Hardy-Poincare' inequalities with boundary singularities

Introduction

where $\mu=\mu_{\lambda}(\Omega)$. If $\mu_{\lambda}(\Omega)$ is achieved, then $\mu_{\lambda}(\Omega)$ is the first eigenvalue of the operator $-\Delta-\lambda$ on $H^{1}_{0}(\Omega)$. Starting from a different point of view, for $0\in\Omega$, $N\geq 3$ and $\mu\leq(N-2)^{2}/4$, Dávila and Dupaigne have proved in the existence of the first eigenfunction ${\varphi}_{1}$ of the operator $-\Delta-\mu|x|^{-2}$ on a suitable functional space $H(\Omega)\supseteq H^{1}_{0}(\Omega)$. Notice that ${\varphi}_{1}$ solves (0.2), where the eigenvalue $\lambda$ depends on the datum $\mu$.

is the best constant in the Hardy inequality for maps with support in a half-space. Indeed in Section 2 we first show that

then we deduce that, provided $\mu_{\lambda}(\Omega)<\mu^{+}$, every minimizing sequence for $\mu_{\lambda}(\Omega)$ converges in $H^{1}_{0}(\Omega)$ to an extremal for $\mu_{\lambda}(\Omega)$.

We recall that $\Omega$ is said to be locally concave at $0\in\partial\Omega$ if it contains a half-ball. That is there exists $r>0$ such that

where $\nu$ is the interior normal of $\partial\Omega$ at 0. Notice that if all the principal curvatures of $\partial\Omega$ at , with respect to $\nu$, are strictly negative, then condition (0.4) is satisfied.

Our first main result is stated in the following theorem.

The ”only if” part, which is the most intriguing, is a consequence of Corollary 3.2 in Section 3, where we provide local nonexistence results for problem

also for negative values of the parameter $\lambda$.

Up to now several questions concerning the infimum $\mu_{\lambda}(\Omega)$ are still open. Put

$i)$ If $\Omega$ is locally convex at , that is, if there exists $r>0$ such that $\Omega\cap B_{r}(0)$ is contained in a half-space, then $\lambda^{*}>-\infty$.

$ii)$ If $\Omega$ is contained in a half-space then

$iii)$ For any $\delta>0$ there exists $\rho_{\delta}>0$ such that, if

The relevance of the geometry of $\Omega$ at the origin is confirmed by Theorem 0.1, by $i)$ and by the existence theorems proved in , and for a related superlinear problem. However, it has to be noticed that also the (conformal) ”size” of $\Omega$ (even far away from the origin) has some impact on the existence of compact minimizing sequences. Actually, no requirement on the curvature of $\Omega$ at is needed in $iii)$. In particular, there exist smooth domains having strictly positive principal curvatures at , and such that the Hardy constant $\mu_{0}(\Omega)$ is achieved.

In Section 1 we point out few remarks on the Hardy inequality on dilation-invariant domains.

In Section 2, Theorem 2.2, we give sufficient conditions for the existence of minimizers for (0.1).

In Section 3 we prove some nonexistence theorems for solutions to (0.5) that might have an independent interest.

In Section 5 we estimate $\lambda^{*}$ from below and form above, under suitable assumptions on $\Omega$.

where $H^{1}(\Omega)$ is the standard Sobolev space of maps on $\Omega$.

Preliminaries

In this section we collect a few remarks on the Hardy inequality on dilation-invariant domains that are partially contained for example in (in case $N=2$) and in .

is an homeomorphism ${\mathcal{C}}_{\Sigma}\to\mathcal{Z}_{\Sigma}$. It induces the Emden-Fowler transform

A direct computation based on the divergence theorem gives

Now we introduce the Hardy constant on the cone ${\mathcal{C}}_{\Sigma}$:

In the next proposition we notice that the Hardy inequality on ${\mathcal{C}}_{\Sigma}$ is equivalent to the Poincaré inequality for maps supported be the cylinder $\mathcal{Z}_{\Sigma}$.

Let ${\mathcal{C}}_{\Sigma}$ be a cone. Then

The result follows by noticing that $\lambda_{1}(\mathcal{Z}_{\Sigma})=\lambda_{1}(\Sigma)$.

The next result is an immediate consequence of the fact that the Dirichlet eigenvalue problem of $-\Delta$ in the strip $\mathcal{Z}_{\Sigma}$ is never achieved. The same conclusion was already noticed in in case $N=2$ and in .

Existence

In this Section we show that the condition $\mu_{\lambda}(\Omega)<\mu^{+}=N^{2}/4$ is sufficient to guarantee the existence of a minimizer for $\mu_{\lambda}(\Omega)$. We notice that throughout this section, the regularity of $\Omega$ can be relaxed to Lipschitz domains which are of class $C^{2}$ at . We start with a preliminary result.

Let $\Omega$ be a smooth domain with $0\in\partial\Omega$. Then

Proof. The proof will be carried out in two steps.

We denote by $\nu$ the interior normal of $\partial\Omega$ at 0. For $\delta>0$, we consider the cone

Now fix ${\varepsilon}>0$. If $\delta$ is small enough then $\mu_{0}({\mathcal{C}}^{\delta}_{-})\geq\mu^{+}-{\varepsilon}$. Since $\Omega$ is smooth at then there exists a small radius $r>0$ (depending on $\delta$) such that $\Omega\cap B_{r_{\delta}}(0)\subset{\mathcal{C}}^{\delta}_{-}$.

Next, let $\psi\in C^{\infty}(B_{r}(0))$ be a cut-off function, satisfying

We write any $u\in H^{1}_{0}(\Omega)$ as $u=\psi u+(1-\psi)u$, to get

where the constant $c$ do not depend on $u$. Since $\psi u\in\mathcal{D}^{1,2}({\mathcal{C}}^{\delta}_{-})$ then

by our choice of the cone ${\mathcal{C}}^{\delta}_{-}$. In addition, we have

Comparing with (2.1) and (2.2) we infer that there exits a positive constant $c$ depending only on $\delta$ such that

Hence we get $(\mu^{+}-{\varepsilon})\leq\mu_{-c}(\Omega)$. Consequently $(\mu^{+}-{\varepsilon})\leq\sup_{\lambda}\mu_{\lambda}(\Omega)$, and the conclusion follows by letting ${\varepsilon}\to 0$.

Step 2: We claim that $\sup_{\lambda}\mu_{\lambda}(\Omega)\leq\mu^{+}$.

As in the first step, for any $\delta>0$ there exists $r_{\delta}>0$ such that ${\mathcal{C}}^{\delta}_{+}\cap B_{r}(0)\subset\Omega$ for all $r\in(0,r_{\delta})$. Clearly by scale invariance, $\mu_{0}({\mathcal{C}}^{\delta}_{+}\cap B_{r}(0))=\mu_{0}({\mathcal{C}}^{\delta}_{+})$. For $\varepsilon>0$, we let $\phi\in H^{1}_{0}({\mathcal{C}}^{\delta}_{+}\cap B_{r}(0))$ such that

Since $\displaystyle\int_{{\mathcal{C}}^{\delta}_{+}\cap B_{r}(0)}|x|^{-2}|\phi|^{2}~{}dx\geq r^{-2}\int_{{\mathcal{C}}^{\delta}_{+}\cap B_{r}(0)}|\phi|^{2}~{}dx$, we get

The conclusion follows immediately, since $\mu_{0}({\mathcal{C}}^{\delta}_{+})\to\mu^{+}$ when $\delta\to 0$.

Notice that if $\Omega$ is bounded then by (2.3) and Poincaré inequality

For $N=2$ this was shown in and for more general domains. We are in position to prove the main result of this section.

Proof. Let $u_{n}\in H^{1}_{0}(\Omega)$ be a minimizing sequence for $\mu_{\lambda}(\Omega)$. We can normalize it to have

We can assume that $u_{n}\rightharpoonup u$ weakly in $H^{1}_{0}(\Omega)$, $|x|^{-1}u_{n}\rightharpoonup|x|^{-1}u$ weakly in $L^{2}(\Omega)$, and $u_{n}\to u$ in $L^{2}(\Omega)$, by (2.4) and by Rellich Theorem. Putting $\theta_{n}:=u_{n}-u$, from (2.5) and (2.6) we get

as $\theta_{n}\to 0$ in $L^{2}(\Omega)$. Testing $\mu_{\lambda}(\Omega)$ with $u$ we get

by (2.7). Therefore $\int_{\Omega}|x|^{-2}|\theta_{n}|^{2}\to 0$, since $\mu_{\lambda}(\Omega)-\mu^{+}+\delta<0$. In particular,

and $u\neq 0$ by (2.7). Thus $u$ achieves $\mu_{\lambda}(\Omega)$.

We conclude this section with a corollary of Theorem 2.2.

Following , for non smooth domains $\Omega$ we can introduce the ”limiting” Hardy constant

Using similar arguments it can be proved that $\sup_{\lambda}\mu_{\lambda}(\Omega)=\hat{\mu}_{0}(\Omega)$, and that $\mu_{\lambda}(\Omega)$ is achieved provided $\mu_{\lambda}(\Omega)<\hat{\mu}_{0}(\Omega)$.

Nonexistence

The main result in this section is stated in the following theorem.

Before proving Theorem 3.1 we point out some of its consequences.

Let $\Omega$ be a smooth bounded domain containing a half-ball and such that $0\in\partial\Omega$. If $\mu_{\lambda}(\Omega)=\mu^{+}$ then $\mu_{\lambda}(\Omega)$ is not achieved.

Proof. Assume that $u$ achieves $\mu_{\lambda}(\Omega)=\mu^{+}$. Then $u$ is a weak solution to

We also point out the following consequence to Theorem 3.1, that holds for smooth domains $\Omega$ with $0\in\partial\Omega$.

for some $\mu>\mu^{+}$, then $u\equiv 0$ in $\Omega$.

it follows that there exists a sequence $r_{h}\to 0$, $r_{h}\in(0,R)$ such that

as $h\to\infty$. Next we introduce the following cut-off functions:

To pass to the limit in the left-hand side we notice that $\nabla\eta_{h}$ vanishes outside the annulus $A_{h}:=\{r_{h}^{2}<|z|<r_{h}\}$, and that $\eta_{h}$ is harmonic on $A_{h}$. Thus

where $c>0$ is a constant that does not depend on $h$, and

by Hölder inequality and by (3.4). In the same way, also

In conclusion, we have proved that $\displaystyle\int{\varphi}\nabla\psi\cdot\nabla\eta_{h}=o(1)$ and therefore (3.5) gives

The same proof gives a similar result for subsolutions.

The next result is crucial in our proof. We state it is a more general form than needed, as it could have an independent interest. Notice that we do not need any a priori knowledge of the sign of $\psi$ in the interior of its domain.

Proof. We fix $R_{\lambda}<1/3$ small enough, in such a way that

Since $\delta>1/2$ then ${\varphi}_{\delta}$ is a smooth solution to

Proof of Theorem 3.1. Without loss of generality, we may assume that $\lambda<0$. Let $\Phi>0$ be the first eigenfunction of $-\Delta_{\sigma}$ on $\Sigma$. Thus $\Phi$ solves

direct computations based on (3.1) lead to

Finally, $\psi\in L^{2}(R^{2}_{R};|z|^{-2}dz)$ as

Thus Lemma 3.7 applies and since $\psi$ is radially symmetric we get $\psi\equiv 0$ in a neighborhood of . Hence $u\equiv 0$ in $B_{r}\cap{\mathcal{C}}_{\Sigma}$, for $r>0$ small enough. To conclude the proof in case $\Omega$ strictly contains $B_{r}\cap{\mathcal{C}}_{\Sigma}$, take any domain $\Omega^{\prime}$ compactly contained in $\Omega\setminus\{0\}$ and such that $\Omega^{\prime}$ intersects $B_{r}\cap{\mathcal{C}}_{\Sigma}$. Via a convolution procedure, approximate $u$ in $H^{1}(\Omega^{\prime})$ by a sequence of smooth maps $u_{\varepsilon}$ that solve

Since $u_{\varepsilon}\geq 0$ and $u_{\varepsilon}\equiv 0$ on $\Omega^{\prime}\cap B_{r}\cap{\mathcal{C}}_{\Sigma}$, then $u_{\varepsilon}\equiv 0$ on $\Omega^{\prime}$ by the maximum principle. Thus also $u\equiv 0$ in $\Omega^{\prime}$, and the conclusion follows.

Remainder terms

We prove here some inequalities that will be used in the next section to estimate the infimum $\lambda^{*}$ defined in (0.6).

Brezis and Vázquez proved in the following improved Hardy inequality:

We show that a Brezis-Vázquez type inequality holds in case the singularity is placed at the boundary of the domain. We start with conic domains

Proof. By homogeneity, it suffices to prove the proposition for $R=1$. Fix $u\in C^{\infty}_{c}({\mathcal{C}}_{1,\Sigma})$ and compute in polar coordinates $t=|x|$, $\sigma=x/|x|$:

Since for every $t\in(0,1)$ it holds that

then by Proposition 1.1 we only have to show that

for any fixed $\sigma\in\Sigma$. For that, we put $w(t)=t^{\frac{N-2}{2}}u(t\sigma)$, and we compute

This gives (4.3) and the proposition is proved.

The main result in this section is contained in the next theorem.

for any $R>0$, $u\in C^{\infty}_{c}(\Omega)$ and the theorem readily follows.

As pointed out by Brezis-Vázquez in , Extension 4.3, the following Hardy-Sobolev inequality holds

for all $p\in\left(2,\frac{2N}{N-2}\right)$, where $c_{p}$ is a positive constant depending on $p$ and $N$.

In this section we provide sufficient conditions to have $\lambda^{*}>-\infty$ or $\lambda^{*}<0$.

In the case where $\Omega$ is locally convex at $0\in\partial\Omega$, the supremum in Lemma 2.1 is attained.

for some constant $c=c(r)>0$. Since $\psi u\in H^{1}_{0}(B_{r}(0)\cap\Omega)$, from the definition of $\mu^{+}$ we infer

Comparing with (5.1), we infer that there exits a positive constant $c$ such that

This proves that $\mu_{-c}(\Omega)\geq\mu^{+}$. Thus $\mu_{-c}(\Omega)=\mu^{+}$ by Lemma 2.1. Finally, noticing that $\mu_{\lambda}(\Omega)$ is decreasing in $\lambda$, we can set

so that $\mu_{\lambda}(\Omega)<\mu^{+}$ for all $\lambda>\lambda^{*}(\Omega)$.

Finally, we notice that by Lemma 2.1, if $\Omega$ is contained in a half-space, then $\mu_{0}(\Omega)=\mu^{+}$, and therefore $\lambda^{*}(\Omega)\geq 0$. Thus, From Theorem 4.2 we infer the following result.

Let $\Omega$ be a bounded smooth domain with $0\in\partial\Omega$. If $\Omega$ is contained in a half-space then

It would be of interest to know if it is possible to get lower bounds depending only on the measure of $\Omega$, as in and .

2 Estimates from above

The local convexity assumption of $\Omega$ at does not necessary implies that $\lambda^{*}(\Omega)\geq 0$. Indeed the following remark holds.

For any $\delta>0$ there exists $\rho_{\delta}>0$ such that if $\Omega$ is a smooth domain with $0\in\partial\Omega$ and

contains a hemispace, then its Hardy constant is smaller than $\mu^{+}$. Thus there exists $u\in C^{\infty}_{c}({\mathcal{C}}_{\delta})$ such that

Assume that the support of $u$ is contained in an annulus of radii $b>a>0$. Then the conclusion in Proposition 5.3 holds, with $\rho:=b/a$.

Notice that $\Omega$ can be locally strictly convex at .

A similar remark holds for the following minimization problem, which is related to the Caffarelli-Kohn-Nirenberg inequalities:

where $2<p<2^{*}$, $b:=N-p(N-2)/2$. In case $0\in\partial\Omega$, the minimization problem (5.3) was studied in , and .

We do not know wether the strict local concavity of $\Omega$ at 0 can implies that $\mu_{0}(\Omega)<\mu^{+}$. See the paper by Ghoussoub and Kang for the minimization problem (5.3).

References