Asymptotics of Robin eigenvalues on sharp infinite cones

Let $\omega\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $\Omega_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1\omega\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^{\alpha}$ acting as the Laplacian $u\mapsto-\Delta u$ on $\Omega_{\varepsilon}$ with the Robin boundary condition $\partial_\nu u=\alpha u$ at $\partial\Omega_\varepsilon$, where $\partial_\nu$ is the outward normal derivative and $\alpha>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^\alpha$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $\omega$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped"cross-sections"$\omega$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes"sharp"and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_\omega:=\dfrac{\mathrm{Vol}_{n-1} \partial\omega }{\mathrm{Vol}_n \omega}$. More precisely, for any fixed $j\in \mathbb{N}$ and $\alpha>0$ the $j$th eigenvalue $E_j(Q^\alpha_\varepsilon)$ of $Q^\alpha_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^\alpha_\varepsilon)=-\dfrac{N_\omega^2\,\alpha^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.


Introduction
Let ω ⊂ R n be a bounded domain (connected open set) with Lipschitz boundary. For ε > 0 consider the open set Geometrically, the set Ω ε is an infinite cone in R n+1 such that the intersection of Ω ε with the hyperplane x 1 = a gives the set εaω. We are interested in some spectral properties of a Robin Laplacian on Ω ε as ε becomes small, i.e. when the cone becomes "sharp" and collapses to the half-line (0, ∞) × {0}. Namely, for α > 0 denote by Q α ε the self-adjoint operator in L 2 (Ω ε ) generated by the closed, densely defined, symmetric bilinear form where dσ stands for the n-dimensional Hausdorff measure. The semiboundedness and the closedness are not completely obvious as Ω ε is unbounded and may have a non-Lipschitz singularity at the origin: we discuss these aspects in detail in the appendices. Informally, the operator Q α ε can be viewed as the positive Laplacian, u → −∆u, with the Robin boundary condition ∂ ν u = αu, where ∂ ν is the outward normal derivative; we refer to [4,10,11] for a discussion of various aspects related to the precise description of the operator domain. Such operators are often referred to as Robin Laplacians with negative parameters [7] due to the negative contribution of the boundary term in the bilinear form. The cone Ω ε is invariant with respect to the dilations x → tx for any t > 0, and standard arguments show the unitary equivalence Q α ε ≃ α 2 Q 1 ε . Hence, it will be convenient to consider α = 1 only and to study the operator and the form For a review of spectral problems with Robin boundary conditions we refer to [7]. In particular, the eigenvalues of Robin Laplacians on infinite cones play a central role in the strong coupling asymptotics of Robin eigenvalues on general domains. Namely, if Ω is an open set in some large class and T Ω,α is the Robin Laplacian on Ω defined as the operator associated with the symmetric bilinear form then the lower edge Λ 1 (T Ω,α ) of the spectrum of T Ω,α satisfies Λ 1 (T Ω,α ) = α 2 inf x∈∂Ω Λ 1 (T Ux,1 ) + o(α 2 ) as α → +∞, where T Ux,1 is the Robin Laplacian on the infinite tangent cone U x at x ∈ ∂Ω. We refer to [6,25] for technical details and precise definitions and to [14-17, 20, 30] for a more precise eigenvalue analysis under more specific regularity assumptions. The function α → Λ 1 (T Ω,α ) plays a role in the study of some non-linear equations as discussed in [24]. Eigenvalues and eigenfunctions of sharp cones can be used to produce counterexamples to spectral gap estimates [22]. In addition, such operators attract some attention as examples of geometric "long-range" configurations producing an infinite discrete spectrum [5,9,29]. Let us summarize the available spectral information for Q ε .
The essential spectrum of Q ε depends in a non-trivial way on ω and ε. If ω has smooth boundary, then in virtue of [29,Thm. 1] the essential spectrum of Q ε is [−1, +∞), as Ω ε is smooth outside the origin. For non-smooth ω the essential spectrum is determined through an iterative procedure and can look differently: see the detailed discussion in [6].
If ω is the unit ball centered at the origin of R n , then Q ε is a round cone whose lateral surface forms the constant angle θ := arctan ε with the central axis, and the bottom of the spectrum of Q ε is the eigenvalue with eigenfunction ψ(x 1 , x ′ ) = exp(−x 1 / sin θ). In fact, only n = 1 and n = 2 were considered explicitly, see e.g. [25,Lem. 2.6] and [19,Prop. 4.2], but the constructions literally hold for arbitrary dimensions n. The case n = 1 (Ω ε is an infinite planar sector) was studied in detail in [21]. The only admissible sets ω are finite intervals, so without loss of generality we can take ω := (−1, 1). In [21] it was shown that the discrete spectrum of Q ε is always finite, but the number of eigenvalues grows unboundedly as ε becomes small, and for each fixed j ∈ N (we use the convention 0 / ∈ N) the jth eigenvalue E j (Q ε ) behaves as Some explicit formulas for eigenpairs of Q ε in this particular case were obtained in [26], but it is unclear if the constructed family exhausts the whole discrete spectrum.
If n ≥ 2, the discrete spectrum of Q ε may be infinite. For example, if n = 2 and ω is simply connected with smooth boundary, then the infiniteness of the discrete spectrum follows by [29,Cor. 8], since the complement of Ω ε is not a convex set (similar arguments apply in higher dimensions: we refer to [29] for details). On the other hand, for polyhedral ω the discrete spectrum can be finite. For example, if one chooses ω in such a way that Ω 1 is an isometric copy of (0, ∞) n+1 , then an easy analysis based on the separation of variables method shows that the discrete spectrum of Q 1 consists of a single eigenvalue −(n+1). For n = 2 and smooth ω, the accumulation rate of eigenvalues at the bottom of the essential spectrum was studied in [5]. Furthermore, in [19] it was shown that round infinite cones maximize the first eigenvalue among all cones with the same perimeter of the spherical cross-section. Various two-sided estimates for the bottom of the spectrum were obtained in [25]. In particular, it was shown that the lowest eigenvalue can be computed explicitly if the spherical cross-section of Ω ε is a spherical polygon admitting an inscribed circle.
In the present work we complement the above results by computing the asymptotics of individual eigenvalues of Q ε for small ε in arbitrary dimensions and for arbitrary cross-sections ω. It turns out that the main term in the asymptotics depends on a single geometric constant N ω given in (3) and, hence, it is rather insensitive to the regularity of ω. Our result reads as follows: Theorem 1.1. Let j ∈ N, then Q ε has at least j discrete eigenvalues below the bottom of the essential spectrum for all sufficiently small ε > 0, and its jth eigenvalue For n = 1 and ω = (−1, 1) one has N ω = 1, and the result follows directly from (2), and all other intervals ω are easily included by applying suitable reparametrizations. Hence, for the rest of the text we explicitly assume n ≥ 2. Remark that if ω is a unit ball centered at the origin, then one has N ω = n, and the exact formula (1) has the form (3) with j = 1 and a more accurate remainder estimate. Based on these observations one may expect that the remainder estimate in (3) is not optimal. We further remark that if the volume Vol n ω or the surface area Vol n−1 ∂ω is fixed, then the quantity N ω is minimized by the ball due to the classical isoperimetric inequality. Hence, the sharp cones Ω ε whose cross-section ω are balls maximize the main term in (3) among all sharp cones with cross-sections of the same volume or surface area.
Our proof is variational and based on the min-max principle, and its main ingredient is a kind of asymptotic separation of the variables x 1 and x ′ , which is quite similar to [21], but the analysis in the x ′ -direction is much more involved and uses some coordinate transforms similar to [23]. Various proof steps are explained in greater detail in Subsection 2.4 below. We intentionally opted for the min-max approach as it allows for an elementary and self-containing proof of the main results concerning the eigenvalues. It should be noted that other more advanced approaches should be applied if one needs more information on eigenfunctions, for example, the method of matched asymptotic expansions [27]. Nevertheless, in view of the expected amount of additional technical work we believe that such an analysis should be the subject of separate study. We also remark that in the appendix we prove some results on Sobolev spaces on Ω ε (which is unbounded and may be non-Lipschitz) that are needed for the spectral analysis: this part of the text may be of its own interest.

Min-max principle
If T is a lower semibounded, self-adjoint operator in an infinite-dimensional Hilbert space H. The spectrum and the essential spectrum of T will be denoted by spec T and spec ess T respectively. Furthermore, denote Σ := inf spec ess T for spec ess T = ∅ and Σ := +∞ otherwise. If T has at least j eigenvalues (counting multiplicities) in (−∞, Σ), then we denote by E j (T ) its jth eigenvalue (when enumerated in the non-decreasing order and counted according to the multiplicities). All operators we consider are real (i.e. map real-valued functions to real-valued functions), and we prefer to work with real Hilbert spaces in order to have shorter expressions.
Let t be the bilinear form for T , with domain D(t), and let D ⊂ D(t) be any dense subset (with respect to the scalar product induced by t). Consider the following "variational eigenvalues" which are independent of the choice of D. One easily sees that j → Λ j (T ) is nondecreasing, and it is known [32, Section XIII.1] that only two cases are possible: • For all j ∈ N there holds Λ j (T ) < Σ. Then the spectrum of T in (−∞, Σ) consists of infinitely many discrete eigenvalues E j (T ) ≡ Λ j (T ) with j ∈ N.
In all cases there holds lim j→∞ Λ j (T ) = Σ, and if for some j ∈ N one has Λ j (T ) < Σ, then E j (T ) = Λ j (T ). In particular, if for some j ∈ N one has the strict inequality Λ j (T ) < Λ j+1 (T ), then E j (T ) = Λ j (T ).

Robin Laplacian on ω
Given r ∈ R denote by B r the self-adjoint operator in L 2 (ω) generated by the closed symmetric bilinear form remark that b r is semibounded from below due to Proposition A.1. Informally, the operator B r is the Laplacian f → −∆f on ω with the Robin boundary condition ∂ ν f = rf , with ∂ ν being the outward normal derivative. We will summarize some important spectral properties of B r as follows.
Lemma 2.1. The following assertions hold true: (a) For any r ∈ R the first eigenvalue E 1 (B r ) is simple, and the corresponding eigenfunction ψ r can be chosen strictly positive with ψ r L 2 (ω) = 1.
Proof. Part (a) is proved for even more general Robin problems in [3,Sec. 4.2]. Both (b) and (d) follow from the fact that the operators B r form a type (B) analytic family with respect to r, see [18,Ch. 7,§4], and (e) is a direct consequence of (b).
To prove (c) we remark first that there exists C > 0 such that the lower bound is proved e.g. in [23,Corol. 2.2], and the upper bound follows from b r (1, 1) < 0 (which holds for all r > 0) by the min-max principle. Furthermore, by Eq. (4.16) in [7] one has d dr and it follows that E 1 (B r ) = −N ω r + O(r 2 ) as r → 0 + . By combining this asymptotics with (6) we arrive at the representation in (c).

One-dimensional model operators
For a fixed λ > 0 consider the symmetric differential operator in L 2 (0, ∞) given by and denote by A λ its Friedrichs extension. Note that n 2 − 2n ≥ 0 due to n ≥ 2.
In [12,Chapter 8.3] the spectrum of A λ was fully determined 1 : the essential spectrum is [0, +∞) and the negative eigenvalues are simple and are explicitly given by In what follows we will need to work with truncated versions of A λ . Namely, for b > 0 we denote by M λ,b and M λ,b the Friedrichs extensions in L 2 (0, b) and respectively. Note that, by construction, the form domain of M λ,b is contained in H 1 0 (0, b), which implies that M λ,b has compact resolvent. We need to relate the eigenvalues of M λ,b to those of A λ . As the bilinear form of A λ extends that of M λ,b , one has, due to the min-max principle, Let us now obtain an asymptotic upper bound for E j (M λ,b ).
Using the identity f 2 ,∞) and the obvious inclusions , we apply the min-max principle as follows:

Scheme of the proof
We first remark that Ω ε is unbounded and, in general, not with Lipschitz boundary near 0, and it does not satisfy the standard assumptions for trace theorems and other important assertions discussed in most books. These aspects are discussed in detail in the appendices, and here we only cite the most important conclusions.
For an open interval I ⊂ (0, ∞) we denote (11) in Appendix A we show: To continue we need suitable coordinates on ∂Ω ε . Consider the diffeomorphism Remark that {0} has zero n-dimensional Hausdorff measure and can be neglected in the integration over ∂Ω ε . By dσ and dτ we will denote the integration with respect to the n-and (n − 1)dimensional Hausdorff measures, respectively. The following technical estimate will play an important role (see Appendix B for a detailed proof): with R := sup t∈ω |t|.
By combining Lemmas 2.3 and 2.4 we show (Proposition B.1) that the trace of any function u ∈ H 1 (Ω ε ) on ∂Ω ε is well-defined and, as a result, that the bilinear form q ε is closed and semibounded from below (Corollary B.2), which shows that Q ε is really a lower semibounded self-adjoint operator. Its spectral study will be mainly based on the min-max principle (Susection 2.1) in which we take D := C ∞ (0,∞) (Ω ε ). In Section 3 we study a Robin Laplacian on a finite part of Ω ε , and it is the most voluminous part of the analysis. Pick some a > 0 (this value will remain fixed through the whole text), and denote then T ε can be informally interpreted as the Laplacian in V ε with the Robin boundary condition ∂ ν u = u on ∂ 0 V ε and the Dirichlet boundary condition on the remaining boundary ∂V ε \ ∂ 0 V ε (which corresponds to x 1 = a). In view of Lemma 2.3 the variational eigenvalues of T ε are defined by Using a suitable change of coordinates and the spectral analysis of B r , the study of eigenvalues of T ε with small ε is reduced to the truncated one-dimensional operators M ε ′ ,a (with suitable ε ′ ∼ ε) from Subsection 2.3. The main result of this reduction is given in Proposition 3.6. The analysis is in the spirit of the Born-Oppenheimer approximation, see e.g. [31,Part 3], with M ε ′ ,a being an "effective operator", and it is essentially an adaptation of the constructions of the earlier paper [23] on Robin eigenvalues in domains with peaks.
The link between the analysis of the truncated operator T ε and the initial operator Q ε is justified in Section 4. We show in Proposition 4.4 that the eigenvalues of Q ε are close to those of T ε , which finishes the proof of Theorem 1.1.

Spectral analysis near the vertex
In this section we study Λ j (T ε ) with small ε. The proof will be based on (13) and on a kind of asymptotic separation of variables.

Change of variables
One observes that This induces the unitary transform (change of variables) Consider the symmetric bi linear form p ε in L 2 (Π, ε n s n ds dt) given by Due to the unitarity of U and Lemma 2.3, the subspace is a core of p ε , and by (13) one has Now we would like to obtain more convenient expressions for p ε (u, u).
For any v ∈ D 0 (t ε ) and u := Uv ∈ D 0 (p ε ), Proof. A standard computation shows that for any u ∈ D 0 (p ε ) there holds where G is the (n + 1) × (n + 1) matrix given by with DX being the Jacobi matrix of X and 1 being the n × n identity matrix. One checks directly that G is a block matrix, We would like to estimate the term ∇u, G ∇u R n+1 from above and from below using simpler expressions. One obtains Using the standard inequality 2|xy| ≤ x 2 + y 2 and |t j | ≤ |t| < R we estimate The substitution into (18) gives a two-sided estimate for ∇u, G ∇u R n+1 , and the substitution into (17) gives the claim.
By applying Lemmas 3.1 and 2.4 to both summands of t ε in (12) and by adjusting various constants we obtain the following two-sided estimate written in a form adapted for the subsequent analysis: There exist c > 0 and ε 0 > 0, with cε 0 < 1, both independent of the choice of a, such that for any ε ∈ (0, ε 0 ) and any u ∈ D 0 (p ε ) there holds In particular, by (15) it follows that for each j ∈ N and any ε ∈ (0, ε 0 ) there holds

Upper bound for the eigenvalues of T ε
We are going to compare the eigenvalues of T ε with those of the truncated onedimensional operators M ε ′ ,a .
Proof. Take first c and ε 0 as in Proposition 3.2. Define a unitary transform and consider the symmetric bilinear form r + ε (u, u) := p + ε (V u, V u). One easily sees that for any u ∈ D 0 (r + ε ) : The substitution u → Vu into the upper bound of Proposition 3.2 shows that .
Using the density, on the right-hand side one can replace r + ε and D 0 (r + ε ) by the closure r + ε and any dense subset D ⊂ D(r + ε ). By Lemma A.3 we can take and we keep the symbol r + ε for r + ε on D, as it is given by the same expression. Therefore, Then integration by parts shows that for u ∈ D one has which implies Note that the functional in the curly brackets is the bilinear form b ερ(s,ε) as defined in is C ∞ , and the derivative (which is smooth and with compact support) coincides with the weak derivative in Π with respect to s. It follows that the function (s, t) → f (s)ψ ερ(s,ε) (t) belongs to the above subspace D.
is a j-dimensional subspace of D. For any u ∈ S one has u L 2 (Π) = f L 2 (0,a) by Fubini's theorem and The substitution into (21) shows that for any u ∈ S there holds By the estimate (5) in Lemma 2.1 we can control the term with ∂ s ψ. Namely, for s ∈ (0, a) and ε ∈ (0, ε 0 ) the values of ερ(s, ε) are contained in some bounded interval, and then one can find some K > 0 such that Hence, for all u ∈ S and ε ∈ (0, ε 0 ) one has Now we apply Lemma 2.1(a,c) to the eigenvalue E 1 (B ερ(s,ε) ): there exists c 0 > 0 such that for all s > 0 and ε > 0 there holds Hence, for all u ∈ S and ε ∈ (0, ε 0 ) we obtain The constants c, c ′ , ε 0 are independent of j and S. By (19), for any j ∈ N and ε ∈ (0, ε 0 ) there holds Corollary 3.4. For any j ∈ N there exist k > 0 and ε 0 > 0 such that Proof. By Lemma 2.2 for any fixed c > 0 and j ∈ N we can choose K ′ > 0 such that if ε is small enough. The substitution into Lemma 3.3 gives the result.

Lower bound for the eigenvalues of T ε
The lower bound for the eigenvalues of T ε is also obtained using a comparison with the operators M ε ′ ,a but requires more work.
Proof. Take c and ε 0 as in Proposition 3.2. For ε ∈ (0, ε 0 ) consider again the unitary transform V : The reparametrization u → Vu in the lower bound of Proposition 3.2 leads to The substitution of Vu into p − ε and the partial integration (20) show that The expression in the curly brackets is the bilinear form b ερ(s,ε) for the Robin Laplacian B ερ(s,ε) on ω as discussed in Subsection 2.2. Denote by ψ ερ(s,ε) the positive eigenfunction for E 1 (B ερ(s,ε) ) with ψ ερ(s,ε) L 2 (ω) = 1, then s → ψ ερ(s,ε) is C ∞ by Lemma 2.1. We decompose each u ∈ D 0 (r − ε ) as By construction we have f ∈ C ∞ c (0, a) and, furthermore, ω w(s, t) ψ ερ(s,ε) (t) dt = 0 for any s ∈ (0, a), The spectral theorem applied to B ερ(s,ε) implies that for any u ∈ D 0 (r − ε ) there holds By Lemma 2.1(c) one can find a constant c 1 > 0 such that for all sufficiently small x > 0.
Let us study the first integral on the right hand side of (27). Using the orthogonality relations (24) we obtain Due to the normalization of ψ, In order to estimate the two last terms in (28) we note that (30) and that, in view of (24), Hence, using |2xy| ≤ x 2 + y 2 , Similarly, As ερ(s, ε) ∈ (0, ε 0 m) for all s ∈ (0, a) and ε ∈ (0, ε 0 ) we can use the estimate (5) of Lemma 2.1: there exists K > 0 such that for all ε ∈ (0, ε 0 ) and s ∈ (0, a) one has ψ s 2 L 2 (ω) ≤ Kε 2 ≤ ε (assuming that ε 0 is sufficiently small). We now use the obtained estimate in (31) and (32), which gives The substitution of these two inequalities into (30) gives We now use the last obtained inequality and (29) in (28), which gives a 0 ω for all ε ∈ (0, ε 0 ) and u ∈ D 0 (r − ε ). Using this lower bound in (27) one arrives at By taking sufficiently large b > k and c ′ > 1 and a smaller value of ε 0 one deduces from the last inequality the simpler lower bound . Using the norm equality (25) this is equivalent to By the norm equality (25), the map u → (f, w) uniquely extends to a unitary map Ψ : L 2 (Π) → L 2 (0, a) ⊕ H, where H is some closed subspace of L 2 (Π). Let h ε be the symmetric bilinear form in L 2 (0, a) ⊕ H defined as the closure of the form then the corresponding self-adjoint operator in , and the lower bound (22) for Λ j (T ε ) implies that for any j ∈ N and any ε ∈ (0, ε 0 ) there holds By Lemma 2.2, for some K 0 > 0 and for all sufficiently small ε > 0 we have for a suitably chosen k ′ > 0 and all sufficiently small ε > 0.
By combining Corollary 3.4 and Lemma 3.5 we obtain the main result of the section: Proposition 3.6. For any j ∈ N there holds

End of proof of Theorem 1.1
Note that the right-hand side of the asymptotics in Proposition 3.6 corresponds to the sought asymptotics for E j (Q ε ) in Theorem 1.1. In order to conclude the proof of Theorem 1.1 it remains to show that the eigenvalues of Q ε and T ε with the same numbers are close to each other. This will be done in several steps.
Proof. Let J : L 2 (V ε ) → L 2 (Ω ε ) be the operator of extension by zero, then J is a linear isometry with JD(t ε ) ⊂ D(q ε ) and with q ε (Ju, Ju) = t ε (u, u) for all u ∈ D(t ε ), and the result follows directly by the min-max principle.
Recall that the subspaces C ∞ I (Ω ε ) were defined in (11). For b > 0 denote and let T ε,b be the self-adjoint operator in L 2 ( V ε,b ) defined by its symmetric bilinear form
The following assertion together with the asymptotics of Λ j (T ε ) from Proposition 3.6 completes our proof of Theorem 1.1: Proposition 4.4. Let j ∈ N be fixed, then: • one can find some ε j > 0 such that Q ε has at least j discrete eigenvalues below inf spec ess Q ε for all ε ∈ (0, ε j ), Proof. Let us fix j ∈ N. By combining the upper bound of Lemma 4.1 and the lower bound of Lemma 4.3 we obtain Λ j (Q ε ) = Λ j (T ε ) + O(1). By Proposition 3.6 we have Λ j+1 (T ε ) − Λ j (T ε ) → +∞ as ε → 0 + . It follows that there exists ε j > 0 such that Λ j (Q ε ) < Λ j+1 (Q ε ) for all ε ∈ (0, ε j ), and then E j (Q ε ) = Λ j (Q ε ) for the same ε due to the min-max principle.
Remark 4.5. By sending j to ∞ in (37) and using the compactness of the resolvent of T ε one also shows that inf spec ess Q ε ≥ inf spec ess T ε,b . Then Lemma 4.2 shows that for small ε > 0 one estimates inf spec ess Q ε ≥ −cε −1 with a fixed c > 0. For each j ∈ N the difference between inf spec ess Q ε and E j (Q ε ) is of order ε −2 , so it is standard to show that the respective eigenfunction u j,ε satisfies an Agmontype exponential decay at infinity [2]. This explains why the analysis of Q ε on the complete infinite cone can be reduced to the analysis of an operator on a finite part of the cone.
book [13]. For an open set Ω ⊂ R m and k ∈ N the k-th Sobolev space H k (Ω) is defined as H k (Ω) := u ∈ L 2 (Ω) : ∂ α u ∈ L 2 (Ω) for all |α| ≤ k} with all derivatives taken in the sense of distributions, and it is a Hilbert space with respect to the scalar product By Most assertions used in the theory of Sobolev spaces (some density and extension results, trace theorems) are usually formulated for bounded open sets with Lipschitz boundaries. On the other hand, the cone Ω ε has in general not even a C 0 boundary: for example, if n = 2 and ω is an annulus, ω = {(x 1 , x 2 ) : 1 < x 2 1 + x 2 2 < 4}, then one easily sees that Ω ε cannot be represented as one of the sides of the graph of a continuous function near the vertex 0. Moreover, further common assumptions used in the theory of Sobolev spaces (e.g. the segment condition or the cone condition) fail as well.
We collect some well known facts about H k (Ω) in the following proposition: We refer to [28, [13,Theorem 1.5.1.10] for (C.2). Note that one usually writes simply u instead of γ 0 u in the integrals over the boundary. Now we pass to the discussion of Sobolev spaces on the infinite cones Ω ε . We start with several preparation steps.
Proof. Let p ∈ ∂Ω, then the following cases are possible.
In order to bring these conditions into the required form we pick θ ∈ (0, π 2 ) and apply a rotation by the angle θ around p in the (x 1 , y n )-plane. Namely, consider the Cartesian coordinates (z, y 1 , . . . , y n−1 , w) with the previous y 1 , . . . , y n−1 and Clearly, the new coordinates are centered at p, and the above inequalities (38) determining Ω near p take the form which can be rewritten as w < H(z, y 1 , . . . , y n−1 ) := min cos θ sin θ z, − sin θ cos θ z + 1 cos θ h(y 1 , . . . , y n−1 ) .
The function H is Lipschitz (since it is the minimum of two Lipschitz functions), hence, one has a required representation of Ω near p. The case 3b: p = (b, p ′ ) with p ′ ∈ ∂ω is considered analogously.
The next step is somewhat technical; it shows that each H 1 -function on a cylinder vanishing identically near the bases can be suitably approximated by test functions vanishing near the bases.
Proof. By Lemma A.2 and Proposition A.1 (part C.1) the function u can be extended to a function v ′ ∈ H 1 (R n+1 ). Choose χ ∈ C ∞ c (R) such that χ(s) = 1 for s ∈ [c, c ′ ] and supp χ ∈ c − δ 2 , c ′ + δ 2 , and, in addition, choose χ 0 ∈ C ∞ c (R n ) with χ 0 = 1 on ω. Then the function v : ( , is an extension of u, has compact support, and v(x 1 , x ′ ) = 0 for all and for t > 0 consider the functions ρ t : Furthermore, the definition of the convolution product implies the inclusion . Now pick any 0 < t < min t 0 , δ 2 and denote ϕ : so ϕ has all the required properties.
Let us finish the proof of Lemma 2.3. First remark that H 1 ∞ (Ω ε ) is dense in H 1 (Ω ε ) by Proposition A.1(A). So we need to show that any function in H 1 ∞ (Ω ε ) can be approximated by functions from C ∞ (0,∞) (Ω ε ) in the H 1 -norm.

B Traces and semiboundedness
Firstly we prove Lemma 2.4 and then move on to show the wellposedness of our spectral problem, i.e. the lower semiboundedness and closedness of q ε .
Proof of Lemma 2.4. As usual for the integration over hypersurfaces, it is sufficient to prove the statement for functions supported in images of local charts, then this statement is extended to general functions using a partition of unity.