Asymptotics of Robin eigenvalues for non-isotropic peaks

Let $\Omega\subset \mathbb{R}^3$ be an open set such that \begin{align*}&\Omega \cap (-\delta,\delta)^3=\left\{(x_1,x_2,x_3)\in \mathbb{R}^2\times(0,\delta): \, \left(\frac{x_1}{x_3^p},\frac{x_2}{x_3^q}\right)\in(-1,1)^2\right\}\subset\mathbb{R}^{3}, \\&\Omega \setminus [-\delta,\delta]^3 \text{ is a bounded Lipschitz domain}, \end{align*} for some $\delta>0$ and $1<p<q<2$. If a set satisfies the first condition one says that it has a non-isotropic peak at $0$. Now consider the operator $Q_\Omega^\alpha$ acting as the Laplacian $u\mapsto-\Delta u$ on $\Omega$ with the Robin boundary condition $\partial_\nu u=\alpha u$ on $\partial\Omega$, where $\partial_\nu$ is the outward normal derivative. We are interested in the strong coupling asymptotics of $Q_\Omega^\alpha$. We prove that for large $\alpha$ the $j$th eigenvalue $E_j(Q_\Omega^\alpha)$ behaves as $E_j(Q_\Omega^\alpha)\approx \mathcal{A}_j\alpha^{\frac{2}{2-q}}$, where the constants $\mathcal{A}_j<0$ are eigenvalues of a one dimensional Schr\"odinger operator which depends on $p$ and $q$.


Introduction
Consider an open set Ω ⊂ R N and the Robin eigenvalue problem −∆u = λu on Ω, where ∂ ν is the outward normal derivative and α > 0 is the so called Robin parameter, which is also referred to as a coupling constant. Numerous results concerning this problem have been published over the last decades. In [1] the authors gave an overview of the current body of knowledge and also presented some open problems. However we are particularly interested in the following question: How do the eigenvalues behave as α tends to infinity? This is often referred to as the strong coupling asymptotics of the eigenvalues and was presumably first studied by Lacey, Ockendon and Sabina [12]. For further discussion we need to define operators more rigorous. So let Ω ⊂ R N be an open set and α > 0 such that the quadratic form q α Ω (u, u) = Ω |∇u| 2 dx − α ∂Ω u 2 dσ, D(q α Ω ) = H 1 (Ω), is semibounded from below and closed, where dσ denotes the integration with respect to the (N − 1)-dimensional Hausdorff measure, and denote by Q α Ω the selfadjoint operator in L 2 (Ω) associated with q α Ω . The asymptotic behavior of the eigenvalues is highly influenced by the regularity of Ω. We list some results about the strong coupling regime for "nice" domains first and then move on to "bad" domains.
If Ω ⊂ R N , with N ≥ 2, is a bounded Lipschitz domain, then it is well known that 0 > E j (Q α Ω ) > −Kα 2 for sufficiently large α > 0. Remark that the second inequality follows immediately from [6, Theorem 1.5.1.10]. Levitin and Parnovski [13,Theorem 3.2] showed that the principal eigenvalue for bounded piecewise smooth domains satisfying the uniform interior cone condition behaves as E 1 (Q α Ω ) ≈ −C Ω α 2 , with C Ω ≥ 1. If ∂Ω is C 1 then Daners and Kennedy [3, Theorem 1.1.] were able to show that C Ω = 1 for every eigenvalue, i.e. E j (Q α Ω ) ≈ −α 2 . The results mentioned above are all one-term asymptotics, but there are also papers which have proven two-term asymptotics. Exner, Minakov and Parnovski [5,Theorem 1.3] showed for planar domains, which have a closed C 4 Jordan curve as their boundary, that the eigenvalues behave as E j (Q α Ω ) ≈ −α 2 −γ * α, where γ * is the maximal curvature of the mentioned Jordan curve. Another two-term asymptotic expansion was obtained for curvilinear polygons by Khalile, Ourmières-Bonafos and Pankrashkin, the corresponding paper is quite voluminous and technical, therefore we refer to [9] for precise statements.
For non Lipschitz domains many different scenarios are possible. If Ω has an outward pointing peak which is "to sharp" the Robin-Laplacian fails to be semibounded from below, see e.g. [15,Lemma 1.2]. However the present paper is motivated by [11], where Kovařík and Pankrashkin looked at isotropic peaks, i.e. there exists δ > 0 such that with 1 < q < 2 and B 1 (0) being the unit ball centered at the origin in R N −1 . They proved that the rate of divergence of the eigenvalues to −∞ is faster than in the pure Lipschitz case. In particular they showed, the eigenvalues behave as E j (Q α Ω ) ≈ E j α for some δ > 0 and 1 < p < q < 2. If a set satisfies condition (1) one says that it has a non-isotropic peak at 0. Since Ω is not Lipschitz, the closedness and semiboundedness of the quadratic form q α Ω are not obvious but will be justified in Appendix B.
Based on the above observation one might expect that the larger power q determines the rate of divergence to −∞, which turns out to be true as described in Theorem 1.1. For a precise statement we need to define a one dimensional Schrödinger operator. Consider the symmetric differential operator given by and denote by A 0,1 its Friedrichs extension in L 2 (0, ∞). Then the result reads as follows:

Remark 1.2.
Since Ω is bounded and has a continuous boundary, the embedding . Therefore Q α Ω has compact resolvent, in particular the essential spectrum is empty and there exist infinitely many discrete eigenvalues. Remark 1.3. It would be desirable to substitute the "cross section" (−1, 1) 2 with an arbitrary Lipschitz domain, but during our analysis this cross section is going to "collapse to an interval". So one has to analyze the behavior of the Robin eigenvalues on such a collapsing cross section, which may have its own interest and requires an independent study.
The proof follows the same scheme as in [11,16] but with several additional technical ingredients, see section 2.5 for further explanations.

Min-max principle
Let T be a lower semibounded, self-adjoint operator in an infinite-dimensional Hilbert space H. The essential spectrum of T will be denoted by spec ess T . 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 realvalued functions to real-valued functions), and we prefer to work with real Hilbert spaces in order to have shorter expressions.
Let t be the quadratic 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 [17, 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 ).
A standard application of the min-max principle shows that for any j ∈ N one has where R N/D,Ω δ α are the self-adjoint operators in L 2 (Ω δ ) defined respectively by the quadratic forms and K N,Θ δ α is the self-adjoint operator in L 2 (Θ δ ) defined by the quadratic form Standard dilation arguments show the unitary equivalence and we remark that

Robin Laplacians on intervals
Given r ∈ R and L > 0 denote by B L,r the self-adjoint operator in L 2 (−L, L) generated by the closed quadratic form The operator B L,r is the Laplacian f → −f ′′ on H 2 (−L, L) with the Robin boundary condition f ′ (±L) = ±rf (±L) and we will summarize some important spectral properties of B L,r as follows.
Lemma 2.1. The following assertions hold true: (d) Let j ∈ N be fixed, then there exists a family of normalized eigenfunctions is C ∞ , which we call a smooth family of eigenfunctions. Furthermore for all r 0 > 0 and all C > 0 there exists C 0 > 0 such that for all L ≤ C and all r ∈ (0, r 0 ).

Proof.
To prove the claim in (a) one can use the unitary operator U : [10,Proposition 2.2] it was shown that there exists φ ∈ C ∞ (R + ) ∩ L ∞ (R + ) such that for all r > 0 there holds E 1 (B 1,r ) = −r + r 2 φ(r). To conclude the proof of (b) we use E 1 (B L,r ) = 1 L 2 E 1 (B 1,rL ) from (a). For (c) we define φ(r) := r −2 (E 2 (B 1,r ) − π 2 4 + 2r). First of all the function R ∋ r → E 2 (B 1,r ) is analytic, since B 1,r is a type (B) analytic family and has compact resolvent and E 2 (B 1,r ) has multiplicity one for all r, which implies that φ is smooth, see e.g. [8,Remark 4.22 in Ch. 7]. To show the boundedness of φ we prove that it admits finite limits, i.e. as r → 0 and r → ∞. For r → 0 we would like to have a Taylor expansion of r → E 2 (B 1,r ) in 0. According to [1,Equation (4.12)] we can use an eigenfunction corresponding to the second Neumann eigenvalue E 2 (B 1,0 ) = π 2 4 to get an expression for ∂ r E 2 (B 1,r )| r=0 . Namely is a corresponding eigenfunction for E 2 (B 1,0 ) and therefore It follows that E 2 (B 1,r ) = π 2 4 − 2r + O(r 2 ) as r → 0 and therefore φ(r) = O(1) as r → 0. For the case r → ∞ we use the result from (a) to get E 2 (B 1,r ) = r 2 E 2 (B r,1 ). Now we can use [7,Proposition A.3.], which states that E 2 (B r,1 ) = −1 + O(r 2 e −2r ) as r → ∞. Plugging these results back into φ yields φ(r) = O(1) as r → ∞. Once again we use the result from (a) to conclude the proof of (c).
Lastly we fix some j ∈ N. The existence of the family (Φ j,L,r ) r∈R mentioned in (d) also follows from B L,r being a type (B) analytic family, see again [8,Remark 4.22 in Ch. 7]. To prove the second claim in (d), choose a family (Φ j,1,r ) r∈R such that r → Φ j,1,r is smooth. Then for all r 0 > 0 there exists C 0 > 0 such that Now let L ≤ C, r 0 > 0 and r 0 := Cr 0 . Use the unitary operator U mentioned in the proof of (a) to get Φ j,L,r = U −1 Φ j,1,Lr = L − 1 2 Φ j,1,Lr ( · L ). By equation (5) there exists holds.
For further reference we need the following Lemma.
Then for all j, k ∈ N and all C > 0 there exists C 0 > 0 such that for all s ∈ [a, b] and all L, L ≤ C. Here (Φ j,L,r ) r∈R and (Φ k, L,r ) r∈R are smooth families of normalized eigenfunctions as defined in Lemma 2.1 (d).
Proof. Let C > 0 and L, L ≤ C. By Young's inequality we have For the equality we used that Φ j,L,r and Φ k, L,r are normalized eigenfunctions, for the second inequality we used that ρ ′ and ρ ′ are bounded and for the last inequality we used that ρ and ρ are bounded combined with Lemma 2.1 (d).

One-dimensional model operators
Let c 1 , c 2 ≥ 0 such that c 1 + c 2 > 0 and consider the symmetric differential operator in L 2 (0, ∞) given by

and the Hardy inequality
the operator is semibounded from below. Denote by A c 1 ,c 2 its Friedrichs extension and by a c 1 ,c 2 its corresponding quadratic form. One can show that A c 1 ,c 2 has infinitely many negative eigenvalues and for c > 0 there holds the unitary equivalence In what follows we will need to work with truncated versions of A c 1 ,c 2 . Namely for b > 0 we denote by M c 1 ,c 2 ,b and M c 1 ,c 2 ,b the Friedrichs extensions in L 2 (0, b) and has compact resolvent. We need to relate the eigenvalues of M c 1 ,c 2 ,b to those of A c 1 ,c 2 . As the quadratic form of A c 1 ,c 2 extends that of M c 1 ,c 2 ,b , one has, due to the min-max principle, Let us now obtain an asymptotic upper bound for E j (M c 1 ,c 2 ,b ).
Proof. The proof is quite standard and uses a so-called IMS partition of unity [2, Sec. 3.1]. Let χ 1 and χ 2 be two smooth functions on R with 0 ≤ χ 1 , ∞ . An easy computation shows that .
Using the identity f 2 ,∞) and the obvious inclusions , we apply the min-max principle as follows:

Scheme of the proof
We are interested in large Robin parameters, so from now on we only consider α ≥ 1. Let us also remark that the same dilation arguments mentioned in Subsection 2.2 can be used to show Q α Ω ≃ α 2 Q 1 αΩ . The peak will determine the asymptotic behavior of the eigenvalues, so we start by restricting the Robin-Laplacian Q 1 αΩ to the peak of αΩ. More precisely, pick some 0 < a < min(δ, 1) (this value will remain fixed through the whole text), and denote Let T α be the self-adjoint operator in L 2 (V α ) associated with the quadratic form then T α can be informally interpreted as the Laplacian in V α with the Robin boundary condition ∂ ν u = u on ∂ 0 V α and the Dirichlet boundary condition on the remain- , so by the min-max principle the variational eigenvalues of T α are given by Using a suitable change of coordinates and the spectral analysis of B L,r , the study of eigenvalues of T α for large α is reduced to the one-dimensional operators A c 1 ,c 2 . In Section 4 we use (4) as a starting point to prove that the eigenvalues of T α and Q 1 αΩ are asymptotically close to each other.

Change of variables
We need suitable coordinates on ∂ 0 V α , therefore we transform coordinates similar to [11]. Consider the diffeomorphism then one checks that Remark that {0} has zero two-dimensional Hausdorff measure and can be neglected in the integration over ∂ 0 V α . This induces the unitary transform (change of variables) where we used the notation t = (t 1 , t 2 ). Consider the quadratic form r α given by in L 2 (Π α , s p+q ds dt). Due to the unitarity of U and Lemma A.1, the subspace is dense in D(r α ), and by (10) one has Now we would like to obtain a more convenient expression for r α (u, u).
Proof. A standard computation shows that for any u ∈ D 0 (r α ) and v := U −1 u there holds where G is the matrix given by with DX being the Jacobi matrix of X. One checks directly that We would like to estimate the term ∇u, G ∇u R 3 from above and from below using simpler expressions. One obtains Using the standard inequalities 2|xy| ≤ x 2 + y 2 , |t 1 | ≤ α 1−p , |t 2 | ≤ α 1−q and 0 < s < a < 1 we estimate The substitution into (13) gives a two-sided estimate for ∇u, G ∇u R 3 , and the substitution into (12) gives the claim.
Also remark that for the boundary we have again for u ∈ D 0 (r α ) and v := U −1 u. By applying Lemma 3.1 and (14) to both summands of t α in (9) and by adjusting various constants we obtain the following two-sided estimate written in a form adapted for the subsequent analysis: There exists c > 0 (this value will remain fixed through the whole text) and α 0 > 1 such that for any u ∈ D 0 (r α ) and all α > α 0 there holds Remark that c > 0 is independent of the choice of 0 < a < min(δ, 1). With the help of the unitary transform we define the following quadratic forms, which we will work with most of the time.
. Using elementary calculations the following Proposition follows.

Proposition 3.2. For all
In particular, by (11) it follows that for each j ∈ N there holds

Upper bound for the eigenvalues of T α
We are going to compare the eigenvalues of T α with those of the one-dimensional operators A c 1 ,c 2 .

Lemma 3.3.
Let j ∈ N then there exist C > 0 and α 0 ≥ 1 such that for any α > α 0 there holds .
We want to have a more convenient expression for p + α to compare it to an one dimensional Robin-Laplacian. There holds Note that the functionals in the curly brackets are the quadratic forms b α 1−k ,ρ k (s) as defined in Subsection 2.3 with Let k ∈ {p, q} and (Φ 1,k,ρ k (s) ) s∈(0,a) be a smooth family of normalized eigenfunctions corresponding to (E 1 (B α 1−k ,ρ k (s) )) s∈(0,a) (see Lemma and For u ∈ S this simplifies the expression of the above quadratic form as follows For the sake of brevity we use the notation Due to the normalization of Φ we have The preceding orthogonality relation and Lemma 2.2 show for appropriate C > 0. Using Lemma 2.1 and taking α sufficiently large, we arrive at for appropriate C and C 1 . One recognizes the quadratic form for M α p−1 1+cα 1−p , α q−1 1+cα 1−p ,a . Also remark that the constants we have chosen, in this proof, to estimate terms are independent of the subspace S. As a result we have This implies 1+cα 1−p ,a + C and concludes the proof.
Proof. By Lemma 3.3, we know 1+cα 1−p ,a + C and, by Lemma 2.3, we get Applying the min-max principle yields Making use of the unitary equivalence (6), we have By adjusting C > 0 it follows that

Lower bound for the eigenvalues of T α
The lower bound for the eigenvalues of T α is also obtained using a comparison with the operators A c 1 ,c 2 but requires more work.
We want to have a more convenient expression for p − α to compare it to an one dimensional Robin-Laplacian. There holds Note that the functionals in the curly brackets are the quadratic forms b α 1−k ,ρ k (s) as defined in Subsection 2.3 with Let k ∈ {p, q} and (Φ 1,k,ρ k (s) ) s∈(0,a) be a smooth family of normalized eigenfunctions corresponding to (E 1 (B α 1−k ,ρ k (s) )) s∈(0,a) (see Lemma 2.1 (d)). We decompose each By construction we have f ∈ C ∞ c (0, a) and, furthermore, ωα w(t, s) v(t, s) dt = 0 for any s ∈ (0, a), The spectral theorem and Lemma 2.1 applied to 1 s 2q B α 1−q ,ρq(s) implies that there exists α 0 such that for all α > α 0 and any u ∈ D 0 (p − α ) we have By Lemma 2.1 we can find constants C i > 0, such that for all u ∈ D 0 (p − α ). Let us now study the integral a 0 ωα |∂ s u| 2 + (p + q) 2 − 2(p + q) 4s 2 |u| 2 dt ds.
Using the orthogonality relation from (15) we obtain a 0 ωα Due to the normalization of Φ we have The preceding orthogonality relation shows the following and, consequently, In order to estimate the two last terms in (18) we note that Due to Lemma 2.2 and Young's inequality, there exists a C > 0 such that for large α there holds Combining these inequalities gives Finally we get an estimation for (17) a 0 ωα for an appropriate C > 0 and α large enough. Using the lower bounds in (16), (19) and adjusting C > 0 we can choose C > 0 such that holds. One recognizes the quadratic form for M α p−1 1−Cα 1−p , α q−1 1−Cα 1−p ,a as well as the quadratic form . As a result of the min-max principle we have where G α,a is the self-adjoint operator in L 2 (Π α ) associated with g α,a . This implies To end the proof we have to show that for large α. Making use of the unitary equivalence (6) and Lemma 2.3 (by suitably adjusting constants) we get for large enough α. It is left to show that the bottom of the spectrum of G α,a is bigger than (21). Since G α,a is a multiplication operator, it is sufficient to show that there exists α 0 > 0 and E(α 0 ) > E j (A 0,1 ) such that for all α > α 0 there holds min s∈(0,a) We estimate the minimum from below in the following way: There exists c 0 > 0 such that Remark that the last inequality holds due to the fact that the exponent 2p q − 2 − ( 4p q −2) q−1 2−q is positive, which also can be shown by an easy calculation. Furthermore we have That means we can choose α 0 large enough to get E j (A 0,1 ) < E(α 0 ). Which in return yields the claim in (20). Proposition 3.6. For any j ∈ N there exist α 0 > and C > 0 such that Proof. Lemma 3.5 and (7) show Due to the unitary equivalence (6), we get We want to compare the eigenvalue on the right hand side with E j (A 0,1 ). Due to the min-max principle it is sufficient enough to estimate the corresponding quadratic form from below. There exists c > 0 such that for any f ∈ C ∞ c (0, ∞) there holds Using this result and applying the unitary equivalence (6) one more time we arrive at for C > 0 large enough. Combining this with (22) and adjusting C gives the desired estimate

End of proof of Theorem 1.1
So far we have shown that E j (T α ) ≈ E j (A 0,1 )α 2 q−1 2−q , which corresponds to the leading term of Q 1 αΩ according to Theorem 1.1, if one uses the unitary equivalence Q α Ω ≃ α 2 Q 1 αΩ . Therefore, in order to conclude the proof of Theorem 1.1 it remains to show that the eigenvalues of Q 1 αΩ and T α with the same numbers are close to each other. We will use (4) as a starting point, to estimate E j (Q 1 αΩ ) by means of E j (T α ).

B Closedness of q α
Ω For a Lipschitz domain it is well known that the quadratic form associated with the Robin-Laplacian is closed. To show this, one can use the trace inequality from [6, Theorem 1.5.1.10]. However we can not use this result immediately, since Ω has a peak. Fortunately this inequality extends to domains with non-isotropic peaks. But let us first show the semiboundedness of q α Ω . Lemma B.1. The quadratic form q α Ω is semibounded from below, for any α > 0 and any open set Ω which satisfies (1) and (2).
In particular the norm induced by q α Ω is equivalent to the standard H 1 -norm and therefore q α Ω is closed.
Proof. Due to Lemma B.1, for every α > 0 we can find a constant K α > 0 such that for all u ∈ H 1 (Ω).
Rearranging this inequality yields for all u ∈ H 1 (Ω).