Rellich inequalities with weights

Paolo Caldiroli, Roberta Musina

Introduction

where $u\in C^{2}(\overline{\Omega})$ runs into suitable classes of functions vanishing on $\partial\Omega$ . More precisely, we consider the following cases:

$u$ vanishes on $\partial\Omega$ and in a neighborhood of and of $\infty$ . We will denote this functional space as $C^{2}_{c}(\overline{\Omega}\setminus\{0\})$ and we will refer to this situation as the Navier case.

$u$ has compact support in $\Omega$ . We will denote this functional space as $C^{2}_{c}(\Omega)$ and we will refer to this situation as the Dirichlet case.

Our first goal is to evaluate the best constants

We provide an explicit formula for the best constant in the Navier case and estimates from below in the Dirichlet case (see Theorem 2.1 and Corollary 2.2).

We also consider cone-like domains, that are of the following kind:

For these domains we will prove inequalities with remainder terms involving logarithmic weights, both in the Navier and in the Dirichlet cases, with optimal constants (see Theorems 4.1 and 4.5).

Inequalities of the form (0.1) are known in the literature as Rellich-type inequalities. Even if they are less studied than the corresponding lower order inequalities, nowadays constitute a fecund field of research. The prototype case is the inequality:

It was proved by Rellich in 1953 (see and the posthumous paper ) with the optimal constant

Subsequently many authors (see , , , ) studied the more general version

under some restrictions on $\alpha$ (see also , , and for related results).

$\alpha<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">-2</annotation></semantics></math>−2-4<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>24<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math>68$ Graph of $\alpha\mapsto\mu_{2,\alpha}$ in dimension $n=2$

We point out that in Rellich-type inequalities on the whole space, the radial and the angular part of $\Delta u$ have independent roles. Actually no symmetrization or rearrangement argument can be used to study the minimizations problems in (0.2). In fact, we can show that considering just radial functions we have

whereas, taking nonradial functions, we have

where $\Lambda(\Sigma)$ is the Dirichlet spectrum of the Laplace-Beltrami operator on $\Sigma$ . In particular $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)>0$ if and only if $-\gamma_{n,\alpha}$ is not an eigenvalue of $-\Delta_{\sigma}$ on $H^{1}_{0}(\Sigma)$ .

We emphasize the fact that the dependence of the constant $\mu_{N}(\Omega;\alpha)$ with respect to the domain $\Omega$ in general exhibits no monotonicity property. Instead, in the Dirichlet case, one easily sees that $\mu_{D}(\Omega;\alpha)\geq\mu_{D}(\Omega^{\prime};\alpha)$ if $\Omega\subset\Omega^{\prime}$ .

A more detailed analysis about Rellich inequalities on cones is developed in Section 2. The special cases previously discussed for the whole space or for a half-space are displayed in Section 3.

Next we deal with cone-like domains. Here, for the sake of simplicity, we limit ourselves to state some of our results when the domain is either the punctured ball or the complement of the ball. Let $\mu_{n,\alpha}$ be given by (0.7) and let

[Navier case] for every $u\in C^{2}_{c}(\overline{\Omega}\setminus\{0\})$ one has

[Dirichlet case] for every $u\in C^{2}_{c}(\Omega)$ one has

For more general cone-like domains we also consider mixed boundary conditions, precisely of Dirichlet type on the “radial” boundary, and of Navier type on the “angular” boundary. A wider discussion is contained in Section 4.

In a forthcoming paper we will study second order interpolation inequalities of Caffarelli-Kohn-Nirenberg type and reletad noncompact semilinear problems.

Auxiliary problems on spherical domains

In the present section we study the following minimization problems:

Notice that $m_{D}(\Sigma;\gamma)\geq m_{N}(\Sigma;\gamma)$ . Our first result concerns the lowest infimum $m_{N}(\Sigma;\gamma)$ .

and $m_{N}(\Sigma;\gamma)$ is always achieved. More precisely, $\varphi\in H^{2}\cap H^{1}_{0}(\Sigma)$ attains $m_{N}(\Sigma;\gamma)$ if and only if $\varphi$ is an eigenfunction relative to the eigenvalue that achieves the minimal distance of $-\gamma$ from $\Lambda(\Sigma)$ .

Proof. Let $\lambda\in\Lambda(\Sigma)$ and let $\varphi$ be an eigenfunction relative to the eigenvalue $\lambda$ . Since $L\varphi=(\lambda+\gamma)\varphi$ , then $m_{N}(\Sigma;\gamma)\leq(\lambda+\gamma)^{2}$ , and thus $m_{N}(\Sigma;\gamma)\leq\textrm{dist}(-\gamma,\Lambda(\Sigma))^{2}$ . Therefore it suffices to show that

If $-\gamma$ is an eigenvalue then clearly $0=m_{N}(\Sigma;\gamma)=\textrm{dist}(-\gamma,\Lambda(\Sigma))^{2}$ and in addition $m_{N}(\Sigma;\gamma)$ is achieved by any corresponding eigenfunction. Thus we can assume that

where $\lambda_{k-1}$ and $\lambda_{k}$ are two consecutive eigenvalues if $-\gamma>\lambda_{\Sigma}$ , while $\lambda_{k-1}=-\infty$ if $-\gamma$ is below the spectrum $\Lambda(\Sigma)$ . If $\lambda_{k-1}$ is finite we split $H^{2}\cap H^{1}_{0}(\Sigma)$ into the direct sum

where $V$ is the finite-dimensional space spanned by the eigenfunctions relative to the eigenvalues $\lambda<\lambda_{k}$ . Otherwise, we agree that $V=\{0\}$ . Since

for any $\varphi\in V^{\perp}$ , then from the Cauchy-Schwarz inequality we readily get that

If $\lambda_{k-1}=-\infty$ then (1.1) is proved. If $\lambda_{k-1}$ is finite, namely $V\neq\{0\}$ , we show that

Indeed, fix an $L^{2}$ -orthonormal basis $\{\varphi_{1},...,\varphi_{h}\}$ of $V$ , made by eigenfunctions. Any function $\varphi\in V$ can be written as

and then (1.4) holds. In order to obtain (1.2) we write any nontrivial $\varphi\in H^{2}\cap H^{1}_{0}(\Sigma)$ as $\varphi=\varphi_{V}+\varphi_{V^{\perp}}$ , with $\varphi_{V}\in V$ and $\varphi_{V^{\perp}}\in V^{\perp}$ . By orthogonality and by (1.3)–(1.4) we get

as desired. Hence (1.1) is proved. The last claim readily follows from (1.1). $\square$

and $m_{D}(\Sigma;\gamma)$ is always achieved in $H^{2}_{0}(\Sigma)$ .

Proof. Clearly $m_{N}(\Sigma;\gamma)\leq m_{D}(\Sigma;\gamma)$ . If $-\gamma\notin\Lambda(\Sigma)$ then $m_{N}(\Sigma;\gamma)>0$ by Proposition 1.1 and hence also $m_{D}(\Sigma;\gamma)$ is positive. Thus it is achieved by some $\varphi\in H^{2}_{0}(\Sigma)$ , because of the compact embedding of $H^{2}_{0}(\Sigma)$ into $L^{2}(\Sigma)$ . By contradiction, assume that $m_{N}(\Sigma;\gamma)=m_{D}(\Sigma;\gamma)$ . Then $\varphi$ achieves $m_{N}(\Sigma;\gamma)$ . Thus, by the last assertion in Proposition 1.1, there exists $\lambda\in\Lambda(\Sigma)$ such that $\varphi\neq 0$ solves

Thus $m_{D}(\Sigma;\gamma)>m_{N}(\Sigma;\gamma)$ . Finally, $m_{D}(\Sigma;\gamma)$ is achieved since it is positive, via standard arguments. $\square$

Rellich inequalities on cones

In this section we investigate Rellich inequalities on cones and we evaluate the best Rellich costant in the Navier case.

One has that $u\in C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ if and only if $\hat{u}\in C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ and $u\in C^{2}_{c}(\mathcal{C}_{\Sigma})$ if and only if $\hat{u}\in C^{2}_{c}(\mathcal{C}_{\Sigma})$ . Moreover

Before stating our first result, let us recall that by $\Lambda(\Sigma)$ we denote the spectrum of $-\Delta_{\sigma}$ in $H^{1}_{0}(\Sigma)$ . Moreover let $\gamma_{n,\alpha}$ be the number defined in (0.8).

Theorem 2.1 will be proved in Subsection 2.2. From the monotonicity property of the mapping $\Sigma\mapsto\mu_{D}(\mathcal{C}_{\Sigma};\alpha)$ we have that

Therefore, from Theorem 2.1, we infer the next result.

In our second main result we show that extremal functions do not exist. This is trivial when $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)$ or $\mu_{D}(\mathcal{C}_{\Sigma};\alpha)$ vanish. When they are positive we need to introduce suitable Sobolev spaces as follows. In particular, when $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)>0$ , we can define a norm on $C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ by setting

The completion of $C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ with respect to this norm will be denoted $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};\alpha)$ .

In the same way, when $\mu_{D}(\mathcal{C}_{\Sigma};\alpha)>0$ we introduce the Sobolev space $\mathcal{D}^{2}(\mathcal{C}_{\Sigma};\alpha)$ as the completion of $C^{2}_{c}(\mathcal{C}_{\Sigma})$ with respect to the norm (2.4).

The infima $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)$ and $\mu_{D}(\mathcal{C}_{\Sigma};\alpha)$ are never attained.

To prove Theorems 2.1 and 2.3 we will use a suitable Emden-Fowler transform, that maps functions defined on $\mathcal{C}_{\Sigma}$ into functions on the cylinder

This will be done in the next subsections.

We denote by $C^{2}_{c}(\overline{\mathcal{Z}_{\Sigma}})$ the space of mappings $w\in C^{2}(\overline{\mathcal{Z}_{\Sigma}})$ such that $w(\cdot,\sigma)=0$ for every $\sigma\in\partial\Sigma$ and $w(s,\cdot)=0$ for $|s|$ large enough. In addition we introduce the differential operator

as in Section 1, with $\gamma_{n,\alpha}$ defined in (0.8).

If $u\in C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ then $Tu\in C^{2}_{c}(\overline{\mathcal{Z}_{\Sigma}})$ . If $u\in C^{2}_{c}(\mathcal{C}_{\Sigma})$ then $Tu\in C^{2}_{c}(\mathcal{Z}_{\Sigma})$ . Moreover, setting $w=Tu$ , one has

Proof. The first two statements and (2.7) are trivial. Let $v:=|x|^{\frac{n-4+\alpha}{2}}u$ and let $w$ be defined as in (2.5). We compute

where $v_{r}=|x|^{-1}(x\cdot\nabla v)$ denotes the radial derivative of $v$ . Now we go from $v$ to $w$ , via the transform

Denoting $w_{s}$ and $w_{ss}$ the partial derivatives with respect to the real variable of $w$ , since

2 Proof of Theorem 2.1

Firstly we prove (2.2). By Proposition 1.1, it suffices to show that

Let $T$ be the Emden-Fowler transform. Fix $u\in C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ and put $w=Tu$ . Then use (2.8) and (2.9). Since $G(w)\geq 0$ , by Proposition 1.1 and by (2.7) we obtain that

Hence $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)\geq m_{N}(\Sigma;\gamma_{n,\alpha})$ . In order to prove the opposite inequality we take a function $w\in C^{2}_{c}(\overline{\mathcal{Z}_{\Sigma}})$ of the form

Taking $t\to 0$ and $h\to\infty$ we immediately obtain that $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)\leq m_{N}(\Sigma;\gamma_{n,\alpha})$ . Hence (2.10) holds true. In the same way one shows that

Then the conclusion follows from Proposition 1.2. $\square$

Assume that $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)>0$ . Let $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ be the Hilbert space endowed with the norm (2.4).

Using an interpolation argument we endow the space $H^{2}\cap H^{1}_{0}(\mathcal{Z}_{\Sigma})$ with the equivalent norm

For every $u\in C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})$ let $w=Tu$ be the Emden-Fowler transform of $u$ . By Lemma 2.4, we have that $Tu\in H^{2}\cap H^{1}_{0}(\mathcal{Z}_{\Sigma})$ .

The operator $T\colon C^{2}_{c}(\overline{\mathcal{C}_{\Sigma}}\setminus\{0\})\to H^{2}\cap H^{1}_{0}(\mathcal{Z}_{\Sigma})$ admits a unique continuous extension on $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ which is an isomorphism between the spaces $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ and $H^{2}\cap H^{1}_{0}(\mathcal{Z}_{\Sigma})$ . Moreover the equalities (2.7) and (2.8)–(2.9) hold true for every function $u$ in $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ .

Proof. Since $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)>0$ an equivalent norm to $\|\cdot\|_{2,\alpha}$ in $\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ is given by

By density, equalities (2.7) and (2.8) hold true for every $u\in\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ . Recalling the definitions of the norms $\|u\|$ and $\|w\|$ given in (2.12) and in (2.11), respectively, and using also (2.7), we have that

Hence if $\gamma_{n,\alpha}\geq 0$ then $\|u\|\geq\|Tu\|^{2}$ for all $u\in\mathcal{N}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ and the conclusion follows. If $\gamma_{n,\alpha}<0$ , using (2.8)–(2.9), we firstly estimate

and, fixing $\varepsilon$ with $1<\varepsilon<1+\gamma_{n,\alpha}^{-2}$ , the conclusion follows as before. $\square$

The Emden-Fowler operator $T$ is an isomorphism between the spaces $\mathcal{D}^{2}(\mathcal{C}_{\Sigma};|x|^{\alpha}dx)$ and $H^{2}_{0}(\mathcal{Z}_{\Sigma})$ .

4 Proof of Theorem 2.3

Assume that $\mu_{N}(\mathcal{C}_{\Sigma};\alpha)$ is attained by some $u\in\mathcal{N}^{2}(\mathcal{C}_{\Sigma},|x|^{\alpha}dx)$ , $u\neq 0$ . By (2.7), (2.8) and by Lemma 2.5, we have that

Therefore the infimum at the right hand side is attained by $w=Tu$ . Notice that $w_{s}\neq 0$ , otherwise $w=0$ and then $u=0$ , too. For $t>0$ let $w^{t}(s,\sigma)=w(ts,\sigma)$ . Then $w^{t}\neq 0$ , $w^{t}\in H^{2}\cap H^{1}_{0}(\mathcal{Z}_{\Sigma})$ for all $t\in(0,1)$ and

a contradiction. A similar argument holds in order to show that $\mu_{D}(\mathcal{C}_{\Sigma};\alpha)$ is not attained in $\mathcal{D}^{2}(\mathcal{C}_{\Sigma},|x|^{\alpha}dx)$ . $\square$

Applications of Theorem 2.1

Equality (3.2) can be immediately obtained via Emden-Fowler transformation. The same holds for (3.3) with the further remark that, arguing as in the proof of Theorem 2.1,

In this remark we take $\alpha=0$ . Clearly we recover the classical Rellich inequality (0.4) with the best constant $\mu_{n}$ defined in (0.5). Moreover we also point out the following inequalities, which hold true in any dimension $n\geq 2$ :

If $\alpha>4-n$ then the weight $|x|^{\alpha-4}$ is locally integrable, and a density argument can be used in order to show that

2 Rellich inequalities on half-spaces

if $k\geq 2$ , whereas $m=n$ if $k=1$ . We claim that there exists an eigenfunction

3 Rellich inequality on cones in low dimension

Firstly consider the dimension $n=2$ , when $\gamma_{2,0}=-1$ . From a direct computation of the spectrum of the Laplace-Beltrami operator on $\Sigma_{\theta}$ (see, e.g., ), it follows that

Notice that $\theta^{*}:=\pi\sqrt{5/2}$ is a local maximum for the map $\theta\mapsto\mu_{N}(\mathcal{C}_{\theta};0)$ . In addition, by (2.3),

When $n=3$ a similar phenomenon appears. In particular, there is exactly one value $\theta^{*}\in(\pi/2,\pi)$ such that $-\gamma_{3,0}=3/4$ is the smallest eigenvalue of $-\Delta_{\sigma}$ on $\Sigma_{\theta^{*}}$ . Thus $\mu_{N}(\mathcal{C}_{\theta^{*}};0)=0$ and $\mu_{N}(\mathcal{C}_{\theta};0)>0$ for $\theta<\theta^{*}$ .

Inequalities with logarithmic weights

In this Section we are concerned with inequalities, with sharp constants, involving the $L^{2}$ norm of $\Delta u$ with a weight $|x|^{\alpha}$ , for mappings $u$ supported by cone-like domains. More precisely, in this section we assume that

As in the previous Sections, we denote by $\lambda_{\Sigma}$ the first eigenvalue of the Laplace-Beltrami operator in $H^{1}_{0}(\Sigma)$ and we define $\gamma_{n,\alpha}$ and $\overline{\gamma}_{n,\alpha}$ as in (0.8) and (0.10), respectively. We have the following result.

[Navier case] For every $u\in C^{2}_{c}(\overline{\Omega}\setminus\{0\})$ it holds that

[Dirichlet case] For every $u\in C^{2}_{c}(\Omega)$ it holds that

with $L$ defined as in (2.6). Each term in the right hand side of (4.5) can be estimated according to the behaviour of $\nabla u$ on $\partial\Omega$ . First of all, observe that for every $s>0$ the mapping $w(s,\cdot)$ belongs to $H^{2}\cap H^{1}_{0}(\Sigma)$ . Hence we can apply Proposition 1.1 and Theorem 2.1 to estimate

Similarly, if $u\in C^{2}_{c}(\Omega)$ then for a.e. $s>0$ the mapping $w(s,\cdot)$ belongs to $H^{2}_{0}(\Sigma)$ and in this case we obtain that

Hence (4.3) follows from (4.6), (4.8), (4.10) and (4.9) whereas (4.4) follows from (4.7), (4.8), (4.10) and (4.9). Thus the theorem is proved when $\Omega$ is the intersection of a cone with the unit ball.

with $R>0$ fixed. In fact the result $(iii)$ can be suitably extended to any bounded domain $\Omega$ with $0\in\partial\Omega$ or to any exterior domain $\Omega$ with $0\not\in\overline{\Omega}$ , with no regularity assumption on $\partial\Omega$ .

In the next corollaries we point out the explicit constants in case $\alpha=0$ , under Navier and Dirichler boundary conditions. For the convenience of the reader we distinguish the case $n=2$ from the higher dimensional one.

for any $u\in C^{2}_{c}(\overline{\Omega}\setminus\{0\})$ , and

In the next result we show that the constants appearing in the right hand side in (4.3) are sharp.

Let $\alpha$ , $\Sigma$ and $\Omega$ as in Theorem 4.1, and Assume that

Since $(\gamma_{n,\alpha}+\lambda_{\Sigma})^{2}=\mu_{N}(\mathcal{C}_{\Sigma};\alpha)$ , (4.11) yields

and thus we obtain the bound on $A$ . Dividing (4.13) by $t^{2}$ and passing to the limit $t\to\infty$ we obtain

Similar results on the optimality of the constants can be proved in the Navier and in the Dirichlet case.