).

The Dirichlet-to-Neumann viewpoint is equally fundamental. For the classical Steklov problem, the boundary trace $p$5 of a harmonic function satisfies

$p$6

where $p$7 is the Dirichlet-to-Neumann operator. The weighted boundary condition

$p$8

is therefore a generalized eigenvalue problem on the boundary. When $p$9, the survey notes the formal rewritings

$(p,q)$0

which place weighted Steklov theory inside the pseudodifferential analysis of first-order elliptic boundary operators (Girouard et al., 2014). This also clarifies why the leading asymptotics are controlled by the boundary geometry together with the weighted boundary measure.

A more explicit operator model appears in optimization over boundary densities. If $(p,q)$1 denotes the Dirichlet-to-Neumann operator for the unweighted problem on mean-zero traces and $(p,q)$2 denotes multiplication by $(p,q)$3, then

$(p,q)$4

is a compact self-adjoint Hilbert–Schmidt operator, and its nonzero eigenvalues are exactly

$(p,q)$5

In a Steklov basis $(p,q)$6, with $(p,q)$7, the matrix representation is

$(p,q)$8

This formulation yields perturbation formulas such as

$(p,q)$9

for simple eigenvalues and a matrix first-order splitting formula for multiple eigenvalues (Kao et al., 19 Sep 2025).

3. Homogenization and realization principles

A major theme in recent work is that weighted spectral data can arise as effective limits of unweighted problems with fine geometric or material microstructure. One mechanism removes many small geodesic balls from a closed manifold $\Omega\subset \mathbb R^N$0. If the radii are chosen by

$\Omega\subset \mathbb R^N$1

then the induced boundary measures satisfy

$\Omega\subset \mathbb R^N$2

and the ordinary Steklov eigenvalues on the perforated domains converge: $\Omega\subset \mathbb R^N$3 where the limit problem is the weighted Laplace equation

$\Omega\subset \mathbb R^N$4

The paper stresses that the limit problem is not a weighted Steklov problem but a weighted Laplace eigenvalue problem on the closed manifold; nevertheless, the construction provides a geometric realization principle in which ordinary Steklov spectra encode weighted interior spectral data (Girouard et al., 2020).

A second mechanism keeps the domain fixed and oscillates the boundary density. For the weighted $\Omega\subset \mathbb R^N$5-Steklov problem with periodic weights

$\Omega\subset \mathbb R^N$6

the weak-* limit is the cell average

$\Omega\subset \mathbb R^N$7

and the eigenvalues satisfy

$\Omega\subset \mathbb R^N$8

In the periodic case the convergence is quantitative: for every $\Omega\subset \mathbb R^N$9,

$\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$0

and for $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$1,

$\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$2

The proof combines boundary oscillatory integral estimates with explicit $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$3 bounds for Steklov eigenfunctions (Salort, 2020).

A third realization principle passes from concentrating interior mass to boundary mass. On admissible domains obtained from the unit ball by trace-compatible Sobolev homeomorphisms, the first nontrivial weighted $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$4-Neumann eigenvalue with concentrating bulk weight $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$5 converges to the corresponding weighted $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$6-Steklov eigenvalue with boundary weight $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$7: $\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$8 Equivalently, the best constants in weighted Poincaré inequalities converge to the best constants in weighted trace inequalities, and in the subcritical trace range one has the quantitative estimate

$\begin{cases} \Delta u=0 & \text{in } \Omega,\[2mm] \dfrac{\partial u}{\partial \mathbf n}=\lambda\,\rho\,u & \text{on } \partial\Omega, \end{cases}$9

Normalized minimizers converge, up to subsequences, strongly in $\rho\in L^\infty(\partial\Omega)$0 to Steklov minimizers (Menovschikov, 10 May 2026).

4. Extremal densities, partitions, and isoperimetric inequalities

Optimization with respect to boundary density has developed in two closely related directions. The first arises from mixed Steklov–Neumann boundary partitions. If $\rho\in L^\infty(\partial\Omega)$1 and $\rho\in L^\infty(\partial\Omega)$2, the characteristic function $\rho\in L^\infty(\partial\Omega)$3 is relaxed to a density $\rho\in L^\infty(\partial\Omega)$4 with

$\rho\in L^\infty(\partial\Omega)$5

The weighted problem

$\rho\in L^\infty(\partial\Omega)$6

has extremal values

$\rho\in L^\infty(\partial\Omega)$7

and both are attained. For $\rho\in L^\infty(\partial\Omega)$8, there exists a bang-bang minimizer $\rho\in L^\infty(\partial\Omega)$9, and $\rho\ge 0$0 is convex on $\rho\ge 0$1. The local optimality conditions are threshold rules for $\rho\ge 0$2; in particular, on any interval where an optimal density takes intermediate values, that sum must be constant. On the disk, the constant density is a maximizer for the relaxed first-eigenvalue problem, while for $\rho\ge 0$3 the maximizing density is numerically nonconstant, and rapidly oscillating Steklov/Neumann partitions approximate the relaxed maximizer through homogenization (Kao et al., 19 Sep 2025).

A second line fixes the domain and optimizes over box-constrained densities

$\rho\ge 0$4

with $\rho\ge 0$5. For every $\rho\ge 0$6, minimizers and maximizers of $\rho\ge 0$7 exist. Minimizers can be chosen bang-bang,

$\rho\ge 0$8

and are characterized by a bathtub threshold condition involving

$\rho\ge 0$9

whenever $\rho\not\equiv 0$0 has multiplicity $\rho\not\equiv 0$1. Maximizers need not be bang-bang. The paper proves that the maps

$\rho\not\equiv 0$2

are generally neither convex nor concave, and introduces the cluster-sum surrogate

$\rho\not\equiv 0$3

with Fréchet derivative

$\rho\not\equiv 0$4

On the disk, the minimization problem has infinitely many minimizers generated by rotational symmetry, while for $\rho\not\equiv 0$5 there are infinitely many distinct maximizers not generated by symmetry, coming from conformal automorphisms of the disk (Kao et al., 26 Sep 2025).

Geometric extremal theory also includes weighted isoperimetric inequalities. For the radial double-density problem

$\rho\not\equiv 0$6

with

$\rho\not\equiv 0$7

the centered ball has explicit first nontrivial eigenvalue

$\rho\not\equiv 0$8

and under balance and isotropy assumptions it maximizes $\rho\not\equiv 0$9 and minimizes the harmonic mean of the first $\rho\in L^\infty(\partial\Omega)$00 positive eigenvalues among domains with fixed weighted volume (Brock et al., 14 Oct 2025). For Steklov-type eigenvalues of the Witten-Laplacian

$\rho\in L^\infty(\partial\Omega)$01

a Brock-type inequality under fixed weighted volume shows that the centered Euclidean or hyperbolic ball maximizes the first nonzero eigenvalue when $\rho\in L^\infty(\partial\Omega)$02 is radial, non-increasing, and concave (Mao et al., 2024).

5. Weighted manifolds, curvature bounds, and Steklov-type geometry

On a compact oriented weighted Riemannian manifold with boundary

$\rho\in L^\infty(\partial\Omega)$03

the weighted Steklov problem is

$\rho\in L^\infty(\partial\Omega)$04

The first nonzero eigenvalue has the weighted Rayleigh characterization

$\rho\in L^\infty(\partial\Omega)$05

The fundamental analytic tool is a weighted Reilly formula involving the Bakry–Émery curvature tensor $\rho\in L^\infty(\partial\Omega)$06 and weighted mean curvature

$\rho\in L^\infty(\partial\Omega)$07

On surfaces, if

$\rho\in L^\infty(\partial\Omega)$08

then the first nonzero weighted Steklov eigenvalue satisfies the sharp Escobar-type estimate

$\rho\in L^\infty(\partial\Omega)$09

with equality if and only if $\rho\in L^\infty(\partial\Omega)$10 is the Euclidean disk of radius $\rho\in L^\infty(\partial\Omega)$11 and $\rho\in L^\infty(\partial\Omega)$12 is constant. In higher dimensions the same paper derives upper and lower bounds under assumptions on $\rho\in L^\infty(\partial\Omega)$13, $\rho\in L^\infty(\partial\Omega)$14, and the second fundamental form (Batista et al., 2015).

A closely related generalization is the weighted Wentzell problem

$\rho\in L^\infty(\partial\Omega)$15

for a compact metric measure space $\rho\in L^\infty(\partial\Omega)$16. When $\rho\in L^\infty(\partial\Omega)$17, this becomes the natural weighted analogue of the Steklov problem. Under

$\rho\in L^\infty(\partial\Omega)$18

the first nonzero weighted Wentzell eigenvalue satisfies a sharp upper bound and an explicit lower bound; in the special case $\rho\in L^\infty(\partial\Omega)$19 and constant $\rho\in L^\infty(\partial\Omega)$20, the latter recovers Escobar’s lower bound for the first classical Steklov eigenvalue (Zhao et al., 2018).

Extrinsic geometry enters the weighted theory through immersed submanifolds. For a compact manifold with boundary, isometrically immersed in $\rho\in L^\infty(\partial\Omega)$21, the weighted $\rho\in L^\infty(\partial\Omega)$22-Steklov problem

$\rho\in L^\infty(\partial\Omega)$23

admits Reilly-type upper bounds expressed by the mean curvature vector $\rho\in L^\infty(\partial\Omega)$24, the density gradient $\rho\in L^\infty(\partial\Omega)$25, and, in tensorial versions, higher-order mean curvatures. In the unweighted equality case these bounds force $\rho\in L^\infty(\partial\Omega)$26 and a minimal immersion into a Euclidean ball; in the weighted equality case they imply self-shrinker geometry (Azami, 2022).

6. Verified computation and finite element lower bounds

The boundary-supported nature of the mass form makes weighted Steklov discretization analytically different from standard interior-mass eigenproblems. An abstract framework for semidefinite boundary forms was developed in the study of guaranteed eigenvalue bounds for the Steklov problem, where

$\rho\in L^\infty(\partial\Omega)$27

and $\rho\in L^\infty(\partial\Omega)$28 is only positive semidefinite on $\rho\in L^\infty(\partial\Omega)$29. The core lower-bound theorem states that if

$\rho\in L^\infty(\partial\Omega)$30

then

$\rho\in L^\infty(\partial\Omega)$31

Using the Crouzeix–Raviart method, explicit interpolation constants, interval arithmetic, Behnke’s method, and INTLAB, the paper produces rigorous two-sided enclosures on square and L-shaped domains. It also observes that the same abstract machinery applies almost verbatim to weighted boundary bilinear forms

$\rho\in L^\infty(\partial\Omega)$32

when $\rho\in L^\infty(\partial\Omega)$33 and the trace compactness assumptions remain valid (You et al., 2018).

A different lower-bound strategy uses the weak Galerkin method for the unweighted Steklov problem

$\rho\in L^\infty(\partial\Omega)$34

The discrete eigenproblem

$\rho\in L^\infty(\partial\Omega)$35

is built from a weak gradient, a stabilizer with $\rho\in L^\infty(\partial\Omega)$36, and a nonconforming space. The method yields arbitrary high-order lower bound estimates, asymptotic lower bounds

$\rho\in L^\infty(\partial\Omega)$37

and, after modifying the stabilizer, guaranteed lower bounds under the criterion

$\rho\in L^\infty(\partial\Omega)$38

Because the scheme is written in terms of an energy form $\rho\in L^\infty(\partial\Omega)$39 and a boundary mass form $\rho\in L^\infty(\partial\Omega)$40, its weighted extension is structurally natural, although weighted boundary projections and weighted trace constants would have to be rederived (Li et al., 2023).

Conforming finite element lower bounds have also been obtained by coupling Liu’s projection-error framework with a hypercircle estimate for a nonhomogeneous Neumann problem. For the generalized eigenproblem

$\rho\in L^\infty(\partial\Omega)$41

the explicit source-problem estimate

$\rho\in L^\infty(\partial\Omega)$42

leads again to

$\rho\in L^\infty(\partial\Omega)$43

The same mechanism extends formally to a weighted form

$\rho\in L^\infty(\partial\Omega)$44

by replacing the boundary projection and source term with weighted analogues (Nakano et al., 2020).

Weighted Steklov eigenvalues therefore form a spectrum of models rather than a single equation. Across these models, several stable principles recur: the denominator of the Rayleigh quotient is a weighted boundary mass; the natural operator is a generalized boundary spectral problem for the Dirichlet-to-Neumann map; homogenization can create effective weights from oscillatory boundaries or concentrating masses; extremal densities often satisfy threshold rules but need not be bang-bang on the maximization side; and curvature, conformal structure, and finite element verification all interact directly with the weighted boundary measure.