Eigenvalues of elliptic operators with density

Abstract

We consider eigenvalue problems for elliptic operators of arbitrary order 2m subject to Neumann boundary conditions on bounded domains of the Euclidean N-dimensional space. We study the dependence of the eigenvalues upon variations of mass density. In particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.

Appendices

Appendices

A Eigenvalues of polyharmonic operators

In this section we shall present some basics of spectral theory for the polyharmonic operators. In particular, we will discuss Neumann boundary conditions, mainly for the Laplace and the biharmonic operator. Then we will characterize the spectrum of the polyharmonic operators subject to Neumann boundary conditions by exploiting classical tools of spectral theory for compact self-adjoint operators. We refer to [21, 34] and to references therein for a discussion on eigenvalue problems for general elliptic operators of order 2m with density subject to homogeneous boundary conditions.

1.1 Neumann boundary conditions

Neumann boundary conditions are usually called ‘natural’ boundary conditions. They are well understood for the Laplace operator. In fact, assume that u is a classical solution of (9). If we multiply the equation $-\varDelta u=\mu \rho u$ by a test function $\varphi \in C^{\infty }(\varOmega )$ and integrate both sides of the resulting identity over $\varOmega$, thanks to Green’s formula we obtain:

$$\begin{aligned} \int _{\varOmega }\nabla u\cdot \nabla \varphi dx=\mu \int _{\varOmega }\rho u\varphi dx+\int _{\partial \varOmega }\frac{\partial u}{\partial \nu }\varphi d\sigma =\mu \int _{\varOmega }\rho u\varphi dx. \end{aligned}$$

Hence (7) with $m=1$ holds for all $\varphi \in C^{\infty }(\varOmega )$ when u is a solution of (8). We can relax our hypothesis on u and just require that $u\in H^1(\varOmega )$ and that (7) holds for all $\varphi \in H^1(\varOmega )$. Hence (7) is the weak formulation of the Neumann eigenvalue problem for the Laplace operator. We note that the boundary condition in (8) arises naturally and is not imposed a priori with the choice of a subspace of $H^1(\varOmega )$ in the weak formulation (as in the case of $H^1_0(\varOmega )$ for Dirichlet conditions): if a weak solution of (7) for $m=1$ exists and is sufficiently smooth, then it solves $-\varDelta u=\mu \rho u$ in the classical sense and satisfies the Neumann boundary condition $\frac{\partial u}{\partial \nu }=0$.

Let us now consider the case of the biharmonic operator in more detail. Assume that u is a classical solution of problem (10). We multiply the equation $\varDelta ^2u=\mu \rho u$ by a test function $\varphi \in C^{\infty }(\varOmega )$ and apply the biharmonic Green’s formula (see [1, Lemma 8.56]). We obtain:

$$\begin{aligned} \int _\varOmega \varDelta ^2 u\varphi dx= & {} \int _{\varOmega }D^2u:D^2\varphi dx\\&+\int _{\partial \varOmega }\left( \mathrm{div}_{\partial \varOmega }\left( D^2u\cdot \nu \right) _{\partial \varOmega }+\frac{\partial \varDelta u}{\partial \nu }\right) \varphi d\sigma -\int _{\partial \varOmega }\frac{\partial ^2 u}{\partial \nu ^2}\frac{\partial \varphi }{\partial \nu } d\sigma \\&=\int _{\varOmega }D^2u{:}\,D^2\varphi dx=\mu \int _{\varOmega }\rho u\varphi dx, \end{aligned}$$

where $(D^2u\cdot \nu )_{\partial \varOmega }$ denotes the tangential component of $D^2u\cdot \nu$. Hence (7) with $m=2$ holds for all $\varphi \in C^{\infty }(\varOmega )$ when u is a solution of problem (10) (we remark that if $\frac{\partial ^2u}{\partial \nu ^2}=0$ then $(D^2u\cdot \nu )_{\partial \varOmega }=(D^2u\cdot \nu )$). We can relax our hypothesis on u and just require that $u\in H^2(\varOmega )$ and that (7) holds for all $\varphi \in H^2(\varOmega )$. This is exactly the weak formulation of the Neumann eigenvalue problem for the biharmonic operator. We note again that the two boundary conditions in (10) arise naturally and are not imposed a priori: if a weak solution of (7) exists and is sufficiently smooth, then it satisfies the two Neumann boundary conditions. We also remark that if $\varOmega$ is sufficiently regular, e.g., if it is of class $C^k$ with $k>4+\frac{N}{2}$ and $\rho$ is continuous, then a weak solution of (7) with $m=2$ is actually a classical solution of (10) (see [26, § 2]). The choice of the whole space $H^2(\varOmega )$ in the weak formulation (7) contains the information on the boundary conditions in (8).

It is natural then to consider problem (7) for any $m\in \mathbb N$ as the weak formulation of an eigenvalue problem for the polyharmonic operator with Neumann boundary conditions. In the case of a generic value of m it is much more difficult to write explicitly the boundary operators $\mathcal N_0,\ldots ,\mathcal N_{m-1}$ (this is already extremely involved for $m=3$). Moreover, if $\varOmega$ is sufficiently regular and $\rho$ is continuous, then weak solutions of (7) are actually classical solution of (8), and the m boundary conditions are uniquely determined and arise naturally from the choice of the whole space $H^m(\varOmega )$ (see [26] for further discussions on higher order elliptic operators and eigenvalue problems).

1.2 Characterization of the spectrum

The aim of this subsection is to prove that problem (7) admits an increasing sequence of non-negative eigenvalues of finite multiplicity diverging to $+\,\infty$, and to provide some additional information on the spectrum. To do so, we will reduce problem (7) to an eigenvalue problem for a compact self-adjoint operator on a Hilbert space.

First we define the following (equivalent) problem: find $u\in H^m(\varOmega )$ and $\varLambda \in \mathbb R$ such that

$$\begin{aligned} \int _{\varOmega }D^mu{:}\,D^m\varphi +\rho u\varphi dx=\varLambda \int _{\varOmega }\rho u\varphi dx\,,\ \ \ \forall \varphi \in H^m(\varOmega ). \end{aligned}$$

(63)

Clearly the eigenfuctions of (63) coincide with the eigenfunctions of (7), while all the eigenvalues $\mu$ of (7) are given by $\mu =\varLambda -1$, where $\varLambda$ is an eigenvalue of (63).

We consider the operator $(-\varDelta )^m+\rho I_d$ as a map from $H^m(\varOmega )$ to its dual $H^m(\varOmega )'$ defined by

$$\begin{aligned} ((-\varDelta )^m+\rho I_d)[u][\varphi ]:=\int _\varOmega D^m u: D^m\varphi +\rho u\varphi dx,\ \ \ \mathcal 8\varphi \in H^m(\varOmega ). \end{aligned}$$

The operator $(-\varDelta )^m+\rho I_d$ is a continuous isomorphism between $H^m(\varOmega )$ and $H^m(\varOmega )'$. In fact it follows immediately that there exist constants $C_1,C_2>0$ such that

$$\begin{aligned} C_1\Vert u\Vert ^2_{H^m(\varOmega )}\le ((-\varDelta )^m+\rho I_d)[u][u]\le C_2 \Vert u\Vert ^2_{H^m(\varOmega )},\ \forall \,u\in H^m(\varOmega ). \end{aligned}$$

(64)

Next we denote by i the canonical embedding of $H^m(\varOmega )$ into $L^2(\varOmega )$ and by $J_\rho$ the embedding of $L^2(\varOmega )$ into $H^m(\varOmega )'$, defined by

$$\begin{aligned} J_\rho [u][\varphi ]:=\int _\varOmega \rho u\varphi dx\ \ \ \mathcal 8 u\in L^2(\varOmega ),\varphi \in H^m(\varOmega ). \end{aligned}$$

Let $T_\rho$ be the operator from $H^m(\varOmega )$ to itself defined by $T_\rho :=((-\varDelta )^m+\rho I_d)^{(-1)}\circ J_\rho \circ i$. Problem (63) is then equivalent to

$$\begin{aligned} T_\rho u =\varLambda ^{-1}u, \end{aligned}$$

in the unknowns $u\in H^m(\varOmega )$, $\varLambda \in \mathbb R$. We now consider the space $H^m(\varOmega )$ endowed with the bilinear form

$$\begin{aligned} \langle u,v\rangle _{\rho }=\int _\varOmega D^mu{:}\,D^mv+\rho u v dx,\ \ \ \mathcal 8 u,v\in H^m(\varOmega ). \end{aligned}$$

(65)

From (64) it follows that (65) is a scalar product on $H^m(\varOmega )$ whose induced norm is equivalent to the standard one. We denote by ${H^m_{\rho }}(\varOmega )$ the space $H^m(\varOmega )$ endowed with the scalar product defined by (65). Then we can state the following theorem:

Theorem 15

Let $\varOmega$ be a bounded domain in $\mathbb R^N$ such that the embedding $H^m(\varOmega )\subset L^2(\varOmega )$ is compact. Let $\rho \in \mathcal R$. Then the operator $T_\rho$ is a compact self-adjoint operator in ${H^m_{\rho }}(\varOmega )$, whose eigenvalues coincide with the reciprocals of the eigenvalues of problem (63) for all $j\in \mathbb N$.

The proof of Theorem 15 is standard, hence we omit it (see e.g., [8, § IX]). As a consequence of Theorem 15 we have the following:

Theorem 16

Let $\varOmega$ be a bounded domain in $\mathbb R^N$ such that the embedding $H^m(\varOmega )\subset L^2(\varOmega )$ is compact. Let $\rho \in \mathcal R$. Then the set of the eigenvalues of (7) is contained in $[0,+\,\infty [$ and consists of the image of a sequence increasing to $+\,\infty$. The eigenvalue $\mu =0$ has multiplicity $d_{N,m}=\left( {\begin{array}{c}N+m-1\\ N\end{array}}\right)$ and the eigenfunctions corresponding to the eigenvalue $\mu =0$ are the polynomials of degree at most $m-1$ in $\mathbb R^N$. Each eigenvalue has finite multiplicity. Moreover the space $H^m_{\rho }(\varOmega )$ has a Hilbert basis of eigenfunctions of problem (7).

Proof

We note that $ker(T_{\rho })=\left\{ 0\right\}$, hence by standard spectral theory it follows that the eigenvalues of $T_{\rho }$ are positive and bounded and form an infinite sequence $\left\{ \lambda _j\right\} _{j\in \mathbb N}$ converging to zero. Moreover to each eigenvalue $\lambda _j$ it is possible to associate an eigenfunction $u_j$ such that $\left\{ u_j\right\} _{j\in \mathbb N}$ is a orthonormal basis of $H^m_{\rho }(\varOmega )$.

From Theorem 15 it follows that the eigenvalues of (63) form a sequence of real numbers increasing to $+\,\infty$ which is given by $\left\{ \varLambda _j=\lambda _j^{-1}\right\} _{j\in \mathbb N}$ and that the space $H^m_{\rho }(\varOmega )$ has a Hilbert basis of eigenfunctions of (63). The eigenvalues $\mu _j$ of (7) are given by $\mu _j=\varLambda _j-1$ for all $j\in \mathbb N$, where $\left\{ \varLambda _j\right\} _{j\in \mathbb N}$ are the eigenvalues of (63) and the eigenfunctions associated with $\varLambda _j$ coincide with the eigenfunctions associated with $\mu _j=\varLambda _j-1$. Moreover, given an eigenvalue $\mu$ of (7) and a corresponding eigenfunction u, we have that

$$\begin{aligned} \int _{\varOmega }|D^m u|^2dx=\mu \int _{\varOmega }\rho u^2 dx, \end{aligned}$$

thus $\mu \in [0,+\,\infty [$. Finally, if $\mu =0$, then $\int _{\varOmega }|D^m u|^2dx=0$, thus u is a polynomial of degree at most $m-1$ in $\mathbb R^N$. The eigenspace associated with the eigenvalue $\mu =0$ has dimension $\left( {\begin{array}{c}N+m-1\\ N\end{array}}\right)$ and coincides with the space of the polynomials of degree at most $m-1$ in $\mathbb R^N$. This concludes the proof. $\square$

B A few useful functional inequalities

In this section we will prove some useful functional inequalities which are crucial in the proof of the results of Sect. 3.3, in particular of Theorem 6. Since we think that they are interesting in their own right, we shall provide all the details of the proofs here. Throughout this section $\varOmega$ will be a bounded domain in $\mathbb R^N$ with Lipschitz boundary. We start this section by recalling the standard Sobolev embeddings.

Theorem 17

Let $\varOmega$ be a bounded domain with Lipschitz boundary. Let $m\in \mathbb N$ and assume that $u\in H^m(\varOmega )$.

• (i) If $N<2m$ then $u\in C^{m-\left[ \frac{N}{2}\right] -1,\gamma }(\varOmega )$, where

  $$\begin{aligned} \gamma = {\left\{ \begin{array}{ll} \left[ \frac{N}{2}\right] +1-\frac{N}{2}, &{} \mathrm{if\ } \frac{N}{2}\notin \mathbb N\\ \mathrm{any\ number\ in\ } ]0,1[,&{} \mathrm{if\ } \frac{N}{2}\in \mathbb N. \end{array}\right. } \end{aligned}$$

  Moreover there exists a positive constant C which depends only on m, N and $\varOmega$ such that

  $$\begin{aligned} \Vert u\Vert _{C^{m-\left[ \frac{N}{2}\right] -1,\gamma }(\varOmega )}\le C\Vert u\Vert _{H^m(\varOmega )}. \end{aligned}$$

  (66)

• (ii) If $N>2m$ then $u\in L^{\frac{2N}{N-2m}}(\varOmega )$ and

  $$\begin{aligned} \Vert u\Vert _{L^{\frac{2N}{N-2m}}(\varOmega )}\le C\Vert u\Vert _{H^m(\varOmega )}, \end{aligned}$$

  (67)

  where the constant C depends only on m, N and $\varOmega$.

• (iii) If $N=2m$ then there exist constants $C_1, C_2>0$ which depend only on m and $\varOmega$ such that

  $$\begin{aligned} \int _{\varOmega }e^{C_1\left( \frac{u(x)}{\Vert u\Vert _{H^{m}(\varOmega )}}\right) ^2}dx\le C_2. \end{aligned}$$

  (68)

We refer to [10, § 4.6-4.7] and [22, § 5.6.3] for the proof of points (i) and (ii) of Theorem 17. We refer to [13, Theorem 1.1] for the proof of (68) (see also [10, § 4.7]).

From Theorem 17 it follows that if $N<2m$ then a function $u\in H^m(\varOmega )$ is (equivalent to) a function of class $C^{m-\left[ \frac{N}{2}\right] -1}(\varOmega )$. In particular if N is odd, we can write $N=2m-2k-1$ for some $k\in \left\{ 0,\ldots ,m-1\right\}$ and a function $u\in H^m(\varOmega )$ is (equivalent to) a function of class $C^{k,\frac{1}{2}}(\varOmega )$. If N is even, we can write $N=2m-2k-2$ for some $k\in \left\{ 0,\ldots ,m-2\right\}$ and a function $u\in H^m(\varOmega )$ is (equivalent to) a function of class $C^{k,\gamma }(\varOmega )$ for any $\gamma \in ]0,1[$.

Assume now that a function $u\in H^{m}(\varOmega )$ has all its partial derivatives up to the kth order vanishing at a point $x_0\in \varOmega$. Then the integral of $u^2$ over $B(x_0,\varepsilon )$ (where $\varepsilon >0$ is such that $B(x_0,\varepsilon )\subset \subset \varOmega$) can be controlled by $\varepsilon ^{2m}\Vert u\Vert ^2_{H^m(\varOmega )}$ if $N<2m$ is odd, and by $\varepsilon ^{2m}(1+|\log (\varepsilon )|)\Vert u\Vert ^2_{H^m(\varOmega )}$ if $N<2m$ is even. This is proved in the following lemma where, without loss of generality, we set $x_0=0$.

Lemma 1

Let $\varOmega$ be a bounded domain in $\mathbb R^N$, $N<2m$, with Lipschitz boundary. Assume that $0\in \varOmega$ and let $\varepsilon >0$ be such that $B(0,\varepsilon )\subset \subset \varOmega$. Let $u\in H^m(\varOmega )$. Then there exists a positive constant C which depends only on m, k and $\varOmega$ such that

• (i) $\int _{B(0,\varepsilon )}{\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right| ^2dx}\le C\varepsilon ^{2m}\Vert u\Vert _{H^m(\varOmega )}^2$ if $N=2m-2k-1$, $\forall k\in \{0,\ldots ,m-1\};$

• (ii) $\int _{B(0,\varepsilon )}{\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right| ^2dx}\le C\varepsilon ^{2m}(1+|\log (\varepsilon )|)\Vert u\Vert _{H^m(\varOmega )}^2$ if $N=2m-2k-2$, $\forall k\in \{0,\ldots ,m-2\}$.

Proof

We start by proving (i). Let $N=2m-2k-1$ for some $k\in \left\{ 0,\ldots ,m-1\right\}$. Actually, we will prove (i) for a function $u\in C^{k+1}(\varOmega )\cap H^m(\varOmega )$. The result for a function $u\in H^m(\varOmega )$ will follow from standard approximation of functions in the space $H^m(\varOmega )$ by smooth functions (see [10, § 2.3] and [22, § 5.3]). Let $u\in C^{k+1}(\varOmega )\cap H^m(\varOmega )$. Throughout the rest of the proof we shall denote by C a positive constant which depends only on m, k and $\varOmega$ and which can be eventually re-defined line by line. From the standard Sobolev embedding Theorem, it follows that $u\in C^{k+1}(\varOmega )\cap C^{k,\frac{1}{2}}(\varOmega )$. From Taylor’s Theorem it follows that

$$\begin{aligned} u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }=\sum _{|\beta |=k+1}\frac{|\beta |}{\beta !}\int _0^1(1-t)^k\partial ^{\beta }u(tx) x^{\beta }dt. \end{aligned}$$

(69)

We now consider the absolute value of the expression in the right-hand side of (69) and integrate each integral which appears in the sum over $B(0,\varepsilon )$. We have

$$\begin{aligned}&\int _{B(0,\varepsilon )}\left| \int _0^1(1-t)^k\partial ^{\beta }u(tx)x^{\beta }dt\right| dx\le \int _0^1(1-t)^k\int _{B(0,\varepsilon )}|\partial ^{\beta }u(tx)||x|^{k+1}dxdt\nonumber \\&\quad =\int _0^1t^{-2m+k}(1-t)^k\int _{B(0,t\varepsilon )}|\partial ^{\beta }u(y)||y|^{k+1}dydt\nonumber \\&\quad \le \int _0^1t^{-2m+k}(1-t)^k \left( \int _{B(0,t\varepsilon )}|\partial ^{\beta }u(y)|^{2 (2m-2k-1)}dy\right) ^{\frac{1}{2(2m-2k-1)}}\nonumber \\&\quad \quad \cdot \left( \int _{B(0,t\varepsilon )}|y|^{\frac{2(2m-2k-1)(k+1)}{2(2m-2k-1)-1}}dy\right) ^{\frac{2(2m-2k-1)-1}{2(2m-2k-1)}}dt\nonumber \\&\quad \le C\int _0^1t^{-2m+k}(1-t)^k \Vert \partial ^{\beta }u\Vert _{L^{2(2m-2k-1)}(\varOmega )}\varepsilon ^{2m-k-\frac{1}{2}}t^{2m-k-\frac{1}{2}}dt\nonumber \\&\quad =C\varepsilon ^{2m-k-\frac{1}{2}}\Vert \partial ^{\beta }u\Vert _{L^{2(2m-2k-1)}(\varOmega )}\le C\varepsilon ^{2m-k-\frac{1}{2}}\Vert \partial ^{\beta }u\Vert _{H^{m-k-1}(\varOmega )} \nonumber \\&\quad \le C\varepsilon ^{2m-k-\frac{1}{2}}\Vert u\Vert _{H^{m}(\varOmega )}, \end{aligned}$$

(70)

where in the last line we have used the Sobolev inequality (67) for functions in $H^{m-k-1}(\varOmega )$ with $\varOmega \in \mathbb R^{2m-2k-1}$.

Next, we estimate the quantity

$$\begin{aligned} \left\| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right\| _{C^{0}(B(0,\varepsilon ))}:=\max _{x\in B(0,\varepsilon )}\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right| \end{aligned}$$

for a function $u\in C^{k,\frac{1}{2}}(\varOmega )$. First we note that there exits $t\in ]0,1[$ such that

$$\begin{aligned} u(x)-\sum _{|\alpha |\le {k-1}}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }=\sum _{|\alpha |=k}\partial ^{\alpha }u(tx)\frac{x^{\alpha }}{\alpha !}, \end{aligned}$$

then

$$\begin{aligned}&\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right| =\left| \sum _{|\alpha |=k}\frac{x^{\alpha }}{\alpha !}(\partial ^{\alpha }u(0)-\partial ^{\alpha }u(tx)\right| \\&\quad \le |x|^k\sum _{|\alpha |=k}\frac{1}{\alpha !}|\partial ^{\alpha }u(0)-\partial ^{\alpha }u(tx)|, \end{aligned}$$

which implies

$$\begin{aligned} \left\| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right\| _{C^{0}(B(0,\varepsilon ))}\le C\varepsilon ^{k+\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}, \end{aligned}$$

(71)

where the constant C depends only on m, k and $\varOmega$ [see also (66)].

Consider now (69). We take the squares of both sides and integrate over $B(0,\varepsilon )$. We have

$$\begin{aligned}&\int _{B(0,\varepsilon )}\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !} x^{\alpha }\right| ^2dx\nonumber \\&\quad \le \left\| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right\| _{C^{0}(B(0,\varepsilon ))} \int _{B(0,\varepsilon )}\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !} x^{\alpha }\right| dx\nonumber \\&\quad \le C \varepsilon ^{2m}\Vert u\Vert _{H^m(\varOmega )}^2, \end{aligned}$$

(72)

where the last inequality follows from (70) and (71) and the constant C depends only on m, k and $\varOmega$. Since inequality (72) holds for all $u\in C^{k+1}(\varOmega )\cap H^m(\varOmega )$, by standard approximation of $H^m(\varOmega )$ functions by smooth functions, we conclude that it holds for all $u\in H^m(\varOmega )$. This proves (i).

Now consider (ii). Let $N=2m-2k-2$ for some $k\in \left\{ 0,\ldots ,m-2\right\}$. Again, we shall prove (ii) for a function $u\in C^{k+1}(\varOmega )\cap H^m(\varOmega )$. The result for a function $u\in H^m(\varOmega )$ follows from standard approximation of functions in the space $H^m(\varOmega )$ by smooth functions.

We first prove the following inequality:

$$\begin{aligned} \Vert f\Vert _{L^2(B(0,\varepsilon ))}^2\le C\varepsilon ^{2m-2k-2}(1+|\log (\varepsilon )|)\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2, \end{aligned}$$

(73)

for all $f\in H^{m-k-1}(\varOmega )$ (the constant $C>0$ depending only on m, k and $\varOmega$). In order to prove (73) we will use the exponential inequality (68) which describes the limiting behavior of the Sobolev inequality (67) when $N=2m-2k-2$ for functions in $H^{m-k-1}(\varOmega )$. Let $f\in H^{m-k-1}(\varOmega )$ and let $C_1,C_2$ be the constants appearing in (68). Then

$$\begin{aligned}&\int _{B(0,\varepsilon )} f(x)^2dx=\frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\int _{B(0,\varepsilon )}C_1\left( \frac{f(x)}{\Vert f\Vert _{H^{m-k-1}(\varOmega )}}\right) ^2\frac{dx}{|B(0,\varepsilon )|}\\&\quad = \frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\int _{B(0,\varepsilon )}\log \left( e^{C_1\left( \frac{f(x)}{\Vert f\Vert _{H^{m-k-1}(\varOmega )}}\right) ^2}\right) \frac{dx}{|B(0,\varepsilon )|}\\&\quad \le \frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\log \left( \int _{B(0,\varepsilon )}e^{C_1\left( \frac{f(x)}{\Vert f\Vert _{H^{m-k-1}(\varOmega )}}\right) ^2}\frac{dx}{|B(0,\varepsilon )|}\right) \\&\quad \le \frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\log \left( \frac{1}{|B(0,\varepsilon )|}\int _{\varOmega }e^{C_1\left( \frac{f(x)}{\Vert f\Vert _{H^{m-k-1}(\varOmega )}}\right) ^2}dx\right) \\&\quad \le \frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\log \left( \frac{C_2}{|B(0,\varepsilon )|}\right) \\&\quad =\frac{|B(0,\varepsilon )|}{C_1}\Vert f\Vert _{H^{m-k-1}(\varOmega )}^2\left( \log (C_2)-\log (|B(0,\varepsilon )|)\right) \\&\quad \le C\varepsilon ^{2m-2k-2}(1+|\log (\varepsilon )|) \Vert f\Vert _{H^{m-k-1}(\varOmega )}^2, \end{aligned}$$

where in the first inequality we have used the concavity of the logarithm and Jensen’s inequality and in the third inequality we have applied (68). Inequality (73) is now proved.

Now let $u\in C^{k+1}(\varOmega )\cap H^m(\varOmega )$. From the Sobolev inequality (66), it follows that $u\in C^{k+1}(\varOmega )\cap C^{k,\gamma }(\varOmega )$ for all $\gamma \in ]0,1[$. From Taylor’s Theorem [see also (69)] it follows that

$$\begin{aligned}&\int _{B(0,\varepsilon )}\left| u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha } u(0)}{\alpha !}x^{\alpha }\right| ^2dx\nonumber \\&\quad \le \sum _{|\beta |=k+1}\frac{(k+1)!}{(\beta !)^2}\int _{B(0,\varepsilon )}\int _0^1(1-t)^{2k}|\partial ^{\beta }u(tx)|^2|x|^{2(k+1)}dtdx. \end{aligned}$$

(74)

We estimate the integrals appearing in the right-hand side of (74) as follows:

$$\begin{aligned}&\int _{B(0,\varepsilon )}\int _0^1(1-t)^{2k}|\partial ^{\beta }u(tx)|^2|x|^{2(k+1)}dtdx\nonumber \\&\quad =\int _0^1(1-t)^{2k}\int _{B(0,\varepsilon )}|\partial ^{\beta }u(tx)|^2|x|^{2(k+1)}dxdt\nonumber \\&\quad =\int _0^1t^{-2m}(1-t)^{2k}\int _{B(0,t\varepsilon )}|\partial ^{\beta }u(y)|^2|y|^{2(k+1)}dxdt\nonumber \\&\quad \le \int _0^1t^{-2m+2k+2}\varepsilon ^{2k+2}(1-t)^{2k}\Vert \partial ^{\beta }u\Vert _{L^2(B(0,t\varepsilon ))}^2dt\nonumber \\&\quad \le C\varepsilon ^{2m}(1+|\log (\varepsilon )|)\Vert u\Vert _{H^m(\varOmega )}^2, \end{aligned}$$

(75)

where the last inequality follows from (73) applied with $f=\partial ^{\beta }u$. From (74) and (75), (ii) immediately follows. This concludes the proof of the lemma. $\square$

Assume now that $u\in H^m(\varOmega )$ is such that $\int _{\varOmega }\tilde{\rho }_{\varepsilon }u(x)x^{\alpha }dx=0$ for all $|\alpha |\le m-1$, where $\tilde{\rho }_{\varepsilon }$ is defined by (36). This means that, for a fixed $\delta \in ]0,1/2[$

$$\begin{aligned} \int _{\varOmega }u(x)x^{\alpha }dx+\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}u(x)x^{\alpha }dx=0 \end{aligned}$$

for all $|\alpha |\le m-1$. In view of Lemma 1 we expect that the quantities $|\partial ^{\alpha }u(0)|$, $\left( \int _{B(0,\varepsilon )}u^2dx\right) ^{\frac{1}{2}}$ and $\int _{\varOmega }u(x)x^{\alpha }dx$ can be bounded by $\Vert u\Vert _{H^m(\varOmega )}$ and a suitable power of $\varepsilon$. In particular, we expect that all these quantities vanish as $\varepsilon \rightarrow 0^+$. The aim of the next lemma is to prove that this is exactly what happens. We shall also provide the correct powers of $\varepsilon$ in the estimates which are crucial in the proof of Theorem 6.

Lemma 2

Let $\varOmega$ be a bounded domain in $\mathbb R^N$, $N<2m$, with Lipschitz boundary. Assume that $0\in \varOmega$ and let $\varepsilon >0$ be such that $B(0,\varepsilon )\subset \subset \varOmega$. Let $\delta \in ]0,1/2[$ be fixed and let $\tilde{\rho }_{\varepsilon }$ be defined by (36). Let $u\in H^m(\varOmega )$ be such that $\int _{\varOmega }\tilde{\rho }_{\varepsilon } u(x) x^{\beta } dx=0$ for all $\beta \in \mathbb N^N$ with $|\beta |\le m-1$. Then there exists a positive constant C which depends only on m, k and $\varOmega$ such that

• (i) $|\partial ^{\alpha }u(0)|\le C\varepsilon ^{k+\frac{1}{2}-|\alpha |}\Vert u\Vert _{H^m(\varOmega )}$ for all $\alpha \in \mathbb N^N$ with $|\alpha |\le k$, if $N=2m-2k-1$, $\forall \,k=0,\ldots ,m-1$;

• (ii) $|\partial ^{\alpha }u(0)|\le C\varepsilon ^{k+1-|\alpha |}(1+|\log (\varepsilon )|)^{\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}$ for all $\alpha \in \mathbb N^N$ with $|\alpha |\le k$, if $N=2m-2k-2$, $\forall \,k=0,\ldots ,m-2$;

• (iii) $\int _{B(0,\varepsilon )}u^2dx\le C\varepsilon ^{2m}\Vert u\Vert _{H^m(\varOmega )}^2$ if $N=2m-2k-1$, $\forall \,k=1,\ldots ,m-1$;

• (iv) $\int _{B(0,\varepsilon )}u^2dx\le C\varepsilon ^{2m} (1+|\log (\varepsilon )|)\Vert u\Vert _{H^m(\varOmega )}^2$ if $N=2m-2k-2$, $\forall \,k=1,\ldots ,m-2$;

• (v) $\left| \int _{\varOmega }u x^{\alpha }dx\right| \le C\varepsilon ^{|\alpha |+\delta -k-\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}$ for all $\alpha \in \mathbb N^N$ with $k+1\le |\alpha |\le m-1$, if $N=2m-2k-1$, $\forall \,k=0,\ldots ,m-2$;

• (vi) $\left| \int _{\varOmega }u x^{\alpha }dx\right| \le C\varepsilon ^{|\alpha |+\delta -k-1}(1+|\log (\varepsilon )|)^{\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}$ for all $\alpha \in \mathbb N^N$ with $k+1\le |\alpha |\le m-1$, if $N=2m-2k-2$, $\forall \,k=0,\ldots ,m-2$.

Proof

Let $u\in H^m(\varOmega )$ be such that $\int _{\varOmega }\tilde{\rho }_{\varepsilon } u(x) x^{\beta } dx=0$ for all $\beta \in \mathbb N^N$ with $|\beta |\le m-1$. We start by proving (i) and (ii). We recall that by Sobolev inequality (66), $u\in C^{k,\frac{1}{2}}(\varOmega )$ if $N=2m-2k-1$, while $u\in C^{k,\gamma }(\varOmega )$ for all $\gamma \in ]0,1[$ if $N=2m-2k-2$. We have, for $|\beta |\le m-1$, that

$$\begin{aligned}&\int _{B(0,\varepsilon )}\tilde{\rho }_{\varepsilon }u(x) x^{\beta }\nonumber \\&\quad =\int _{B(0,\varepsilon )}\tilde{\rho }_{\varepsilon }\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx+\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}\int _{B(0,\varepsilon )}\tilde{\rho }_{\varepsilon }x^{\alpha } x^{\beta }dx.\qquad \quad \end{aligned}$$

(76)

From (76) and from the fact that $\int _{\varOmega }\tilde{\rho }_{\varepsilon } u(x) x^{\beta } dx=0$ it follows that

$$\begin{aligned}&\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}x^{\alpha } x^{\beta }dx\nonumber \\&\quad =-\int _{\varOmega }u(x)x^{\beta }dx-\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx. \end{aligned}$$

(77)

We now compute $\int _{B(0,\varepsilon )}\varepsilon ^{-2m+\delta }x^{\alpha }x^{\beta }dx$. It is convenient to pass to the spherical coordinates $(r,\theta )=(r,\theta _1,\ldots \theta _{N-1})\in [0,+\,\infty [\times \partial B$, where $\partial B$ denotes the unit sphere in $\mathbb R^N$ endowed with the $N-1$ dimensional volume element $d\sigma (\theta )$. With respect to these new variables, we write the coordinate functions $x_i$ as $x_i=r H_i(\theta )$, where $H_i(\theta )$ are the standard spherical harmonics of degree 1 in $\mathbb R^N$. We then have

$$\begin{aligned}&\int _{B(0,\varepsilon )}\varepsilon ^{-2m+\delta }x^{\alpha }x^{\beta }dx\\&\quad =\int _{\partial B}\int _0^{\varepsilon }H_1^{\alpha _1+\beta _1}\cdots H_N^{\alpha _N+\beta _N}r^{|\alpha |+|\beta |}r^{N-1}drd\sigma (\theta )=K_{\alpha ,\beta }\varepsilon ^{N+|\alpha |+|\beta |-2m+\delta }, \end{aligned}$$

where

$$\begin{aligned} K_{\alpha ,\beta }=\frac{1}{N+|\alpha |+|\beta |}\int _{\partial B}H_1^{\alpha _1+\beta _1}\cdots H_N^{\alpha _N+\beta _N}d\sigma (\theta ). \end{aligned}$$

(78)

From (77) it follows that for all $|\beta |\le k$

$$\begin{aligned}&\sum _{|\alpha |\le k}\frac{K_{\alpha ,\beta }}{\alpha !}\varepsilon ^{|\alpha |}\partial ^{\alpha }u(0)\nonumber \\&\quad =-\varepsilon ^{2m-N-|\beta |-\delta }\int _{\varOmega }u(x)x^{\beta }dx-\varepsilon ^{-N-|\beta |}\int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx.\nonumber \\ \end{aligned}$$

(79)

Clearly

$$\begin{aligned} \left| \int _{\varOmega }u(x)x^{\beta }dx\right| \le C_{\varOmega ,\beta }\Vert u\Vert _{L^2(\varOmega )}, \end{aligned}$$

(80)

where $C_{\varOmega ,\beta }$ depends only on $\varOmega$ and $|\beta |$. Moreover, from Lemma 1 and Hölder’s inequality it follows that

$$\begin{aligned} \left| \int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx\right| \le C\varepsilon ^{|\beta |+2m-k-\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}, \end{aligned}$$

(81)

if $N=2m-2k-1$, while

$$\begin{aligned} \left| \int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx\right| \le C\varepsilon ^{|\beta |+2m-k-1}(1+|\log (\varepsilon )|)^{\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )},\nonumber \\ \end{aligned}$$

(82)

if $N=2m-2k-2$. From (79), (80), (81), (82) and since $\delta \in ]0,1/2[$, we deduce that

$$\begin{aligned}&\sum _{|\alpha |\le k}\frac{K_{\alpha ,\beta }}{\alpha !}\varepsilon ^{|\alpha |}\partial ^{\alpha }u(0)\le C (\varepsilon ^{2k+1-|\beta |-\delta }+\varepsilon ^{k+\frac{1}{2}})\Vert u\Vert _{H^m(\varOmega )}\\&\quad \le 2C \varepsilon ^{k+\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}, \end{aligned}$$

if $N=2m-2k-1$, while

$$\begin{aligned}&\sum _{|\alpha |\le k}\frac{K_{\alpha ,\beta }}{\alpha !}\varepsilon ^{|\alpha |}\partial ^{\alpha }u(0)\le C (\varepsilon ^{2k+2-|\beta |-\delta }+\varepsilon ^{k+1}(1+|\log (\varepsilon )|^{\frac{1}{2}}))\Vert u\Vert _{H^m(\varOmega )}\\&\quad \le 2C \varepsilon ^{k+1}(1+|\log (\varepsilon )|^{\frac{1}{2}})\Vert u\Vert _{H^m(\varOmega )}, \end{aligned}$$

if $N=2m-2k-2$. Since for all $|\alpha |\le k$, $K_{\alpha ,\alpha }>0$ by definition (see (78)), necessarily inequalities (i) and (ii) must hold with a constant $C>0$ which depends only on m, k and $\varOmega$.

Now consider (iii). We have

$$\begin{aligned}&\int _{B(0,\varepsilon )}u^2dx=\int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }+\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) ^2dx\\&\quad \le 2 \int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) ^2dx\\&\qquad +\,\,2\left( {\begin{array}{c}N+k\\ k\end{array}}\right) \sum _{|\alpha |\le k}\frac{|\partial ^{\alpha }u(0)|^2}{(\alpha !)^2}\int _{B(0,\varepsilon )}|x|^{2\alpha }dx\\&\quad \le C\varepsilon ^{2m}\Vert u\Vert _{H^m(\varOmega )}^2+C\sum _{|\alpha |\le k}\varepsilon ^{2k+1-2\alpha }\int _{B(0,\varepsilon )}|x|^{2\alpha }dx\\&\quad \le C\varepsilon ^{2m}\Vert u\Vert _{H^m(\varOmega )}^2, \end{aligned}$$

where in the second inequality we have used point (i) of Lemma 1 and in the last inequality we have used point (i) of the present lemma and the fact that

$$\begin{aligned} \left| \int _{B(0,\varepsilon )}x^{2\alpha }dx\right| \le \varepsilon ^{2|\alpha |}|B(0,\varepsilon )|=\omega _{2m-2k-1}\varepsilon ^{2|\alpha |+2m-2k-1}. \end{aligned}$$

This concludes the proof of (iii). Point (iv) is proved exactly as point (iii), by using point (ii) of Lemma 1 and point (ii) of the present lemma.

We now consider point (v). Let $N=2m-2k-1$, with $0\le k\le m-2$ and let $\beta \in \mathbb N^N$ such that $k+1\le |\beta |\le m-1$. We have

$$\begin{aligned}&\int _{\varOmega }u(x)x^{\beta }dx=-\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}u(x)x^{\beta }dx\nonumber \\&\quad =-\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}\left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }dx\nonumber \\&\qquad -\,\varepsilon ^{-2m+\delta }\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}\int _{B(0,\varepsilon )}x^{\alpha }x^{\beta }dx. \end{aligned}$$

(83)

From Lemma 1 point (i) and from Hölder’s inequality, we have that

$$\begin{aligned}&\varepsilon ^{-2m+\delta }\int _{B(0,\varepsilon )}\left| \left( u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha }\right) x^{\beta }\right| dx\nonumber \\&\quad \le \varepsilon ^{-2m+\delta }\left( \int _{B(0,\varepsilon )}(u(x)-\sum _{|\alpha |\le k}\frac{\partial ^{\alpha }u(0)}{\alpha !}x^{\alpha })^2dx\right) ^{\frac{1}{2}}\left( \int _{B(0,\varepsilon )}|x|^{2|\beta |}dx\right) ^{\frac{1}{2}}\nonumber \\&\quad \le C\varepsilon ^{|\beta |+\delta -k-\frac{1}{2}}\Vert u\Vert _{H^m(\varOmega )}. \end{aligned}$$

(84)

Moreover, from point (i) of the present lemma we have that for all $|\alpha |\le k$

$$\begin{aligned} \varepsilon ^{-2m+\delta }\frac{|\partial ^{\alpha }u(0)|}{\alpha !}\int _{B(0,\varepsilon )}|x|^{|\alpha |+|\beta |}dx\le C\varepsilon ^{|\beta |+\delta -k-\frac{1}{2}}. \end{aligned}$$

(85)

The proof of (v) follows by combining (83) with (84) and (85). The proof of (vi) is identical to that of (v) and follows form point (ii) of Lemma 1 and point (ii) of the present lemma. This concludes the proof. $\square$