A Serrin-type problem with
partial knowledge of the domain

The precise problem that we consider may be stated as follows. Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain – that is a bounded, open, connected set, whose boundary will be denoted by $\Gamma$, and let $\omega\Subset\Omega$ be an open (not necessarily connected) subset of $\Omega$ with boundary denoted by $\partial\omega$. We consider the following problem:

under the overdetermined condition

for some $c\in\mathbb{R}$. Here and in what follows $\nu$ denotes the outward unit normal of $\Omega\setminus\overline{\omega}$ and $u_{\nu}$ the derivative of $u$ in the direction $\nu$. Concerning the setting in (1.2), we remark that even without explicitly imposing any regularity assumptions on $\Gamma$, [Vo, Theorem 1] guarantees that

therefore the notation $u_{\nu}$ on $\Gamma$ is well posed in the classical sense, being $u\in C^{2,\alpha}\left(\left(\Omega\setminus\overline{\omega}\right)\cup\Gamma\right)$ by standard elliptic regularity theory.

We will further assume $u$ to be of class $C^{2}$ up to $\partial\omega$, and hence

(see Section 3 for details on this).

Concerning the regularity of the domain taken into account, to avoid unessential technicalities we will first assume that

and that

We recall, for instance, that domains with $C^{1,1}$ boundaries satisfy (1.6), see e.g. [ROV, Lemma A.1].

To state our main result we introduce some notation. For a given domain $D\subset\mathbb{R}^{N}$, we denote by $|D|$ and $|\partial D|$ the $N$-dimensional Lebesgue measure of $D$ and the surface measure of $D$, respectively. Our main result aims at considering a convenient point $z$ and at obtaining suitable bounds on the “pseudo-distance”

and on the “asymmetry”

where $\Omega\Delta B_{Nc}(z)$ denotes the symmetric difference of $\Omega$ and the ball $B_{Nc}(z)$ of radius $Nc$ centered at $z$. In addition, we provide a “geometric” bound on the set by estimating the difference between the largest ball centered at $z$ contained in $\Omega$ and the smallest ball centered at $z$ that contains $\Omega$.

The precise result that we have here goes as follows:

We point out that Theorem 1.1 collects three different aspects on the problem: indeed, the statement in (1.10) is an “integral” stability result in $L^{2}$ (see Section 5), the estimate in (1.11) provides a stability “in measure”, and the result in (1.14) deals with a “pointwise” notion of stability.

The exponent $\tau_{N}$ in (1.14) can be better precised. Indeed, we can take

Moreover, $\tau_{3}$ can be taken arbitrarily close to one, in the sense that for any $\theta>0$, we have that (1.14) holds with $\tau_{3}:=1-\theta$ and $C$ depending also on $\theta$.

Furthermore, for $N\geq 4$, we can take

We think that it is an interesting open problem to detect whether these choices of exponents $\tau_{2}$, $\tau_{3}$, $\tau_{N}$ in (1.14), as well as the exponents appearing in (1.10) and (1.11), are optimal in Theorem 1.1. It would also be very interesting to have explicit examples to check the optimality of the structural assumptions in Theorem 1.1.

We also point out that condition (1.12) is naturally satisfied in many concrete situations (see also Remark 7.2): in particular, for “small” $\omega$, the point $z$ is “close” to the baricenter of $\Omega$, hence condition (1.12) is fulfilled in this case when the baricenter of $\Omega$ lies in $\Omega$ (as it happens, for instance, for convex sets).

Of course, when $\omega=\varnothing$, we have that (1.14) reduces to $\rho_{e}=\rho_{i}$ and therefore (1.13) gives that $\Omega$ is a ball: in this sense Theorem 1.1 recovers the classical results of [Se, We] for overdetermined problems. The main difference here is that, differently from the existing literature, the equation is supposed to hold possibly only outside a subdomain $\omega$: as a counterpart, Theorem 1.1 does not prove that the full domain is a ball, but only that it is geometrically close to a ball whenever the subdomain $\omega$ has a small Lebesgue measure.

In the present setting, Theorem 1.1 will be in fact a particular case of more general quantitative results, presented in Section 7, and relying on a number of auxiliary integral identities. For more details, see Theorems 7.1, 7.3, 7.4, and 7.6.

In this spirit, Theorem 1.1 falls within the broad stream of research aiming at obtaining quantitative rigidity results, see e.g. [ABR, BNST, CMV, CV1, Fe, Ma, MP1, MP2, MP3, Pog2] and the references therein. More generally, overdetermined problems have been also considered e.g. in [AB, CV2, EP, FV1, FV2, FV3, FV4, FG, FGK, FGLP, MR1808686, MR2002730, GL, GS, GX, PS, Pog] and in the references therein.

We remark that assumption (1.6) gives a lower bound on the distance not only between $\partial\omega$ and $\Gamma$ (being $\mathop{\mathrm{dist}}(\partial\omega,\Gamma)\geq 2r_{i}$), but also between the boundaries of the different connected components of $\omega$ (if any).

Interestingly, assumptions (1.5) and (1.6) can be relaxed, as explained in Section 8.

In particular, suitable counterparts of (1.10) and (1.11) can be obtained if (1.5) and (1.6) are replaced by the weaker assumptions

and

Moreover, a counterpart of the pointwise estimate (1.14) can be obtained if (1.5) and (1.6) are dropped and replaced by (1.15), (1.16), and the assumption

We stress that (1.17) is equivalent to assume a lower bound only for $\mathop{\mathrm{dist}}(\partial\omega,\Gamma)$. Indeed, being $\Gamma$ of class $C^{2,\alpha}$, the set $\Omega$ surely satisfies the uniform interior sphere condition on $\Gamma$.

In this situation, Theorem 1.1 remains valid, with the following structural modifications:

• •

  The boundary measure of $\partial\omega$ is replaced by its perimeter, namely by the $(N-1)$-dimensional Hausdorff measure ${\mathcal{H}}^{N-1}(\partial^{*}\omega)$ of its reduced boundary $\partial^{*}\omega$. In turn, $\nu$ on $\partial^{*}\omega$ has to be intended as the (measure-theoretic) outer unit normal (see Section 8).

• •

  The constants $C$ depend on $r_{i}$ defined in (1.17) and on the structural constant $b_{0}$ of the given $b_{0}$-John domain;

• •

  The explicit expression of the exponents $\tau_{N}$ in the pointwise estimate (1.14) is possibly worse than the ones obtained in Theorem 1.1.

The definition and details for $b_{0}$-John domains can be found in Subsection 8.1. Here, we just stress that the class of John domains is huge: in particular, if (1.6) is satisfied then $\Omega\setminus\overline{\omega}$ is surely a $b_{0}$-John domain with $b_{0}\leq d_{\Omega}/r_{i}$ (see [Pog2, (iii) of Remark 3.12]).

The precise statement that we have in this framework is stated next, and can be deduced from more general results presented in Theorems 8.3, 8.6, 8.9 and 8.10.

We point out that in Theorem 1.2 we maintained the “same” choice (1.9) for the point $z$.

The dependence of the constants $C$ on $c$ in (1.18) and (1.19) could be replaced with the dependence on the surface measure $|\Gamma|$, as explained in Remark 8.11.

The explicit values of $\tau_{N}$ in (1.20) are the following ones. We have that $\tau_{2}$ can be taken as close as we wish to $1$, namely one can fix any $\theta>0$ and take $\tau_{2}:=1-\theta$ (in this case, the constant $C$ in (1.20) will also depend on $\theta$). When $N\geq 3$, one can take $\tau_{N}:=\frac{2}{N}$.

We notice that these exponents are all smaller (i.e., “worse”) than the ones obtained in Theorem 1.1. Nevertheless, it is possible to get the pointwise estimate (1.20) with the better exponents $\tau_{N}$ obtained in (1.14), and by removing the John condition (1.15), provided that we make a different choice of $z$.

The precise statement, that can be deduced from more general results obtained in Theorems 8.15 and 8.16, is the following:

We stress that the approach used in Theorem 1.3 does not need the assumption in (1.15) that $\Omega\setminus\overline{\omega}$ is a John domain and hence the dependence on $b_{0}$, present in (1.20), has been dropped.

Interestingly, the values of the exponents $\tau_{N}$ in Theorem 1.3 are the same as those in Theorem 1.1 (and therefore they are “better” than the ones obtained in (1.20), though they rely on a different choice of $z$).

We think that it would be interesting to investigate the possible optimality of these exponents also in the framework provided by Theorem 1.3.

We remark that the setting of $z$ in (1.21) is modeled on the annular set $\Omega^{c}_{r_{i}}$ rather than on $\Omega\setminus\overline{\omega}$ as in (1.9). It would be interesting to further investigate the impact of different possible choices for $z$.

The rest of this paper is organized as follows. Section 2 contains some detailed motivations from shape optimization, fluid dynamics and mechanics which naturally lead to the problem considered in this paper.

In Section 3 we make some notation precise.

In Section 4, we present some integral identities of Rellich-Pohozaev-type for solutions of (1.1). In these computations, one does not need to impose the additional condition in (1.2) from the beginning, and aims at comparing a weighted “deficit” on $\Omega\setminus\overline{\omega}$ (measuring “how far from rotational invariant” the solution is) with suitable surface integrands on $\Gamma$ and $\partial\omega$. From these identities, the auxiliary information in (1.2) provides a more precise, and simpler information.

In Section 5 we collect useful estimates and we use them to obtain a suitable stability bound on the spherical pseudo-distance defined in (1.7) (see Theorem 5.7). We also put in relation this pseudo-distance with the asymmetry defined in (1.8) (see Lemma 5.1).

The estimates collected in Section 5 are then also used in Section 6 to bound the difference $\rho_{e}-\rho_{i}$.

In Section 7, from the information obtained in Sections 5 and 6, we obtain a number of quantitative results, that also contain Theorem 1.1 as a particular case.

Finally, in Section 8 we obtain generalizations of the quantitative results presented in Section 7, by relaxing the regularity assumptions (1.5) and (1.6), and hence we establish Theorems 1.2 and 1.3 as particular cases.

A natural motivation for problem (1.1)-(1.2) comes from the optimal heating theory. In this setting, a region $\Omega$ is given which is in direct contact with an external environment having constant (say, zero) temperature. In this setting, at the equilibrium, the temperature on the boundary of $\Omega$ is set to be zero. The region $\Omega$ is also provided by a fine set of heating devices. All these devices are the same and produce the same heating effect with the exception of those placed in a small subregion $\omega$ which are malfunctioning, see Figure 1.

In this setting, at the equilibrium, the local heat flow, normalized by the flow surface, is constant, say equal to $1$, in $\Omega\setminus\omega$, but it may be different from $1$ in $\omega$. In a nutshell, the mathematical description of this situation is given by

with $f(x)=1$ for every $x\in\Omega\setminus\omega$.

A natural question in this setting is to optimize the shape of $\Omega$ in order to store in the domain the biggest possible amount of caloric energy. Concretely, given $u=u^{(f)}$ satisfying (2.1), one may try to select $\Omega$ in order to maximize (among the domains of fixed measure) the functional

This variational problem is well posed (see e.g. Theorem 4.5.2 in [HP]) and it is naturally related to the boundary derivative prescription

thus leading to the problem described in (1.1)-(1.2). We refer to Proposition 6.1.10 in [HP] and Appendix A here for a direct computation relating the shape optimization of (2.2) and the Neumann condition in (2.3).

In this framework, our result¹¹1Strictly speaking, (2.1) with a source $f\geq 0$ should be described as an optimal cooling, rather than heating, problem: speaking of heating problem should require to add a minus sign in front of $\Delta$ in (2.1). Nevertheless, we preferred to keep the sign convention in accordance with (1.1) and the rest of the paper. in Theorem 1.1 states that, for a small region $\omega$ of malfunctioning of the heat source, the optimal domain $\Omega$ is necessarily close to a ball (with a quantitative information on the proximity between $\Omega$ and a suitable ball). This result is close to intuition, since jagged domains end up dissipating most of the caloric energy from their boundaries.

Another motivation of the problem in (1.1)-(1.2) comes from laminar flows, as modeled by a Navier-Stokes equation of the type

where $v$ is the vectorial velocity of the fluid, $\rho$ is its density, $\mu$ is its viscosity coefficient, $p$ is the pressure, $g$ is the gravity acceleration and $\otimes$ denotes the outer product (i.e., given two vectors $a$ and $b$, $a\otimes b$ is the matrix whose $(i,j)$th entry is $a_{i}b_{j}$) see e.g. [Da]. The incompressibility condition

leads to

In this way, one obtains from (2.4) that

One assumes that the flow is “vertical”, namely $v=(0,0,u)$ for some scalar function $u$, thus obtaining that

We also suppose that the laminar flow occurs in a “vertical tube” of the type $\Omega\times\mathbb{R}$, with $\Omega\subset\mathbb{R}^{2}$, and that the density of the fluid only depends on the horizontal position (that is, the fluid maintains the same density along its vertical flow). In this setting, we have that $\rho=\rho(x_{1},x_{2})$ and accordingly, by (2.5) and (2.7),

Hence, we deduce from the third component of (2.6) that

The case of a “steady state” flow (that is $\partial_{t}u=0$) with constant pressure (hence $\partial_{3}p=0$) reduces (2.8) to

If the fluid has constant (say, up to changing the inertial reference frame, zero) velocity at the boundary of the pipe, equation (2.9) is complemented by the boundary condition

Also, if the fluid presents a constant tangential stress on the pipe, we have that

The case described in (1.1)-(1.2) is, in this setting, a byproduct of (2.9), (2.10) and (2.11) in which the density of the fluid (as well as its viscosity and the gravity acceleration) is constant in the region $\Omega\setminus\overline{\omega}$, but it is possibly unknown in $\omega$. In this framework, our result in Theorem 1.1 says that if a laminar fluid is homogeneous out of a small region and presents a constant tangential stress on the pipe, then necessarily the pipe is close to a right circular cylinder.

In the theory of elasticity, one can consider the displacement vector $U=(U_{1},U_{2},U_{3})$ that describes the deformation of some material. Also, it is customary to introduce the stress tensor

describing the force (per unit area) in the $i$th direction along the infinitesimal surface orthogonal to $e_{j}$, being $e_{1}:=(1,0,0)$, $e_{2}:=(0,1,0)$ and $e_{3}:=(0,0,1)$, see e.g. formula (129) in [Hj] (here, we are supposing the strain and the stress to be proportional, setting the proportionality constant equal to $1$ for the sake of simplicity).

In this framework, the equilibrium configurations are those for which the forces are infinitesimally balanced, that is

see e.g. formula (189) in [Hj].

More specifically, we focus here on the torsion of a vertical beam ${\mathcal{B}}$ (see e.g. [So, pages 100-120]). The fact that the beam is vertical means, in our setting, that

for a suitable (bounded and smooth) family of domains $\Omega_{x_{3}}\subset\mathbb{R}^{2}$. We assume that each point of the beam lying on a given horizontal plane $\{x_{3}=\bar{x}_{3}\}$ performs a horizontal rotation of a small angle $\vartheta=\vartheta(x)$, plus a vertical movement that is the same for every height of the bar (we will denote this vertical movement by $v=v(x_{1},x_{2})$). In this setting, the torsion of the bar moves a point $x=(x_{1},x_{2},\bar{x}_{3})$ to the point

and so, in this approximation, we can write the displacement vector as

From this and (2.12), we can write

As a result, exploiting the latter equation and (2.13) with $i:=3$,

We then fix $x_{3}=1$ and consider the $1$-form on $\mathbb{R}^{2}$ given by

and we deduce from (2.16) that

This gives that there exists a “warping potential” $u$ such that $\alpha=du$ as $1$-forms in $\mathbb{R}^{2}$. Accordingly, by (2.17), we have that

Combining this with (2.15), one sees that

and therefore

with these functions evaluated at $x_{3}=1$. As a special case, one can take into account the situation in which the angle $\vartheta$ depends linearly on the height of the beam, say $\vartheta(x)=\theta(x_{1},x_{2})\,x_{3}$ (this is physically reasonable, for instance, if the beam is constrained at $\{x_{3}=0\}$ and some torque is applied from the top of the beam). In this setting, (2.19) reduces to

If we suppose that the beam is built by two different materials, one occupying $\Omega\setminus\overline{\omega}$ and the other occupying $\omega$ (where $\Omega$ here represents the domain $\Omega_{x_{3}}$ in (2.14) with $x_{3}=1$), the expression in (2.20) takes two different forms in $\Omega\setminus\overline{\omega}$ and $\omega$. In particular, if we know that the material in $\Omega\setminus\overline{\omega}$ is homogeneous we can suppose that the horizontal rotation is uniform there and thus $\theta$ is independent on the point (i.e., $\partial_{1}\theta=\partial_{2}\theta=0$), hence deducing from (2.20) that

Also, the surface traction, as a force per unit of area, at a boundary point of the beam is defined as the normal component of the vertical stress, that is

being $\bar{\nu}\in\mathbb{R}^{3}$ the normal to the beam (see e.g. the third component in formula (174) of [Hj]). Recalling (2.15), we have that

being $\nu\in\mathbb{R}^{2}$ normal to $\Omega$. Thus, in view of (2.18),

being $\tau:=(-\nu_{2},\nu_{1})$ a unit tangent vector to $\Omega$. Therefore, if the traction vanishes, the tangential derivative of $u$ vanishes as well, hence $u$ is constant along $\partial\Omega$. Since $u$ was introduced as a potential, it is defined up to an additive constant, hence we can rephrase these considerations by saying that if the traction vanishes, then

We also remark that, if $\sigma_{3}=(\sigma_{31},\sigma_{32},\sigma_{33})$, then

thanks to (2.15) and (2.18).

Hence, for constant stress intensity $|\sigma_{3}|$, we deduce from (2.22) that

We thus observe that problem (1.1)-(1.2) arises naturally from (2.21), (2.22) and (2.23), and, in this framework, Theorem 1.1 says that if a beam is homogeneous out of a small region and presents zero traction and constant stress intensity, then necessarily the horizontal section of the beam is close to a disk.