Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity

Abstract

We consider a mixed boundary value problem in a domain $\Omega$ contained in a half-ball $B_+$ and having a portion $\overline{T}$ of its boundary in common with the curved part of $\partial B_+$. The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution u satisfies a Steklov condition on T and a homogeneous Dirichlet condition on $\Sigma =\partial \Omega \setminus \overline{T} \subset B_+$. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in $\Omega$ to its normal derivative $u_\nu$ on $\Sigma$. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for $u_\nu$ on $\Sigma$: in fact, it turns out that $\Sigma$ must be a spherical cap that meets T orthogonally. This result returns the one obtained by Guo and Xia under the stronger pointwise condition that the values of $u_\nu$ be constant on $\Sigma$. A second important consequence is a set of stability bounds, which quantitatively measure how $\Sigma$ is far uniformly from being a spherical cap, if $u_\nu$ deviates from a constant in the norm $L^1(\Sigma )$.

Appendix A: Remarks on the uniform cone condition

In this appendix, we detail some geometrical facts and amend an inaccuracy contained in [18].

1.1 Some geometrical facts

As already mentioned, the uniform $(\theta , a )$-interior cone condition adopted in the present paper is equivalent to the strong local Lipschitz property of Adams [1, p. 66] and to the uniform Lipschitz regularity in [5, Section III] and [22, Definition 2.1]. By putting together [22, Proposition 4.1 in the Appendix] and [5, Proposition III.1], we easily infer the following result.

Lemma A.1

Let $\Omega$ be a bounded domain satisfying the uniform $(\theta , a )$-interior cone condition. There exists a positive constant $\delta _0$ depending on $a, \theta$, and $d_\Omega$ such that, for any $\sigma \le \delta _0$, the parallel set $\Omega _\sigma =\left\{ x\in \Omega \,: \, \delta _{\Gamma }(x) > \sigma \right\}$ is connected.

A domain $\Omega$ in ${\mathbb {R}}^N$ is a b-John domain, with $b \ge 1$, if each pair of distinct points $x_1$ and $x_2$ in $\Omega$ can be joined by a curve $\psi : [0,1] \rightarrow \Omega$ such that $\psi (0)=x_1$, $\psi (1)=x_2$, and

$$\begin{aligned} b\,\delta _{\Gamma } (\psi (t)) \ge \min \left\{ |\psi (t) - x_1|, |\psi (t) - x_2| \right\} . \end{aligned}$$

A curve satisfying the previous inequality is called a John curve. By using the previous lemma, we now prove that domains satisfying the uniform $(\theta , a )$-interior cone condition are b-John domains and provide an explicit estimate for b in terms of $\theta , a, d_\Omega$.

Lemma A.2

Let $\Omega$ be a bounded domain satisfying the uniform $(\theta , a )$-interior cone condition. Then, $\Omega$ is a b-John domain with

$$\begin{aligned} b \le \max \left\{ \frac{1}{\sin (\theta )}, \frac{d_\Omega }{\min \left\{ \frac{a}{2} \frac{\sin \theta }{1+ \sin \theta }, \, \delta _0 \right\} } \right\} , \end{aligned}$$

where $\delta _0$ is the constant appearing in Lemma A.1.

Proof

Set $\sigma = \min \left\{ \frac{a}{2} \frac{\sin \theta }{1+ \sin \theta }, \, \delta _0 \right\}$. Lemma A.1 guarantees that any two points $x_1, x_2\in \Omega _\sigma$ can be joined by a curve $\psi : [0,1]\rightarrow \Omega _\sigma$. Also, we easily compute that

$$\begin{aligned} \frac{\min \left\{ |\psi (t) - x_1|, |\psi (t) - x_2| \right\} }{\delta _{\Gamma } (\psi (t))} \le \frac{d_\Omega }{\delta _{\Gamma } (\psi (t))} \le \frac{ d_\Omega }{\min \left\{ \frac{a}{2} \frac{\sin \theta }{1+ \sin \theta }, \, \delta _0 \right\} }. \end{aligned}$$

On the other hand, if $x_j$ (for $j=1$ and/or 2) is a point in $\Omega \setminus \Omega _\sigma$, then we can find a point $y_j \in \Omega _\sigma$ and another curve $\phi _j$, joining $x_j$ to $y_j$, such that

$$\begin{aligned} \frac{\min \left\{ |\phi _j (t) - x_1|, |\phi _j (t) - x_2| \right\} }{\delta _{\Gamma } (\phi _j (t))} \le \frac{1}{\sin \theta }. \end{aligned}$$

In fact, we have that $\delta _{\Gamma } (x_j) \le \sigma \le \frac{a}{2} \frac{\sin \theta }{1+ \sin \theta } \le a/4$. Hence, if $x^j$ is the projection of $x_j$ on $\Gamma$, (3.1) gives that $x_j + {\mathcal {C}}_{\omega } \subset \Omega$ with $\omega =\omega _{x^j}$. If we set $y_j = x_j + \frac{a}{1+ \sin \theta } \omega$ (which is a point on the axis of the cone $x_j + {\mathcal {C}}_{\omega }$), by some trigonometry we have that $\delta _{\Gamma } (y_j) \ge \delta _{\partial (x_j + {\mathcal {C}}_\omega )}(y_j) = a \frac{\sin \theta }{ 1 +\sin \theta } > \frac{a}{2} \frac{\sin \theta }{ 1 +\sin \theta }$. In particular, $y_j \in \Omega _\sigma$.

For $\ell >0$, the choice

$$\begin{aligned} \phi _j(t)= {\left\{ \begin{array}{ll} x_1 + \frac{t}{\ell } (y_1 - x_1) \quad &{} \text {if } j=1,\\ y_2 + \frac{t}{\ell } (x_2 - y_2) \quad &{} \text {if } j=2, \end{array}\right. } \quad \quad t \in [0, \ell ], \end{aligned}$$

is clearly admissible. Moreover, for any $x_1, x_2 \in \Omega$, it allows to create a suitable curve from $x_1$ to $x_2$ by joining together $\phi _1$ (if $x_1 \in \Omega \setminus \Omega _\sigma$), a curve contained in $\Omega _\sigma$, and $\phi _2$ (if $x_2 \in \Omega {\setminus } \Omega _\sigma$).

In any case, for any $x_1, x_2 \in \Omega$ we can always find a John curve $\psi$ from $x_1$ to $x_2$ such that

$$\begin{aligned} \frac{\min \left\{ |\psi (t) - x_1|, |\psi (t) - x_2| \right\} }{\delta _{\Gamma } (\psi (t))} \le \max \left\{ \frac{1}{\sin (\theta )}, \frac{d_\Omega }{\min \left\{ \frac{a}{2} \frac{\sin \theta }{1+ \sin \theta }, \, \delta _0 \right\} } \right\} , \end{aligned}$$

and the conclusion follows. $\square$

We now prove the following useful result.

Lemma A.3

Let $\Omega$ satisfy the $(\theta , a )$-uniform interior cone condition. Then, the parallel set $\Omega _\sigma =\left\{ x\in \Omega \,: \, \delta _{\Gamma }(x) > \sigma \right\}$ satisfies the $(\theta , a/2 )$-uniform interior cone condition, for any $\sigma \le a/4$.

Proof

Let x be any point on $\partial \Omega _\sigma$ and let y be a point in $\Gamma$ (not necessarily unique) such that $\delta _{\Gamma }(A)=|x-y| = \sigma$. Since $\Omega$ satisfies the $(\theta , a )$-uniform interior cone condition, we set ${\mathcal {C}}_{\omega }$ to be a cone satisfying (3.1) (with $x=y$). Since $B_\sigma (x)\subset \Omega$, by using (3.1) we can easily verify that $x+{\mathcal {C}}_{\omega }\cap B_{a/2} \subset \Omega _\sigma$ (Fig. 2).

Moreover, we can also check that

$$\begin{aligned} w +{\mathcal {C}}_{\omega }\cap B_{a/2} \subset \Omega _\sigma \ \text { for every } \ w \in B_{a/2} (x) \cap \overline{\Omega }_\sigma . \end{aligned}$$

Since x is chosen arbitrarily in $\partial \Omega _\sigma$, the last inclusion gives that $\Omega _\sigma$ satisfies the $(\theta , a/2 )$-uniform interior cone condition. The last inclusion holds by noting that, for any $w\in B_{a/2}(x) \cap \overline{\Omega }_\sigma$, we have that $B_\sigma (w)\subset \Omega$ (by definition of $\Omega _\sigma$) and $B_\sigma (w) \subset B_a (y)$ (being as $\sigma \le a/4$). Hence, we can argue as above to get that $w +{\mathcal {C}}_{\omega }\cap B_{a/2} \subset \Omega _\sigma$. $\square$

Fig. 2

The construction of Lemma A.3. Here, $x\in \partial \Omega _\sigma$ and $y\in \Gamma =\partial \Omega$ is such that $|x-y|=\delta _\Gamma (x)=\sigma \le a/4$. The shaded region is the cone $x+{\mathcal {C}}_{\omega }\cap B_{a/2}$. By (3.1), the region bounded by the dashed lines and containing the smallest disk is contained $\Omega$

1.2 Errata corrige of [18, Corollary 2.3 and Theorems 2.4 and 2.7]

In [18], we assumed the following notion of cone condition, which is strictly weaker than the one adopted in the present paper. A bounded domain $\Omega \subset {\mathbb {R}}^N$ with boundary $\Gamma$ satisfies the $(\theta , a)$-uniform interior cone condition if, for every $x \in \overline{\Omega }$, there is a cone ${\mathcal {C}}_x$ with vertex at x, opening width $\theta$, and height a, such that ${\mathcal {C}}_{x} \subset \Omega$ and $\overline{{\mathcal {C}}}_{x} \cap \Gamma = \{ x \}$, whenever $x \in \Gamma$. We will refer to this definition as the old cone condition. It is easy to check that this condition is verified (with same $\theta$ and a), if $\Omega$ satisfies the (new) $(\theta ,a)$-uniform interior cone condition adopted in Sect. 3.

It is a classical result [1, 22] that if $\Omega$ is a bounded domain satisfying the old cone condition, then there exists a positive constant $C_p(\Omega )$ — the (p, p)-Poincaré constant — such that

$$\begin{aligned} \Vert f - f_\Omega \Vert _{p,\Omega }\le C_p(\Omega ) \Vert \nabla f\Vert _{p,\Omega } \ \text { for any } \ f \in W^{1,p}(\Omega ). \end{aligned}$$

We realized that the proof of [18, Corollary 2.3] contains a mistake. Here, we correct that proof. The amended proof below shows that the constant c in [18, Corollary 2.3] depends not only on N, p, $\theta$, a, but also on $C_p(\Omega )$. As a consequence, the dependence on $C_p(\Omega )$ should be added also in the constants c of [18, Theorems 2.4 and 2.7]. Since, when $\Omega$ is of class $C^2$, $C_p(\Omega )$ can be estimated in terms of the radius $r_i$ of the uniform interior sphere condition and the diameter $d_\Omega$ (see [16, item (iii) of Remark 2.4]), [18, Lemma 3.2] remains true with a constant $c=c(N, p, r_i, d_\Omega )$ and the rest of the paper remains unchanged.

Amended proof of [18, Corollary 2.3]

By using [18, (2.3)], we have that

$$\begin{aligned} | f(x) - f_{{\mathcal {C}}_x}| \le c_{N,p}\,a \left( \frac{1}{|{\mathcal {C}}_x|} \int _{{\mathcal {C}}_x} | \nabla f |^p \, dx \right) ^{1/p} \le c_{N,p}\,\frac{a\ }{|{\mathcal {C}}_x|^{1/p}} \Vert \nabla f\Vert _{p,\Omega }. \end{aligned}$$

(Note that in [18], differently from the present paper, the $L^p$ norms were normalized by the Lebesgue measure of the domain.)

Next, we easily infer that

$$\begin{aligned}{} & {} |f_{{\mathcal {C}}_x} - f_\Omega | \le \frac{1}{|{\mathcal {C}}_x|} \int _{{\mathcal {C}}_x} | f - f_\Omega | \, dx \le \frac{1}{|{\mathcal {C}}_x|^{1/p}} \left( \int _{{\mathcal {C}}_x} | f - f_\Omega |^p \, dx \right) ^{1/p} \\{} & {} \quad \le \frac{1}{|{\mathcal {C}}_x|^{1/p}} \Vert f - f_\Omega \Vert _{p,\Omega } \le \frac{C_p(\Omega )}{|{\mathcal {C}}_x|^{1/p}}\, \Vert \nabla f\Vert _{p,\Omega }. \end{aligned}$$

All in all, we conclude that

$$\begin{aligned} | f(x) - f_{\Omega }|\le | f(x) - f_{{\mathcal {C}}_x}|+|f_{{\mathcal {C}}_x} - f_\Omega | \le c \, \Vert \nabla f\Vert _{p,\Omega }, \end{aligned}$$

for some constant c that depends on $N, p, \theta , a$, and $C_p(\Omega )$. $\square$

Remark A.4

As pointed out in Remark 3.3, if $\Omega$ is a bounded b-John domain, $C_p(\Omega )$ can be estimated in terms of b and $d_\Omega$. In turn, if $\Omega$ satisfies the new cone condition of the present paper, the John parameter b, and hence $C_p(\Omega )$, can be estimated in terms of the parameters $\theta$, a of the relevant definition, and $d_\Omega$. From this observation, the statement of Lemma 3.4 easily follows.

On the contrary, the old cone condition adopted in [18] is not sufficient to give an estimate of the (p, p)-Poincaré constant (see, e.g., [22]), and hence neither of the John parameter. In fact, reasoning as in [22, Example 2.6], one can construct a family of (uniformly) bounded domains $\Omega ^\varepsilon$ sharing the same (fixed) parameters of the old cone condition and a sequence $u_\varepsilon \in W^{1,2}(\Omega ^\varepsilon )$ such that

$$\begin{aligned} \int _{\Omega ^\varepsilon } u_\varepsilon \, dx = 0, \quad \int _{\Omega ^\varepsilon }|\nabla u_\varepsilon |^2 \, dx \rightarrow 0, \end{aligned}$$

while $\int _{\Omega ^\varepsilon } u_\varepsilon ^2 \, dx$ remains bounded away from zero.