On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

Abstract

We construct two-term asymptotics $\lambda ^\varepsilon _k= \varepsilon ^{m-2}(M+\varepsilon \mu _k+ O(\varepsilon ^{3/2}) )$ of eigenvalues of a mixed boundary-value problem in $\Omega \subset {{\mathbb {R}}}^2$ with many heavy ($m>2$) concentrated masses near a straight part $\Gamma$ of the boundary $\partial \Omega$. $\varepsilon$ is a small positive parameter related to size and periodicity of the masses; $k\in {\mathbb N}$. The main term $M>0$ is common for all eigenvalues but the correction terms $\mu _k$, which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on $\Gamma$, exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.

A Appendix

The material of this appendix complements Sect. 2 and supports the estimate (5.49) which lead us to Proposition 5.5.

1.1 A.1 The homogeneous Kondratiev norms

Let us consider the model mixed boundary-value problem in the half-plane

$$\begin{aligned} \displaystyle -\Delta v= & {} f \quad \text{ in } \,{{\mathbb {R}}}^2_+, \end{aligned}$$

(A.1)

$$\begin{aligned} \displaystyle v= & {} 0 \quad \text{ on } \{x:\, x_2=0 , \, x_1>0\} ,\quad \frac{\partial v}{\partial x_2} =0\quad \text{ on } \{x:\, x_2=0,\, x_1>0\}\qquad \quad \end{aligned}$$

(A.2)

within the Kondratiev theory [10]. By $V^l_\beta ({{\mathbb {R}}}^2_+)$, with the indexes of smoothness $l\in {\mathbb N}_0$ and weight $\beta \in {{\mathbb {R}}}$, we denote the completion of the linear space $C^\infty _c\left( \overline{{{\mathbb {R}}}^2_+}{\setminus }{{\mathcal {O}}}\right)$ in the homogeneous weighted norm

$$\begin{aligned} \Vert u;V^l_\beta ({{\mathbb {R}}}^2_+)\Vert =\left( \,\sum \limits _{j=0}^l \left\| r^{\beta -l+j}\nabla ^j u;L^2({{\mathbb {R}}}^2_+)\right\| ^2\right) ^{1/2}, \end{aligned}$$

where $(r,\varphi )$ is the polar coordinate system centered at the coordinate origin ${\mathcal {O}}$, the collision point, and $\nabla ^j u$ denotes all partial derivatives of u of order j. It is known, see [10] and, e.g., Ch. 2 in [31], that, for any $l\in {\mathbb N}$, the operator

$$\begin{aligned} \displaystyle {\mathcal A}^l_\beta :\,\Big \{v\in V^{l+1}_\beta ({{\mathbb {R}}}^2_+):\, v\,\,\hbox { satisfies}\,\,(\hbox {A}.2)\Big \} \quad \rightarrow \quad V^{l-1}_\beta ({{\mathbb {R}}}^2_+) \end{aligned}$$

(A.3)

of problem (A.1), (A.2) is Fredholm if and only if $\beta -l\not =\beta _{\pm j}:=\pm (j+1/2)$ with $j\in {\mathbb N}_0$; otherwise, the range of (A.3) is not closed in $V^{l-1}_\beta ({{\mathbb {R}}}^2_+)$. The forbidden indexes $\beta _{\pm j}$ are closely connected to exponents (2.17) in harmonics (2.16).

If $v\in V^{l+1}_\beta ({{\mathbb {R}}}^2_+)$ is a solution of problem (A.1), (A.2) with the right-hand side $f\in V^{l-1}_{\beta -2}({{\mathbb {R}}}^2_+)$ and

$$\begin{aligned}&\displaystyle |\beta -l|<\frac{1}{2}, \end{aligned}$$

(A.4)

$$\begin{aligned}&\displaystyle \hbox { supp}\,f\subset \left\{ x\in {{\mathbb {R}}}^2_+:\,r\le 1\right\} , \end{aligned}$$

(A.5)

then

$$\begin{aligned} \displaystyle v(x)=\varsigma (r)\Big (Kr^{1/2}\sin \frac{\varphi }{2}+ K^1r^{3/2}\sin \frac{3\varphi }{2}\Big )+{\widetilde{v}}(x) \end{aligned}$$

(A.6)

where the coefficients K, $K^1$ and the remainder $\widetilde{u} \in V^{l+1}_{\beta -2}({{\mathbb {R}}}^2_+)$ satisfy the estimate

$$\begin{aligned} \displaystyle |K|+|K^1|+\Vert {\widetilde{v}};V^{l+1}_{\beta -2}({{\mathbb {R}}}^2_+)\Vert \le c \Vert f;V^{l-1}_{\beta -2}({{\mathbb {R}}}^2_+)\Vert . \end{aligned}$$

(A.7)

We emphasize that mapping (A.3) becomes an isomorphism under restriction (A.4) and the inclusion $f\in V^{l-1}_{\beta -2}({{\mathbb {R}}}^2_+)$ implies a faster decay rate as $r\rightarrow 0^+$ than the decay rate of $\Delta v\in V^{l-1}_\beta ({{\mathbb {R}}}^2_+)$ prescribed by the original inclusion $v\in V^{l+1}_\beta ({{\mathbb {R}}}^2_+)$. In the same way, formula (A.6) gives the asymptotics of the solution v in the radial variable r.

1.2 A.2 The multi-scaled weighted norms

Considering the Rayleigh principle for the spectral problem

$$\begin{aligned} - \Phi ''(t)=\Lambda \Phi (t)\, \text{ for } \, t\in (0,\pi ), \quad \Phi (0)=0, \Phi '(\pi )=0, \end{aligned}$$

and the angular variable $t=\varphi$, a function in $\{v\in V^1_\beta ({{\mathbb {R}}}^2_+):\,v(x_1,0)=0,\,x_1>0\}$ satisfies

$$\begin{aligned} \begin{aligned}&\int \limits _{{{\mathbb {R}}}^2_+}\, r^{2\beta -2}\left| v(x)\right| ^2dx= \int \limits _{{{\mathbb {R}}}_+}\int \limits _0^\pi \, r^{2\beta -2}\left| v(x)\right| ^2rdrd\varphi \\&\quad \le \,4\int \limits _{{{\mathbb {R}}}_+}\int \limits _0^\pi \, r^{2\beta }\Big | \frac{1}{r}\, \frac{\partial v}{\partial \varphi }(x)\Big |^2rdrd\varphi \le 4\int \limits _{{{\mathbb {R}}}^2_+}\, r^{2\beta }\left| \nabla v(x)\right| ^2dx. \end{aligned} \end{aligned}$$

This apparent observation was suggested in [24] to introduce multi-scaled weighted space ${\mathcal V}^{l,0}_{\beta ,\gamma }({{\mathbb {R}}}^2_+)$ in two-dimensional angular domains with the norm

$$\begin{aligned} \displaystyle \Vert u;{\mathcal V}^{l,0}_{\beta ,\gamma }({{\mathbb {R}}}^2_+)\Vert =\left( \,\sum \limits _{j=0}^l \left\| r^{\beta -l+j}\varphi ^{\gamma -l+j}\nabla ^j u;L^2({{\mathbb {R}}}^2_+)\right\| ^2\right) ^{1/2} \end{aligned}$$

(A.8)

involving two weights, radial and angular, with different weight exponents $\beta$ and $\gamma$. Such function spaces are convenient in the investigation of different perturbations of the boundary of the angular domain, cf. [24, 29].

If restrictions (A.4) and

$$\begin{aligned} \displaystyle \gamma -l\in \Big (-\frac{1}{2},\frac{1}{2}\Big ) \end{aligned}$$

(A.9)

are satisfied, the operator

$$\begin{aligned} \displaystyle {\mathcal A}^l_{\beta ,\gamma }:\,\Big \{v\in {\mathcal V}^{l+1,0}_{\beta ,\gamma } ({{\mathbb {R}}}^2_+):\,\frac{\partial v}{\partial x_2}(x_1,0){ = 0}, \,\,x_1<0\Big \} \rightarrow {\mathcal V}^{l-1}_{\beta ,\gamma }({{\mathbb {R}}}^2_+) \end{aligned}$$

(A.10)

of problem (A.1), (A.2) is an isomorphism and in the case of the right-hand side $f\in V^{l-1}_{\beta -2,\gamma ^1}({{\mathbb {R}}}^2_+)$ with the compact support (A.5) and the second weight index $\gamma ^1\in (0,\gamma ]$ the asymptotic representation (A.6) is valid where the remainder ${\widetilde{v}}\in {\mathcal V}^{l+1,0}_{\beta ,\gamma ^1}({{\mathbb {R}}}^2_+)$ and the coefficients K, $K^1$ satisfy appropriately modified estimate (A.7).

We emphasize that the Dirichlet condition at the semi-axis ${{\mathbb {R}}}_+\ni x_1$ does not appear explicitly in the domain of the operator (A.10) because, by virtue of the restriction $\gamma <1/2$, the assumption $v(r,0)\not =0$ leads to the divergent integral

$$\begin{aligned} \displaystyle \int \limits _0^\pi \varphi ^{2(\gamma -l-1)}|v(r,\varphi )|^2d\varphi . \end{aligned}$$

(A.11)

1.3 A.3 Weighted spaces with detached asymptotics

Under condition (A.9) operator (A.10) stays Fredholm for any $\beta \in {{\mathbb {R}}}$ with the exception of the forbidden indexes $\beta _{\pm j}$ indicated in Section A.1. However, denying (A.9) deprives the operator of the Fredholm property. For example, in the case $\gamma -l<-1/2$ a function $v\in {\mathcal V}_{\beta ,\gamma }^{l+1}({{\mathbb {R}}}^2_+)$ has a finite norm only under the two conditions $v(x_1,0)=0$ and $\displaystyle \frac{\partial v}{\partial x_2}(x_1,0)=0$ on the semi-axis ${{\mathbb {R}}}_+$. The latter is not possible for a non-trivial harmonics due to the theorem on unique extension. To vary the second weight index $\gamma$ requires detaching asymptotics, cf. Ch. 12 in [31], namely to deal with functions in the form

$$\begin{aligned} \displaystyle v(x)= {\mathcal {K}}(r) r^{1/2}\sin \frac{\varphi }{2} +{\mathcal {K}}^1(r) r^{3/2}\sin \frac{3\varphi }{2} +{\widetilde{v}}(x). \end{aligned}$$

(A.12)

The remainder ${\widetilde{v}}$ must belong to the space ${\mathcal V}^{l+1}_{\beta ,\gamma }({{\mathbb {R}}}_+^2)$ with the weighted norm

$$\begin{aligned} \displaystyle \Vert {\widetilde{v}};{\mathcal V}^{l+1}_{\beta ,\gamma }({{\mathbb {R}}}_+^2)\Vert = \left( \,\sum \limits _{j=0}^{l+1} \left\| r^{\beta -l-1+j}\varphi ^{\gamma -l-1+j}(\pi -\varphi )^{\gamma -l-1+j}\nabla ^j {\widetilde{v}};L^2({{\mathbb {R}}}^2_+)\right\| ^2\right) ^{1/2} \end{aligned}$$

(A.13)

and the weight indexes

$$\begin{aligned} \displaystyle \beta -l\in \left( -\frac{1}{2},\frac{1}{2}\right) ,\quad \gamma -l \in \left( -\frac{3}{2},-\frac{1}{2}\right) . \end{aligned}$$

(A.14)

We emphasize that now, in contrast to (A.8), weights are introduced in (A.13) at both endpoints of the arc $(0,\pi )$ while the restriction on $\gamma$ in (A.14) demands that the remainder ${\widetilde{v}} \in { {\mathcal V}}^{l+1}_{\beta ,\gamma }({{\mathbb {R}}}_+^2)$ satisfies formulas

$$\begin{aligned} {\widetilde{v}}(x_1,0)=0,\,\,\frac{\partial {\widetilde{v}}}{\partial x_2}(x_1,0)=0\,\, \quad \hbox { for}\,\, x_1\in {{\mathbb {R}}}{\setminus }\{0\}; \end{aligned}$$

otherwise, norm (A.13) cannot be finite because of the divergence of the integral [similar to (A.11)]

$$\begin{aligned} \int \limits _0^\pi \varphi ^{2(\gamma -l-1)} (\pi -\varphi )^{2(\gamma -l-1)}|{\widetilde{v}}(r,\varphi )|^2d\varphi . \end{aligned}$$

Hence, according to (A.12), we have

$$\begin{aligned} \begin{aligned} \displaystyle v(r,0)&=0,\,\,\frac{\partial v}{\partial x_2}(r,0)=\frac{1}{2}\, r^{-1/2}{\mathcal {K}}(r) + \frac{3}{2}\, r^{1/2}{\mathcal {K}}^1(r) \quad \hbox {for}\quad x_1=r>0,\\ \displaystyle v(r,0)&= r^{1/2}{\mathcal {K}}(r)- r^{3/2}{\mathcal {K}}^1(r),\,\,\frac{\partial v}{\partial x_2}(r,0)= 0 \,\,\, \quad \hbox { for}\,\, x_1=-r<0. \end{aligned} \end{aligned}$$

(A.15)

Roughly speaking, to compose from functions (A.12) a weighted space with detached asymptotics by means of a procedure in Ch. 12 of [31] requires setting the coefficient functions ${\mathcal {K}}{ (r)}$, ${\mathcal {K}}^1{ (r)}$ in a certain weighted Kondratiev space and incorporating their norms together with norm (A.13) into the norm of the whole function v. Additional difficulties originate in insufficient smoothness properties of the coefficients: according to (A.15) none of the traces $v\big |_{{{\mathbb {R}}}_\pm }$ and $\displaystyle \frac{\partial v}{\partial x_2}\Big |_{{{\mathbb {R}}}_\pm }$ and, therefore, none of ${\mathcal {K}}$ and ${\mathcal {K}}^1$ belongs to the proper space $H^{l+1}({{\mathbb {R}}}_\pm )$. The latter requires the introduction of special extension operators into the asymptotic forms of type (A.12) (cf. [11, 34] and Ch. 12 in [31]). To avoid unnecessary complications, we consider a particular case with an infinitely differentiable right-hand side f vanishing near the coordinate origin, we deal with the model differential equation corresponding to the original problem at collision points

$$\begin{aligned} \displaystyle -\Delta v -\mu H v = f \quad \text{ in } \quad {{\mathbb {B}}}^+_R, \end{aligned}$$

(A.16)

and we write only an asymptotic formula for a solution of (A.16), (A.2) near the point ${{\mathcal {O}}}$. In (A.16), $\mu \in {{\mathbb {R}}}_+$, $H=H(x_1)$ is the Heaviside unit step function and ${{\mathbb {B}}}^+_R=\{x:\, |x|<R, x_2>0\}$ is the upper half-disk of radius $R>0$. We have the following result.

Proposition A.1

Let $v\in H^1({{\mathbb {B}}}^+_R)\cap H^2_{loc}(\overline{{{\mathbb {B}}}^+_R}{\setminus }{{\mathcal {O}}})$ satisfy equation (A.16) with

$$\begin{aligned} f\in L^2({{\mathbb {B}}}^+_R)\,\, \text{ and }\,\, f(x)=0 \text{ for } r<R/2, \end{aligned}$$

and the boundary conditions (A.2) for $r\in (0,R)$. Then v falls into ${\mathcal V}^2_{1, \gamma }({{\mathbb {B}}}^+_{2R/3})$ with any $\gamma \in (1/2,3/2)$ and admits the asymptotic form

$$\begin{aligned} \displaystyle v(x)=\varsigma (r)\left( {\mathcal {K}}(r) r^{1/2}\sin \frac{\varphi }{2} +{\mathcal {K}}^1(r) r^{3/2}\sin \frac{3\varphi }{2}\right) +{\widetilde{v}}(x) \,\,\,\,\textit{ for}\,\,\, r\le \frac{1}{3}R, \end{aligned}$$

(A.17)

where the remainder ${\widetilde{v}}$ and the coefficients

$$\begin{aligned} {\mathcal {K}}(r)=K+\widetilde{{\mathcal {K}}}(r), \quad {\mathcal {K}}^1(r)=K^1+\widetilde{{\mathcal {K}}}^{\,1}(r) \end{aligned}$$

fulfill the estimate

$$\begin{aligned} \begin{aligned}&\sum \limits _{j= 0,1}\Big (r^{-2+j}\big |\partial ^j_r \widetilde{{\mathcal {K}}}(r)\big | +r^{-1+j}\big |\partial ^j_r \widetilde{{\mathcal {K}}}^{\,1}(r)\big | +r^{-2+j}\varphi ^{-2+j} (\pi -\varphi )^{-2+j}\big |\nabla ^j {\widetilde{v}}(x)\big |\Big ) \\&\quad + |K|+|K^1| \le c\Big (\Vert f;L^2({{\mathbb {B}}}^+_R{\setminus }{{\mathbb {B}}}^+_{{ R}/2})\Vert + \Vert v;H^1({{\mathbb {B}}}^+_R)\Vert \Big ) \,\,\quad \textit{ for}\,\, r\le \frac{1}{3}R. \end{aligned} \end{aligned}$$

(A.18)

Formulas (A.17) and (A.18) suffice to support all the calculations and estimations for the quotient function (2.29) in Sects. 2.4 and 5.2.