Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −ΔΘ,Ω in L²(Ω;dnx) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω⊂Rn, n∈N, n?2. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems.     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

Hardy inequality and asymptotic eigenvalue distribution for discrete Laplacians

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.     

Eigenvalues of Robin Laplacians in infinite sectors

For , let denote the infinite planar sector of opening 2α, and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as and admits a full asymptotic expansion in powers of α². The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.     

Weighted Hardy inequalities

For bounded Lipschitz domains D in it is known that if 1<p<∞, then for all β[0,β₀), where β₀=p−1>0, there is a constant c<∞ with

for all . We show that if D is merely assumed to be a bounded domain in that satisfies a Whitney cube-counting condition with exponent λ and has plump complement, then the same inequality holds with β₀ now taken to be . Further, we extend the known results (see [H. Brezis, M. Marcus, Hardy's inequalities revisited, Dedicated to Ennio De Giorgi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997–1998) 217–237; M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal. 189 (2002) 537–548; J. Tidblom, A geometrical version of Hardy's inequality for W^1,p(Ω), Proc. Amer. Math. Soc. 132 (2004) 2265–2271]) concerning the improved Hardy inequality

c=c(n,p), by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains.     

Pointwise Hardy inequalities

If is an open set with the sufficiently regular boundary, then the Hardy inequality holds for and , where . The main result of the paper is a pointwise inequality , where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy-Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.

     

Finsler Hardy inequalities

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved.     

Generalized Hardy inequalities

We show that the generalized Hardy inequality holds for f∈H₁ and certain (b_k)∈l_r (2?q?∞) whenever satisfies appropriate growth conditions dependent on r.     

The Hardy inequalities

Translated from Matematicheskie Zametki, Vol. 52, No. 4, pp. 106–111, October, 1992.     

The Robin and Wentzell-Robin Laplacians on Lipschitz Domains

Let $\Omega\subset{\Bbb R}^N$ be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ generates a holomorphic $C_0$ -semigroup of angle $\pi/2$ on $C(\overline{\Omega})$ if $0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)$ . With the same assumption on $\Omega$ and assuming that $0\le\beta\in L^{\infty}(\partial \Omega)$ , we show in the second part that one can define a realization of the Laplacian on $C(\overline{\Omega})$ with Wentzell-Robin boundary conditions $\Delta u+\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ and this operator generates a $C_0$ -semigroup.     

Hardy inequalities for weighted Dirac operator

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r ^−b for functions in \mathbbRⁿ{\mathbb{R}^n}. The exact Hardy constant c _b  = c _b (n) is found and generalized minimizers are given. The constant c _b vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in \mathbbR²{\mathbb{R}^2}. Analogous inequalities are proved in the case c _b  = 0 under constraints and, with error terms, for a bounded domain.     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Weighted bilinear Hardy inequalities

We characterize the weights w, $w_1$, $w_2$ such that the weighted bilinear Hardy inequality$\left(\underset{a}{\overset{b}{\int }}\right(\underset{a}{\overset{x}{\int }}f{)}^q(\underset{a}{\overset{x}{\int }}g{)}^{q}w(x)dx{)}^{\frac{1}{q}}?C(\underset{a}{\overset{b}{\int }}{f}^{p_1}{w}_1{)}^{\frac{1}{p_1}}(\underset{a}{\overset{b}{\int }}{g}^{p_2}{w}_2{)}^{\frac{1}{p_2}}$ holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, $p_1$ and $p_2$ with $1<q,p_1,p_2<\infty$. We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.     

Vector-valued Hardy inequalities andB-convexity

Inequalities of the form for allf∈H ¹, where {m _k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that satisfies the previous inequality for vector valued functions inH ¹ (X), defined as the space ofX-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {m _k }={2^k} but not for {m _k }={k ^a } for any α ∈ N. The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0291.     

Hardy inequalities in Orlicz spaces

We establish a sharp extension, in the framework of Orlicz spaces, of the (-dimensional) Hardy inequality, involving functions defined on a domain , their gradients and the distance function from the boundary of .

     

Some Hardy type integral inequalities

We present new kinds of Hardy integral inequalities involving some generalization and improvement.     

Hardy inequalities and Assouad dimensions

We establish both sufficient and necessary conditions for weighted Hardy inequalities in metric spaces in terms of Assouad (co)dimensions. Our sufficient conditions in the case where the complement is thin are new, even in euclidean spaces, while in the case of a thick complement, we give new formulations for previously known sufficient conditions which reveal a natural duality between these two cases. Our necessary conditions are rather straight-forward generalizations from the unweighted case but, together with some examples, indicate the essential sharpness of our results. In addition, we consider the mixed case in which the complement may contain both thick and thin parts.     

Weighted Hardy and Opial-type inequalities

An integral condition on weights u and v is given which is equivalent to the boundedness of the Hardy operator between the weighted Lebesgue spaces L_u^p and L_v^q with 0 < q < 1 < p < ∞. The Hardy inequalities are applied to give easily verified weight conditions which imply inequalities of Opial type.     

Weighted inequalities of Hardy type

Khabarovsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 13–22, January–February, 1989.