Existence and non-existence of minimizers for Poincaré–Sobolev inequalities

Abstract

In this paper we study the existence and non-existence of minimizers for a type of (critical) Poincaré–Sobolev inequalities. We show that minimizers do exist for smooth domains in ${\mathord {{\mathbb {R}}}}^d$, an also for some polyhedral domains. On the other hand, we prove the non-existence of minimizers in the rectangular isosceles triangle in ${\mathord {{\mathbb {R}}}}^2$.

Appendix A: an associated variational principle

In this appendix, we consider the generalized problem

$$\begin{aligned} G_p(\Omega , d) = \inf _{H^1(\Omega )} \frac{\int _\Omega \left| \nabla u\right| ^2 \left( \int _\Omega u^2 \right) ^{p}}{\int _\Omega \left| u-u_\Omega \right| ^{2+2p}}, \quad 0 \le p \le 2/(d-2) . \end{aligned}$$

(15)

While the problem for $p = 2/d$ was discussed in the main body of the manuscript we study the cases $p \ne 2/d$ in this appendix.

Theorem A.1

Let $\Omega \subset {\mathord {{\mathbb {R}}}}^d$ be a bounded domain with locally Lipschitz boundary. For all $p\in [0,2/d)$ a minimizer for $G_p(\Omega , d)$ exists. For all $p \in (2/d, 2/(d-2)]$, $G_p(\Omega , d) =0$ and no minimizer exists.

Proof

Non-existence for $p>2/d$. By scaling. Take the origin in the interior of $\Omega$, $\phi \in C_c^{\infty }$ with zero average. For sufficiently large $\lambda >0$, the support of $u_\lambda \equiv \phi (\lambda \cdot )$ is in $\Omega$. We compute

$$\begin{aligned} \frac{\int _\Omega \left| \nabla u_\lambda \right| ^2 \left( \int _\Omega u_\lambda ^2 \right) ^{p}}{\int _\Omega \left| u_\lambda \right| ^{2+2p}} = \frac{\lambda ^{2-d}\int _{{\mathord {{\mathbb {R}}}}^d} \left| \nabla \phi \right| ^2 \left( \lambda ^{-d}\int _{{\mathord {{\mathbb {R}}}}^d} \phi ^2 \right) ^{p}}{\lambda ^{-d}\int _{{\mathord {{\mathbb {R}}}}^d} \left| \phi \right| ^{2+2p}} = \lambda ^{2-pd} C_\phi . \end{aligned}$$

This tends to zero when $\lambda$ increases if $p>2/d$.

Existence for $p<2/d$. In this case, we prove existence of a minimizer sequence by showing that minimizing sequences can not concentrate in small sets. As in the proof of Theorem 1.1, if a minimizing sequence has no convergent subsequence, we obtain from Lemma 2.1,

$$\begin{aligned} G_p(\Omega , d) \ge \liminf _{\delta \rightarrow 0} G_p(\Omega , d,\delta ). \end{aligned}$$

Now, we show that

$$\begin{aligned} G_p(\Omega , d,\delta ) \ge C_{p, \Omega } \delta ^{d p -2}, \end{aligned}$$

(16)

with some constant $C_{p, \Omega }>0$ depending only on p, the dimension d and the Lipschitz constant of $\Omega$.

First of all, by using Hölder’s inequality for $f= \left| u\right| ^{2+2p}$, $g= 1$ in the denominator,

$$\begin{aligned} \inf _{u \in H^1_0(B(0,1)) } \frac{\int \left| \nabla u\right| ^2 \left( \int u^2 \right) ^{p}}{\int \left| u \right| ^{2+2p}}\ge \left( \frac{\mu _1}{\omega _d} \right) ^{\frac{2-pd}{2+d}}G(d)^{\frac{1+p}{1+2/d}}>0, \end{aligned}$$

with $\mu _1$ the first Dirichlet eigenvalue of B(0, 1) and $\omega _d$ the volume of B(0, 1).

By scaling, if $B(s,\delta ) \subset \Omega$,

$$\begin{aligned} \inf _{u \in H^1_0(B(s,\delta )) } \frac{\int _\Omega \left| \nabla u\right| ^2 \left( \int _\Omega u^2 \right) ^{p}}{\int \left| u_\Omega \right| ^{2+2p}}\ge \delta ^{d p -2} \left( \frac{\mu _1}{\omega _d} \right) ^{\frac{2-pd}{2+d}}G(d)^{\frac{1+p}{1+2/d}}. \end{aligned}$$

Otherwise, by replacing $\delta$ with $2 \delta$, we may assume that s is on the boundary. In this case we assume that $\delta$ is small enough such that $B(s,2\delta ) \cap \partial \Omega$ is the graph of a Lipschitz function over some hyperplane passing through s. We chose coordinates such that this hyperplane coincides with $x_1 =0$. and write

$$\begin{aligned} (x_1, x_t) \in \Omega \cap B(s,2\delta ) \Leftrightarrow (x_1,x_t) \in B(s, 2 \delta ) \text { and } x_1 > h(x_t), \end{aligned}$$

for some Lipschitz function h. We define f with support in $B(s, 2 \delta ) \cap {\mathord {{\mathbb {R}}}}^d_+$ by $f(x_1, x_t) = v(x_1 + h(x_t), x_t)$. As before, this change of variables has unit Jacobian and, by definition of a Lipschitz domain [1, Chapter IV], the distributional derivative of h is bounded by the Lipschitz constant L. Therefore,

$$\begin{aligned} \frac{\int _\Omega \left| \nabla v\right| ^2 \left( \int _\Omega v^2 \right) ^{p}}{\int _\Omega \left| v\right| ^{2+2p}}&\ge \frac{1}{(2+2L)^2} \frac{\int _{{\mathord {{\mathbb {R}}}}^d_+} \left| \nabla f\right| ^2 \left( \int _{{\mathord {{\mathbb {R}}}}^d_+} f^2 \right) ^{p}}{\int _{{\mathord {{\mathbb {R}}}}^d_+} \left| f\right| ^{2+2p}} \\&\ge \frac{1}{(2+2L)^2} 2^{-p} \delta ^{d p -2}\left( \frac{\mu _1}{\omega _d} \right) ^{\frac{2-pd}{2+d}}G(d)^{\frac{1+p}{1+2/d}}. \end{aligned}$$

This proves (16). Thus, if a minimizing sequence does not have a convergent subsequence,

$$\begin{aligned} G_p(\Omega , d) \ge \liminf _{\delta \rightarrow 0} G_p(\Omega , d,\delta ) = + \infty , \end{aligned}$$

which is clearly a contradiction. $\square$