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For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.
Given a smooth compact hypersurface
$M$
with boundary
$\unicode[STIX]{x1D6F4}=\unicode[STIX]{x2202}M$
, we prove the existence of a sequence
$M_{j}$
of hypersurfaces with the same boundary as
$M$
, such that each Steklov eigenvalue
$\unicode[STIX]{x1D70E}_{k}(M_{j})$
tends to zero as
$j$
tends to infinity. The hypersurfaces
$M_{j}$
are obtained from
$M$
by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of
$M$
, while the principal curvatures of the boundary remain unchanged.
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form
$\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$
, and
$\unicode[STIX]{x1D6FC}$
lies between
$-2\unicode[STIX]{x1D70B}$
and
$2\unicode[STIX]{x1D70B}$
. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.
The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.
We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux, we establish a Hardy-type inequality. In the regime with an infinite discrete spectrum, we obtain sharp spectral asymptotics with a refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.
In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator
$\unicode[STIX]{x039B}$
is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of
$\unicode[STIX]{x039B}$
and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of
$\unicode[STIX]{x039B}$
are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of
$p$
-forms on the boundary of a
$2p+2$
-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.
We study bound states in weakly deformed and heterogeneous waveguides, and compare analytical predictions using a recently developed perturbative method with precise numerical results for three different configurations: a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken strip. We have found excellent agreement between the analytical and numerical results in all the examples; this provides a numerical verification of the analytical approach.
In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.
Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface
$S$
of finite type and Euler characteristic
$\unicode[STIX]{x1D712}(S)<0$
has at most
$-\unicode[STIX]{x1D712}(S)$
small eigenvalues.
We apply a mean-value inequality for positive subsolutions of the
$f$
-heat operator, obtained from a Sobolev embedding, to prove a nonexistence result concerning complete noncompact
$f$
-maximal spacelike hypersurfaces in a class of weighted Lorentzian manifolds. Furthermore, we establish a new Calabi–Bernstein result for complete noncompact maximal spacelike hypersurfaces in a Lorentzian product space.
We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.
The energy of a type
$\text{II}$
superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a special (thermodynamic) limit of a non-linear energy. In this note, we give a new formula for this reference energy. In particular, we obtain it as a special limit of a linear energy defined over configurations normalized in the
${{L}^{4}}$
-norm.
In this work we study the homogenisation problem for nonlinear elliptic equations involving
$p$
-Laplacian-type operators with sign-changing weights. We study the asymptotic behaviour of variational eigenvalues which consist of a double sequence of eigenvalues. We show that the
$k$
th positive eigenvalue goes to infinity when the average of the weights is nonpositive, and converges to the
$k$
th variational eigenvalue of the limit problem when the average is positive for any
$k\geq 1$
.
In this paper, we study a finite connected graph which admits a quasi-monomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasi-monomorphism.
We prove that on any compact manifold
$M^{n}$
with boundary, there exists a conformal class
$C$
such that for any Riemannian metric
$g\in C$
of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies
${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$
. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of
$(M,C)$
is
$\text{Vol}(S^{n},g_{\text{can}})$
, and that the Friedlander–Nadirashvili invariant and the Möbius volume of
$M$
are equal to those of the sphere. If
$M$
is a domain in a space form,
$C$
is the conformal class of the canonical metric.
In this article the instabilities appearing in a liquid layer are studied numerically by means of the linear stability method. The fluid is confined in an annular pool and is heated from below with a linear decreasing temperature profile from the inner to the outer wall. The top surface is open to the atmosphere and both lateral walls are adiabatic. Using the Rayleigh number as the only control parameter, many kind of bifurcations appear at moderately low Prandtl numbers and depending on the Biot number. Several regions on the Prandtl-Biot plane are identified, their boundaries being formed from competing solutions at codimension-two bifurcation points.
Many engineering structures exhibit frequency dependent characteristics and analyses of these structures lead to frequency dependent eigenvalue problems. This paper presents a novel perturbative iteration (PI) algorithm which can be used to effectively and efficiently solve frequency dependent eigenvalue problems of general frequency dependent systems. Mathematical formulations of the proposed method are developed and based on these formulations, a computer algorithm is devised. Extensive numerical case examples are given to demonstrate the practicality of the proposed method. When all modes are included, the method is exact and when only a subset of modes are used, very accurate results are obtained.
Let
$D$
be a connected bounded domain in
${{\mathbb{R}}^{n}}$
. Let
$0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $
be the eigenvalues of the following Dirichlet problem:
We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.
For a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.