Convex duality for principal frequencies

1.   Introduction

1.1.   Overview

Let $\Omega\subset\mathbb{R}^N$ be an open set, we denote by $W^{1, 2}_0(\Omega)$ the completion of $C^\infty_0(\Omega)$ with respect to the Sobolev norm

In what follows, we will always consider for simplicity sets with finite measure. This guarantees that we have at our disposal the Poincaré inequality

Consequently, the space $W^{1, 2}_0(\Omega)$ can be equivalently endowed with the norm

For $1\le q\le 2$, we consider the generalized principal frequency

already considered in ^[26] and more recently in ^[4,13,35], among others. The fact that the minimum above is attained in $W^{1, 2}_0(\Omega)$ follows from the compactness of the embedding $W^{1, 2}_0(\Omega)\hookrightarrow L^q(\Omega)$. The latter holds true since we are assuming that $\Omega$ has finite measure, see [5,Theorem 2.8].

Two important cases deserve to be singled-out from the very beginning: $q = 1$ and $q = 2$. In the first case, this quantity actually coincides with the reciprocal of the so-called torsional rigidity of $\Omega$

For $q = 2$, on the other hand, the quantity $\lambda_1(\Omega; 2)$ is nothing but the first eigenvalue of the Dirichlet-Laplacian or principal frequency of $\Omega$. This is the smallest real number $\lambda$ such that the Helmholtz equation

admits a nontrivial weak solution in $W^{1, 2}_0(\Omega)$. For simplicity, we will simply denote this quantity by $\lambda_1(\Omega)$.

In general, the quantity $\lambda_1(\Omega; q)$ can not be explicitly computed. It is then quite useful to seek for (possibly sharp) estimates in terms of geometric quantities of the set $\Omega$. Some particular instances of results in this direction are given by:

● the Faber-Krahn inequality (see for example [26,Theorem 2])

which is valid for every open set $\Omega\subset\mathbb{R}^N$ with finite measure. Here $B$ is any $N-$dimensional open ball;

● the Hersch-Makai–type inequality (see [8,Theorem 1.1])

which is valid for every open bounded convex set $\Omega\subset\mathbb{R}^N$. Here $R_\Omega$ is the inradius, i.e., the radius of a largest ball contained in $\Omega$ and $\pi_{2, q}$ is the one-dimensional constant defined by

This inequality is the extension to the range $1\le q\le 2$ of [28,equation (3')] by Makai (case $q = 1$) and of [22,Théorème 8.1] by Hersch (case $q = 2$);

● the Pólya–type inequality (see [6,Main Theorem])

which is again valid for open bounded convex sets. Here $P(\Omega)$ stands for the perimeter of $\Omega$. This inequality generalizes the original result by Pólya ^[31] for the cases $q = 1$ and $q = 2$.

We point out that all the previous estimates are sharp. All the exponents appearing above are of course dictated by scale invariance.

As a general rule, we can assert that lower bounds on $\lambda_1(\Omega; q)$ are harder to obtain with respect to upper bounds, since every generalized principal frequency is defined as an infimum. It would then be interesting to investigate whether $\lambda_1(\Omega; q)$ admits a sort of "dual'' equivalent formulation, in terms of a supremum. This is the main goal of the present paper.

1.2.   Towards duality

At this aim, it is interesting to have a closer look at the case $q = 1$. It is well-known that the torsional rigidity can be equivalently rewritten as an unconstrained concave maximization problem, i.e.,

As such, it admits in a natural way a dual convex minimization problem

which gives yet another equivalent definition of torsional rigidity. Here the divergence constraint has to be intended in distributional sense, i.e.,

By means of a standard density argument, it is easily seen that we can enlarge the class of competitors in (1.3) to the whole $W^{1, 2}_0(\Omega)$, since $\phi\in L^2(\Omega; \mathbb{R}^N)$.

For a better understanding of the contents of this paper, it may be useful to recall the proof of (1.2). At first, one observes that for every admissible vector field and every $\varphi\in W^{1, 2}_0(\Omega)$, we have

by virtue of (1.3). We can now use Young's inequality

By integrating this inequality, from the identity above we get

The arbitrariness of both $\varphi$ and $\phi$ automatically gives

On the other hand, by taking $\varphi = w$ to be the optimal function for the problem on the left-hand side, this satisfies the relevant Euler-Lagrange equation. The latter is given by

Thus $\phi_0 = \nabla w$ is an admissible vector field and we have

This proves that

as well. Thus (1.2) holds true and we also have obtained that the unique (by strict convexity) minimal vector field $\phi_0\in L^2(\Omega; \mathbb{R}^N)$ must be of the form

with $w$ being the unique $W^{1, 2}_0(\Omega)$ solution of $-\Delta w = 1$. In conclusion, getting back to the notation $\lambda_1(\Omega; 1)$, we obtain the following dual characterization of the relevant generalized principal frequency

1.3.   Main results

The main result of the present paper asserts that the dual characterization (1.4) is not an isolated exception. Actually, it is possible to prove that

coincides with the minimum of a constrained convex minimization problem, for the whole range $1\le q\le 2$. The deep reason behind this result is a hidden convex structure of the problem which defines $\lambda_1(\Omega; q)$, see Remark 4.2 below. Such a convex structure, which apparently is still not very popular, fails for $q > 2$ and this explains why our result has $1\le q\le 2$ as the natural range of validity.

In order to precisely state the result, we need at first to introduce the following convex lower semicontinuous function $G_q:\mathbb{R}\times\mathbb{R}^N\to [0, +\infty]$, defined for $1 < q\le 2$ by:

see [30,Lemma 5.17]. We then distinguish between the cases $q < 2$ and $q = 2$.

Theorem 1.1 (Sub-homogeneous case). Let $1 < q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. If we set

then we have

where $G_q$ is defined in $(1.5)$. Moreover, if $w\in W^{1, 2}_0(\Omega)$ denotes the unique positive solution of

we get that the pair $(f_0, \phi_0)$ defined by

is a minimizer for the problem in $(1.6)$.

Remark 1.2. Observe that the previous result is perfectly consistent with the case $q = 1$. Indeed, by formally taking the limit as $q$ goes to $1$ in the statement above, the role of the dual variable $f$ becomes immaterial and we get back (1.2), together with the optimality condition $\phi_0 = \nabla w$.

For the limit case $q = 2$, corresponding to the first eigenvalue of the Dirichlet-Laplacian, we have the following dual characterization.

Theorem 1.3 (Homogeneous case). Let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. If we set

then we have

where $G_2$ is defined in $(1.5)$. Moreover, if we denote by $U\in W^{1, 2}_0(\Omega)$ any positive first eigenfunction of $\Omega$, we get that the pair $(f_0, \phi_0)$ defined by

is a minimizer for the problem in $(1.7)$.

Remark 1.4. It may be worth recalling that the existence of a (sort of) dual formulation for $\lambda_1$ is not a complete novelty. A related result can be traced back in the literature and attributed to the fundamental contributions of Protter and Hersch. This is called maximum principle for $\lambda_1$ and reads as follows

under suitable regularity assumptions on $\Omega$ and on the admissible vector fields. It is not difficult to see that

is a maximizer for the previous problem, at least formally. Here $U$ is again any positive first eigenfunction of $\Omega$. We refer to the paper ^[21] by Hersch for a presentation of this result and for a detailed discussion about its physical interpretation. \end{oss}

Remark 1.5. As a last observation, we wish to point out the interesting papers ^[18] and ^[19], where yet another equivalent characterization for the torsional rigidity $T(\Omega)$ is obtained, when $\Omega\subset\mathbb{R}^2$ is a simply connected open set. Such a characterization is in terms of a minimization problem among holomorphic functions (see [18,Theorem 1.2]) and thus it is suitable for giving upper bounds on $T(\Omega)$ (see ^[19]).

1.4.   Plan of the paper

We start by exposing some preliminary facts in Section 2. In Section 3 we consider a certain convex function and show that its Legendre-Fenchel transform is related to the function $G_q$ above. The core of the paper is Section 4, where Proposition 4.1 permits to rewrite the value $\lambda_1(\Omega; q)$ as an unconstrained concave maximization problem, exactly as in the case of the torsional rigidity. We can then prove our main results in Section 5. Finally, in the last section we briefly show some applications of our results to geometric estimates for principal frequencies.

2.   Preliminaries

We first recall that it is possible to rewrite the minimization problem which defines $\lambda_1(\Omega; q)$ as an unconstrained optimization problem, in the regime $1\le q < 2$. This generalizes formula (1.1). The proof is standard, we include it for completeness.

Proposition 2.1. Let $1\le q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. Then we have

Moreover, the maximization problem on the left-hand side admits a unique non-negative solution $w$, which has the following properties

Proof. Existence of a maximizer follows by a standard application of the Direct Method in the Calculus of Variations. The fact that a non-negative maximizer exists is a consequence of the fact that the functional is even, thus we can always replace $\varphi$ by $|\varphi|$ without decreasing the energy.

We also observe that for $\varphi\in W^{1, 2}_0(\Omega)\setminus\{0\}$ and $t > 0$, the quantity

is strictly positive for $t$ sufficiently small. This shows that

and thus $\varphi\equiv 0$ can not be a maximizer. Observe that the same argument, together with the locality of the functional, imply that for any maximizer $w$ we must have $w\not\equiv0$ on every connected component of $\Omega$. By coupling this information with the optimality condition, we get that any non-negative maximizer $w$ must be a nontrivial weak solution of the Euler-Lagrange equation

In particular, $w$ is a weakly superharmonic function and by the strong minimum principle, we get that $1/w\in L^\infty_{\rm loc}(\Omega)$. The fact that $w\in L^\infty(\Omega)$ follows from standard Ellipic Regularity.

Finally, uniqueness of the positive maximizer can be found in [7,Lemma 2.2], where the uniqueness result of [9,Theorem 1] is extended to the case of open sets, not necessarily smooth.

In order to prove the claimed equality between the extremum values, it is sufficient to exploit the different homogeneities of the two integrals and the fact that the maximum problem is equivalently settled on $W^{1, 2}_0(\Omega)\setminus\{0\}$. We then have

It is easily seen that, for every $\varphi\in W^{1, 2}_0(\Omega)\setminus\{0\}$ the function

is maximal for

With such a choice of $t$, we get

By recalling the definition of $\lambda_1(\Omega; q)$, we get the desired conclusion.

We also record the following technical result: this will be useful somewhere in the paper. More sophisticated results about the dependence on $q$ of the quantity $\lambda_1(\Omega; q)$ can be found in [1,Theorem 1] and ^[17].

Lemma 2.2. Let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. Then we have

Proof. For $1 < q < 2$ and for every $\varphi\in W^{1, 2}_0(\Omega)\setminus\{0\}$, we have by Hölder's inequality

By taking the infimum over $\varphi$, this leads to

On the other hand, if $U\in W^{1, 2}_0(\Omega)$ is any minimizer for $\lambda_1(\Omega)$, then we have

The last two displays eventually prove the desired result for $q$ converging to $2$. The other result can be proved in exactly the same way.

Remark 2.3. The assumption on the finiteness of the measure is sufficient, but in general not necessary, for the previous result to hold. However, as observed in [6,Remark 2.2], for a general open set $\Omega\subset\mathbb{R}^N$ it may happen that

3.   A convex function

In order to prove the main result of this paper, we will need to study a particular convex function $F_q:\mathbb{R}\times\mathbb{R}^{N}\to[0, +\infty]$ and its Legendre-Fenchel transform

We refer to the classical monograph ^[32] for the basic properties of this transform.

Lemma 3.1. Let $1 < q < 2$, we consider the convex lower semicontinuous function $F:\mathbb{R}\times \mathbb{R}^N\to \mathbb{R}\cup \{+\infty\}$ defined by

Then its Legendre-Fenchel transform is given by the convex lower semicontinuous function

where the constant $\alpha_q$ is given by

Proof. We divide the proof in various parts, according to the claim that we are going to prove.

Lower semicontinuity. In order to verify the semicontinuity of $F_q$, we need to prove that the epigraph

is a closed set. We take $\{((t_n, x_n); \ell_n)\}_{n\in\mathbb{N}}\subset \mathrm{epi\, }(F_q)$ such that

By using the definition of $F_q$ and that of epigraph, the fact that

automatically entails that

This in particular implies that the limit point $t$ is such that $t\ge 0$. The same can be said for $\ell$, since $F_q$ always takes positive values.

We now observe that if $t > 0$, we would have $t_n > 0$ for $n$ large enough. In this case, we can simply pass to the limit in (3.1) and get

thus proving that $((t, x);\ell)\in \mathrm{epi\, }(F_q)$.

Let us now suppose that $t = 0$ and assume by contradiction that $((0, x);\ell)\not \in \mathrm{epi\, }(F_q)$. This means that

By recalling that $\ell\ge 0$ and that $F_q(0, 0) = 0$, this would automatically gives that $x\not = 0$. On the other hand, by (3.1), we get that

This entails that

which gives a contradiction. This finally proves that the epigraph is closed.

Convexity. We need to prove that for every $t_0, t_1\in\mathbb{R}$, $x_0, x_1\in\mathbb{R}^N$ and $\lambda\in [0, 1]$, we have

We observe that for $t_0, t_1\le 0$, every $x_0, x_1\in\mathbb{R}^N\setminus\{0\}$ and every $\lambda\in[0, 1]$, we trivially have (3.2), since both terms on the right-hand side are equal to $+\infty$. We are thus confined to prove (3.2) for

Moreover, if at least one between $(t_0, x_0)$ and $(t_1, x_1)$ coincides with $(0, 0)$, then again the desired inequality follows by a straighforward computation. Finally, we can assume that

We introduce the function

and we observe that for every $1 < q < 2$ we have

By using that $t\mapsto F_2(t, x)$ is decreasing, that $t\mapsto t^{2-\frac{2}{q}}$ is concave (since $1 < q < 2$) and that $(t, x)\mapsto F_2(t, x)$ is convex (see for example [30,Lemma 5.17]), we get

as desired.

Computation of $F^*_q$. We now come to the computation of the Legendre-Fenchel transform. This is lengthy but elementary. We first observe that $F_q$ is positively $2/q-$homogeneous, that is for every $\tau > 0$ we have

Correspondingly, $F^*_q$ will be positively $2/(2-q)-$homogeneous, by standard properties of the Legendre-Fenchel transform. Thanks to this remark, it is sufficient to compute for $\xi\in\mathbb{R}^N$

By definition, we have

where we set^*

^* For notational simplicity, we use the convention that $m^2\, t^{\frac{2}{q}-2} = 0$ when both $t = 0$ and $m = 0$.

We thus easily get for $\xi\in\mathbb{R}^N$

and

Moreover, for $\xi\not = 0$ we have

as can be seen by taking

and letting $n$ go to $+\infty$. We are left with computing for $\xi\in\mathbb{R}^N$

We observe at first that we easily have

We thus take $\xi\in\mathbb{R}^N\setminus\{0\}$ and we make a preliminary observation: if $(t, m)\in\mathcal{E}$ are such that

we get

and the last quantity can be made strictly positive, for $m > 0$ small enough, thanks to the fact that $2/q > 1$. On the contrary, every point $(t, 0)\in\mathcal{E}$ can not be a maximizer for the problem which defines $F^*_q(-1, \xi)$, since on these points

This simple observation implies that we can rewrite the maximization problem for $F^*_q(-1, \xi)$ as

Moreover, this quantity is strictly positive. In order to explicitly compute it, we will exploit the homogeneity of the function $(t, m)\mapsto m^2\, t^{\frac{2}{q}-2}$. Indeed, we first observe that by taking $\lambda > 0$ and replacing $(t, m)$ by $(\lambda\, t, \lambda\, m)$ we get

Now we observe that the derivative of the function

is given by

We now distinguish two cases: if $m\, |\xi|-t\le 0$, the previous computation shows that $h$ is decreasing on $(0, +\infty)$ and thus

On the other hand, if $m\, |\xi|-t > 0$, then we get that $h$ has a unique maximum point at

thus

This discussion entails that

We are left with computing such a supremum. We may notice that the objective function only depends on the ratio $t/m$, indeed we have

Thus, if we set $\tau = t/m$, we finally arrive at the problem

It is easily seen that the function

is maximal in the interval $(0, |\xi|)$ for

Thus we obtain

which eventually leads to

Thanks to the positive homogeneity of $F_q^*$ already discussed, we get the desired conclusion.

Remark 3.2 (Relation between $F^*_q$ and $G_q$). From the previous result, we get more generally that for every $C > 0$, we have

This easily follows from the properties of the Legendre-Fenchel transform, together with the fact that $F_q^*$ is $2/(2-q)-$positively homogeneous. In particular, by taking $C = 1/(2\, q)$, we have

By recalling the definition (1.5) of $G_q$, we easily see that

and thus

Finally, by using that

we get the relation

We are going to use this identity in the proof of the main result.

4.   A concave maximization problem

By combining Proposition 2.1 and the convexity of the function $F_q$ above, we can rewrite the variational problem which defines $\lambda_1(\Omega; q)$ as a concave optimization problem. This property is crucial for the proof of Theorem 1.1.

Proposition 4.1. Let $1 < q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. We define the following subset of $W^{1, 2}_0(\Omega)$

Then $X_q(\Omega)$ is convex and we have

Finally, the last supremum is attained by a function $v\in X_q(\Omega)$ of the form

where $w$ is the same as in Proposition 2.1.

Proof. Convexity of $X_q(\Omega)$ immediately follows from the convexity of the function $F_q$. We now come to the proof of (4.1). The first identity is already contained in Proposition 2.1. Let us take $w\in W^{1, 2}_0(\Omega)\cap L^\infty(\Omega)$ to be the positive maximizer of

We now set $v = w^q$ and observe that $v\in W^{1, 2}_0(\Omega)\cap L^\infty(\Omega)$, since $v$ is the composition of a function in $W^{1, 2}_0(\Omega)\cap L^\infty(\Omega)$ with a locally Lipschitz function, vanishing at the origin. From the chain rule in Sobolev spaces, we get

where we also used the relation between $w$ and $v$, to replace $w^{q-1}$. Since $w > 0$ in $\Omega$, we have the same property for $v$, as well. Thus we can infer

By raising to the power $2$ and integrating, we get

which shows that $v\in X_q(\Omega)$. By recalling that $w$ is optimal, this also shows that

On the other hand, let $\psi\in X_q(\Omega)$. Thanks to the form of the function $F_q$, this in particular implies that

For every $\varepsilon > 0$, we introduce the $C^1$ function

Then we set $\varphi_\varepsilon = g_\varepsilon\circ \psi$ and observe that $\varphi_\varepsilon\in W^{1, 2}_0(\Omega)$, thanks to the fact that $g_\varepsilon$ is $C^1$ with bounded derivative and $g_\varepsilon(0) = 0$. Again by the chain rule, we have

By raising to the power $2$ and integrating, we get

In the last inequality, we used the well-known fact that $\nabla \psi$ vanishes almost everywhere on the zero set of the Sobolev function $\psi$ (see for example [27,Theorem 6.19]). This in turn implies that

It is only left to pass to the limit as $\varepsilon$ goes to $0$ in the integral containing $g_\varepsilon(\psi)$. This can be done by a standard application of the Lebesgue Dominated Convergence Theorem. This finally leads to

By arbitrariness of $\psi\in X_q(\Omega)$, we eventually get the desired conclusion (4.1).

The above discussion also prove the last statement, about a maximizer of the problem settled over $X_q(\Omega)$.

Remark 4.2. The previous result is crucially based on the fact that the Dirichlet integral, apart from being convex in the usual sense, enjoys a suitable form of "hidden'' convexity. In other words, for $\varphi$ positive we have that

remains convex also with respect to the new variable $\psi = \varphi^q$, for the whole range $1\le q\le 2$. In the limit case $q = 2$, this remarkable fact has been proved by Benguria, see [2,Theorem 4.3] and [3,Lemma 4]. For $1 < q < 2$ this property seems to have been first detected in [25,Proposition 4], see also [29,Proposition 1.1] and [34,Example 5.2].

Actually, we can restrict the maximization to smooth compactly supported functions, without affecting the value of the supremum. This is the content of the following result.

Lemma 4.3. With the notation of Proposition 4.1, we have

Proof. We just need to prove that

We take $\varphi\in C^\infty_0(\Omega)$ not identically zero and set $\psi = |\varphi|^q\in C^1_0(\Omega)$. As above, we have

which holds everywhere on $\Omega$. This in particular implies that $\nabla \psi$ vanishes on every point where $\psi$ vanishes. Thus we have

By integrating and recalling the relation above between $\nabla\psi$, $\psi$ and $\nabla\varphi$, we then obtain

This in turn implies

By arbitrariness of $\varphi\in C^\infty_0(\Omega)$, we get

On the other hand, by density of $C^\infty_0(\Omega)$ in $W^{1, 2}_0(\Omega)$, it is easily seen that

The desired conclusion now follows by combining Proposition 4.1 and (4.2).

5.   Proof of the main results

5.1.   Proof of Theorem 1.1

We recall the definition

where the condition $-\mathrm{div\, } \phi+f\ge 1$ has to be intended in distributional sense, i.e.,

In particular, for every $(f, \phi)\in\mathcal{A}(\Omega)$ and every $\psi\in X_q(\Omega)\cap C^1_0(\Omega)$, we can write

By Lemma 3.1 and Eq (3.3) from Remark 3.2, the following inequality holds almost everywhere

This simply follows from the definition of Legendre-Fenchel transform. By integrating this inequality and taking the supremum over $\psi$, we obtain

By combining Proposition 4.1 and Lemma 4.3 and taking the infimum over admissible pairs $(f, \phi)$, we get

In order to prove the reverse inequality and identify a minimizing pair, we take

where $w$ is the function of Proposition 2.1. In light of the properties of $w$, both $\phi_0$ and $f_0$ have the required integrability properties. Moreover, it is not difficult to see that these are admissible, since it holds

in distributional sense. It is sufficient to use the equation solved by $w$. By recalling the definition (1.5) of $G_q$, when $\nabla w\not = 0$ we have

On the other hand, since both $f_0$ and $\phi_0$ vanish when $\nabla w = 0$, in this case we have that $G_q(f_0, \phi_0)$ would vanish, as well. In conclusion, we get

Thus we obtain

We can now use that by Proposition 2.1

The first identity above follows by testing the Euler-Lagrange equation for $w$, i.e.,

with $\varphi = w$ itself. This finally proves that the reverse inequality holds

as well. The second part of the proof also proves that $(f_0, \phi_0)$ is an optimal pair.

5.2.   Proof of Theorem 1.3

In order to prove the inequality

we will go through an approximation argument, for simplicity. Let $(f, \phi)\in \mathcal{A}(\Omega)$ be an admissible pair, we can suppose that

Thanks to the definition of $G_2$, this implies in particular that the pair $(f, \phi)$ has the following properties:

and

We thus obtain for every $1 < q < 2$

and thus

This estimate guarantees that for every open set $\Omega'$ compactly contained in $\Omega$, we have

By Theorem 1.1 applied to $\Omega'$, we have for every $1 < q < 2$

By using (5.2) on the right-hand side and taking the limit as $q$ goes to $2$, we get

by virtue of Lemma 2.2. We can now take an increasing sequence of open sets $\{\Omega_n\}_{n\in\mathbb{N}}$ compactly contained in $\Omega$ and invading it, i.e., such that

By using that^†

^† This simply follows from the properties of the sequence $\{\Omega_n\}_{n\in\mathbb{N}}$ and the fact that

and applying (5.3) to each $\Omega_n$, we get

By arbitrariness of $(f, \phi)\in\mathcal{A}(\Omega)$, we get (5.1) as desired.

In order to prove the reverse inequality, we take

where $U$ is any positive first eigenfunction of $\Omega$. It is easily seen that this pair is admissible for the variational problem

Moreover, we have

where we also used that $f_0$ vanishes if and only if $\phi_0$ vanishes and in this case $G_2(f_0, \phi_0) = 0$. This gives

thus concluding the proof.

6.   Applications to geometric estimates

In this section, we briefly sketch some geometric estimates for the generalized principal frequencies, that can be inferred from our main result.

6.1.   Diaz-Weinstein–type estimates

We start by recalling the Diaz-Weinstein inequality for the torsional rigidity. This is given by the following estimate^‡

^‡ In ^[16] the case $N = 2$ is considered and a slightly different proof is given. The definition of torsional rigidity in ^[16] coincides with ours for simply connected sets, up to a multiplicative factor $4$.

see [16,formula (11)], which is valid for open sets $\Omega\subset\mathbb{R}^N$ such that

The quantity $\mathcal{I}_2(\Omega)$ is sometimes called polar moment of inertia of $\Omega$. It is easily seen that the minimum in its definition is uniquely attained at the centroid of $\Omega$, i.e., at the point

The Diaz-Weinstein inequality can be proved by appealing to the dual formulation for the torsional rigidity. Indeed, for every $x_0\in\Omega$, it is sufficient to use the admissible vector field

in the dual problem (1.2). This automatically gives

and thus (6.1) follows by arbitrariness of $x_0\in\mathbb{R}^N$. We observe that this estimate is sharp, as equality is attained for a ball. Indeed, recall that the unique $W^{1, 2}_0(\Omega)$ solution of $-\Delta u = 1$ in a ball of radius $R$ and center $x_0$ is given by

Thus, by observing that $\phi_0 = \nabla w$ and recalling the discussion in Subsection 1.2, we get the claimed optimality.

We now show how the previous argument can be extended to the case $1 < q < 2$. We fix again a point $x_0\in\mathbb{R}^N$ and take a constant $\alpha > 1/N$, then we choose the pair

Observe that this solves

Thus the pair $(f_0, \phi_0)$ is admissible for the dual problem (1.6), for every $1 < q < 2$. By Theorem 1.1 we immediately get

We now observe that the quantity

is minimal for $\alpha = q/N$. By making such a choice for $\alpha$ and using the arbitrariness of $x_0$, we then get the following

Corollary 6.1 (Diaz-Weinstein–type estimate). Let $1 < q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open set such that

Then we have

Remark 6.2. We notice that for $1 < q < 2$ the estimate (6.2) does not appear to be sharp. In order to get the sharp constant, it seems unavoidable the use of more sophisticated arguments, based on radially symmetric decreasing rearrangements. These permit to show that both quantities

are minimal for a ball, among sets with given measure. By combining these two facts, then we get that the sharp constant in (6.2) is given by

where $B_1 = \{x\in\mathbb{R}^N\, :\, |x| < 1\}$.

Nevertheless, we believe that the duality-based proof exposed above is interesting anyway: this gives a cheap way to get a scale invariant geometric estimate with a simple explicit constant, by means of an elementary argument.

6.2.   Cheeger–type estimates

We recall that the Cheeger constant for an open set $\Omega\subset\mathbb{R}^N$ is given by

Here $P(E)$ is the distributional perimeter of a set $E$. The Cheeger constant has the following dual characterization

This characterization seems to have first appeared in [33,Section 4]. The fact that the minimum in (6.3) is attained easily follows from the Direct Method in the Calculus of Variations.

We take an optimal vector field $\phi_\Omega$ in (6.3) and then for every $\varepsilon > 0$ we make the choice

By observing that $(f_0, \phi_0)\in\mathcal{A}(\Omega)$, from Theorem 1.1 we get

We now notice that the quantity

is minimal with the choice $\varepsilon = (q-1)$. Then such a choice leads to the estimate

Thus, we proved the following

Corollary 6.3. Let $1 < q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open set, with finite measure. Then we have the Cheeger-type inequality

Remark 6.4. By taking the limits as $q$ goes to $1$ and as $q$ goes to $2$ in (6.4), we recover

respectively. The first estimate has been proved in [11,Theorem 2], while the second one is the classical Cheeger inequality for the Laplacian, see ^[14]. Both inequalities are sharp in the following sense: by taking the $N-$dimensional unit ball $B_1(0)$, one may prove that

The first fact can be easily seen, by recalling that

The second fact has been recently observed in [20,Theorem 1.3] and is based on asymptotics for zeros of Bessel functions.

This somehow suggests that the general estimate (6.4) should be sharp, as well, by using a similar argument. However, the task of computing the exact asymptotics for $\lambda_1(B_1(0);q)$, as the dimension $N$ goes to $\infty$, does not seem easy.

6.3.   Hersch-Makai–type estimates

We now take $\Omega\subset\mathbb{R}^N$ to be an open bounded convex set. We will employ in a dual way a trick by Kajikiya (see ^[23,24] and also ^[8]), in conjunction with our duality result. This will give us a sharp lower bound on $\lambda_1(\Omega; q)$ in terms on the inradius and the perimeter of the set.

We indicate by $d_\Omega:\overline\Omega\to\mathbb{R}$ the distance function from the boundary $\partial\Omega$, while $R_\Omega$ will be the inradius of $\Omega$. We recall that this coincides with the supremum of the distance function, that is

We take $g$ to be the unique positive solution of

This satisfies the equation

This is a concave even function, which is increasing on $(-1, 0)$ and decreasing on $(0, 1)$. We then "transplant'' this function to $\Omega$, by setting

By using the equation solved by $g$, the fact that $|\nabla d_\Omega| = 1$ almost everywhere and the weak superharmonicity^$\S$ of $d_\Omega$, we get that

^$\S$ This follows from the convexity of $\Omega$.

in weak sense. This entails that the pair

is admissible for the dual problem (1.6). By Theorem 1.1, we then get

By using the explicit form of $u$ and again the fact that $|\nabla d_\Omega| = 1$ almost everywhere in $\Omega$, the previous estimate can be rewritten as

We observe that this is already a geometric estimate in nuce, since the right-hand side only depends on elementary geometric quantities of $\Omega$ (i.e., the distance function and the inradius) and on the universal one-dimensional function $g$. Moreover, it is not difficult to see that (6.5) is sharp (see Remark 6.6 below).

Let us try to derive from (6.5) a more explicit estimate. At this aim, we can use the Coarea Formula with respect to the distance function, so to get

where $\Omega_t = \{x\in\Omega\, :\, d_\Omega(x) > t\}$. We now recall that $t\mapsto P(\Omega_t)$ is monotone decreasing, in a convex set (see [10,Lemma 2.2.2]). Thus we automatically get

A simple change of variable then leads to

We can insert this estimate in (6.5) to obtain

By using that $g$ is even and the identity

we have

In the last equality we used the optimality of $g$ and Proposition 2.1, for the one-dimensional set $\Omega = (-1, 1)$. This gives

If we now recall the definition

and use the scaling properties of Sobolev-Poincaré constants, we get

Thus, we finally obtain the following

Corollary 6.5. Let $1 < q < 2$ and let $\Omega\subset\mathbb{R}^N$ be an open bounded convex set. Then we have

Remark 6.6. When compared with the Hersch-Makai–type inequality

already recalled in the Introduction, we see that the estimate (6.6) is slightly weaker. Indeed, the former implies the latter, by recalling that for a open bounded convex set we have

Nevertheless, inequality (6.6) is still sharp: it is sufficient to take the "slab–type'' sequence

with $L$ diverging to $+\infty$. For this family of sets we have (see [8,Lemma A.2])

and $R_{\Omega_L} = 1/2$, for $L > 1$.

We also observe that this slight discrepancy between (6.6) and (6.7) is lost in the limit as $q$ converges to $2$: in both cases the estimates boil down to

which is the original Hersch sharp inequality from [22,Théorème 8.1]. We also refer to [12,Theorem 5.5], [15,Theorem 5.1] and [23,Theorem 2.1] for other proofs and extensions of this result.

Acknowledgments

We wish to thank Francesco Maggi for first introducing us to the hidden convexity of the Dirichlet integral. We also thank Rafael Benguria, Guillaume Carlier and Eleonora Cinti for some comments on a preliminary version of the paper.

Conflict of interest

The author declares no conflict of interest.