Linear stability estimates for Serrin’s problem via a modified implicit function theorem

We study the shape of a bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, in which a solution $u$ to the Dirichlet problem

(1.1a)		$\displaystyle\begin{aligned} -\Delta u&=1\quad\text{in}\ \Omega,\\ u&=0\quad\text{on}\ \Gamma=\partial\Omega\end{aligned}$
satisfies the overdetermined boundary condition
(1.1b)		$\displaystyle-\frac{\partial u}{\partial\nu}=f\quad\text{on}\ \Gamma,$

where $\nu$ is the outer unit normal vector to $\Gamma$ and $f$ is a prescribed positive function defined on $\mathbb{R}^{n}$.

The overdetermined problem (1.1) arises in a shape optimization problem called the Saint-Venant problem, in which one maximizes the torsional rigidity

$$P(\Omega)=\sup_{u\in H_{0}^{1}(\Omega)\setminus\{0\}}\frac{\left(\int_{\Omega}u\,dx\right)^{2}}{\int_{\Omega}|\nabla u|^{2}\,dx}$$

of a bar with cross section $\Omega$, among all sets $\Omega$ of equal weighted volume

$$V(\Omega)=\int_{\Omega}f^{2}\,dx.$$

The Euler-Lagrange equation, after multiplying a normalizing constant, consists in (1.1). In the case where $f$ is a constant, Pólya [17] proved that the maximizer $\Omega$ of $P$ must be a ball with the prescribed volume $V$, using the symmetric rearrangement of a function. This applies to a more general situation, where $f$ is radially symmetric and non-decreasing in the radial direction.

In fact, the same symmetry result also holds for all critical points, namely, if $f$ is a constant, then (1.1) has a solution $u$ if and only if $\Omega$ is a ball. In particular, for the normalized constant $f=\tfrac{1}{n}$, $\Omega$ is a ball of radius one and $u(x)=\tfrac{1}{2n}(1-|x-c|^{2})$ with $c$ being the center of the ball. This well-known symmetry result is due to Serrin [18]. The proof introduces the method of moving planes motivated by Alexandrov’s reflection principle [2] originally used to establish the soap bubble theorem. This symmetry result can be alternatively proven by an ingenious combination of the Rellich-Pohozaev integral identity and elementary inequalities (see Weinberger [19], and Brandolini, Nitsch, Salani, and Trombetti [5]), or by a continuous version of the Steiner symmetrization (see Brock and Henrot [7]).

The objective of this paper is the stability of a domain $\Omega$ under a perturbation of the Neumann boundary condition (1.1b), which naturally arises if one considers the torsional rigidity in anisotropic media. Namely, setting $\Omega_{0}:=\mathbb{B}$, the $n$-dimensional unit ball centered at the origin, and

(1.2)

$$f(x)=\frac{1}{n}+g\left(\frac{x}{|x|}\right)\quad(x\in\mathbb{R}^{n}\setminus\{0\})$$

with a prescribed function $g$ defined on $\Gamma_{0}:=\mathbb{S}$, where $\mathbb{S}=\partial\mathbb{B}$, we prove the existence and local uniqueness of $\Omega$ admitting a solution $u$ to (1.1), and establish a quantitative estimate of the deviation of $\Omega$ from $\Omega_{0}$ in terms of the perturbation $g$.

The domain deviation is measured by a function $\rho=\rho(\zeta)\in(-1,\infty)$ which defines the star-shaped bounded domain $\Omega_{\rho}$ enclosed by

(1.3)

$$\Gamma_{\rho}:=\left\{\zeta+\rho(\zeta)\zeta\mid\zeta\in\mathbb{S}\right\}.$$

A domain $\Omega$ admitting a solution to (1.1) will also be referred to as a solution of the problem.

In what follows, $h^{k+\alpha}(\overline{\Omega})$ denotes the little Hölder space defined as the closure of the Schwartz space $\mathcal{S}$ of rapidly decreasing functions in $C^{k+\alpha}(\overline{\Omega})$, and similarly $h^{k+\alpha}(\Gamma)$ for a hypersurface $\Gamma$ (see Lunardi [13]).

In order to motivate our study, let us mention several related results concerning existence and stability of solutions to (1.1). The existence of $\Omega$ for non-constant $f$ is known (see Bianchini, Henrot and Salani [4]) in the case where $f$ is positively homogeneous, i.e.,

(1.4)

$$f(tx)=t^{\gamma}f(x)\quad(t>0,\ x\in\mathbb{R}^{n})$$

for $\gamma>0$ with $\gamma\neq 1$ with $f$ being Hölder continuous on $\mathbb{R}^{n}\setminus\{0\}$. This condition ensures the existence of a maximizer $\Omega$ of the Saint-Venant problem, and a solution $u$ to (1.1a) then satisfies $-\partial_{\nu}u=\lambda f$ on $\Gamma$ with a Lagrange multiplier $\lambda>0$. The $\gamma$-homogeneity of $f$ allows us to control $\lambda$ by considering $t\Omega:=\{tx\mid x\in\Omega\}$, and indeed $t=\lambda^{1/(1-\gamma)}$ gives a desired domain. However, the $0$-homogeneous case (1.2) cannot be treated by this variational approach, since the dichotomy of a maximizing sequence cannot be excluded in the concentration-compactness alternative.

Most of the existing stability results in the literature for (1.1), fitted into our context by translation and dilation, take inequalities of the form

(1.5)

$$\|\rho\|_{L^{\infty}(\mathcal{S}^{n-1})}\leq C\left[\frac{\partial u_{\Omega}}{\partial\nu}+R\right]_{X}^{\tau},$$

where $u_{\Omega}$ is a solution to (1.1a) in $\Omega=\Omega_{\rho}$ with $C^{2+\alpha}$-boundary, $0<\tau\leq 1$ and $[\,\cdot\,]_{X}$ denotes a norm or seminorm which measures the deviation of $-\partial_{\nu}u_{\Omega}$ from a constant $R>0$. Aftalion, Busca and Reichel [1] adopted a quantitative version of the method of moving planes and obtained a logarithmic version of (1.5) with $X=C^{1}(\Gamma)$. The method was further developed by Ciraolo, Magnanini and Vespri [8], and they obtained (1.5) for some $0<\tau<1$ in terms of the Lipschitz seminorm of $X=\text{Lip}(\Gamma)$. In fact, these results also hold for semilinear equations $-\Delta u=f(u)$ with $u>0$. On the other hand, Brandolini, Nitsch, Salani and Trombetti [6] made use of integral identities and proved (1.5) with $X=L^{\infty}(\Gamma)$ for some $0<\tau<1$. Moreover, they obtained an estimate of the volume of the symmetric difference of $\Omega$ and a union of balls by a weaker norm, i.e., $X=L^{1}(\Gamma)$. Note that the problem (1.1) admits a domain $\Omega$ composed of a finite number of balls joined by tiny tentacles if we only control the extra boundary condition (1.1b) by the $L^{1}$-norm. Following this approach, Feldman [9] obtained the sharp estimate

$$|\Omega\triangle\mathbb{B}|\leq C\left\|\frac{\partial u_{\Omega}}{\partial\nu}+R\right\|_{L^{2}(\Gamma)},$$

where $|\Omega\triangle\mathbb{B}|$ is the volume of the symmetric difference of $\Omega$ and $\mathbb{B}$ and is considered as $\|\rho\|_{L^{1}(\mathbb{S})}$ for star-shaped $\Omega=\Omega_{\rho}$. The linear (i.e., $\tau=1$) stability estimate has also been expected in (1.5). Recently, Magnanini and Poggesi proved (1.5) with $X=L^{2}(\Gamma)$ and $\tau=1$ for $n=2$, $\tau$ arbitrarily close to $1$ for $n=3$, and $\tau=\frac{2}{n-1}$ for $n\geq 4$ [14].

In general, for overdetermined problems, the super-subsolution method based on the maximum principle provides an existence criterion. In our setting, a bounded domain $\Omega$ is called a supersolution to (1.1) if the unique solution $u=u_{\Omega}$ to (1.1a) satisfies

$$-\frac{\partial u_{\Omega}}{\partial\nu}\leq f\quad\text{on}\ \Gamma,$$

and a subsolution is defined analogously with the opposite inequality. The existence of a solution, i.e., a bounded domain $\Omega$ in which $u_{\Omega}$ satisfies (1.1b), is guaranteed provided there are a supersolution $\Omega_{\sup}$ and a subsolution $\Omega_{\text{sub}}$ satisfying $\Omega_{\text{sub}}\subset\Omega_{\sup}$. Typically, balls $\mathbb{B}_{r}$ with large or small radii $r>0$ give super- or subsolutions. Indeed, for $\Omega=\mathbb{B}_{r}$,

$$u_{\mathbb{B}_{r}}(x)=\frac{r^{2}-|x|^{2}}{2n}$$

with $-\partial_{\nu}u_{\mathbb{B}_{r}}=\frac{r}{n}$ on $\partial\mathbb{B}_{r}$ solves (1.1), and we see that, in the $\gamma$-homogeneous setting (1.4) with $\gamma>1$, $\mathbb{B}_{r}$ with large (resp. small) $r>0$ is a supersolution (resp. subsolution); while for $0\leq\gamma<1$, $\mathbb{B}_{r}$ with large (resp. small) $r>0$ is a subsolution (resp. supersolution). Hence these balls provide an appropriate pair of super- and subsolutions only if $\gamma>1$.

We therefore take another approach in this paper based on an implicit function theorem, yielding linear stability estimates with Hölder norms on both sides of the estimate, as well as the existence and local uniqueness of $\Omega$ for a given perturbation $g$ in (1.2). We will need to exploit detailed properties of the linearized equation

(1.6a)		$\displaystyle\begin{aligned} -\Delta p&=0&&\text{in}\ \Omega_{\rho_{0}},\\ \left(H-\frac{1}{f}\right)p+\frac{\partial p}{\partial\nu}&=-\varphi&&\text{on}\ \Gamma_{\rho_{0}},\end{aligned}$
(1.6b)		$\displaystyle p=f\tilde{\rho}\quad\,\text{on}\ \Gamma_{\rho_{0}},$

where $H=H_{\Gamma_{\rho_{0}}}$ is the mean curvature of $\Gamma_{\rho_{0}}$ normalized such that $H=n-1$ for $\Omega=\mathbb{B}$. The linearized equation (1.6) is derived by substituting a solution pair $(\Omega_{\rho_{0}+\varepsilon\tilde{\rho}},u_{\rho_{0}+\varepsilon\tilde{\rho}})$ with formal expansions

	$\displaystyle\Gamma_{\rho_{0}+\varepsilon\tilde{\rho}}$	$\displaystyle=\{\zeta+\left(\rho_{0}(\zeta)+\varepsilon\tilde{\rho}(\zeta)\nu(\zeta)\right)+o(\varepsilon)\mid\zeta\in\mathbb{S}\},$
	$\displaystyle u_{\rho_{0}+\varepsilon\tilde{\rho}}$	$\displaystyle=u_{\rho_{0}}+\varepsilon p+o(\varepsilon)$

into (1.1) for a right hand side $f+\varepsilon\varphi$, and equating functions of order $\varepsilon$. Note that (1.6) is a decoupled system for $p$ and $\tilde{\rho}$, and we may consider only (1.6a) for the solvability of (1.6). Then (1.6b) with known $p$ yields a solution $\tilde{\rho}$.

Recall that the implicit function theorem states that the nonlinear equation $F(\rho,g)=0$ has for each $g$ close to $g_{0}$ a unique solution $\rho$ near $\rho_{0}$ with $F(\rho_{0},g_{0})=0$, if

1. (i)

   the mapping $F\colon X\times Y\to Z$ is $C^{1}$ in a neighbourhood of $(\rho_{0},g_{0})$ and if

2. (ii)

   the partial derivative $\partial_{\rho}F(\rho_{0},g_{0})\in\mathscr{L}(X,Z)$ is bijective.

Here, $X$, $Y$ and $Z$ are Banach spaces with $X\subset Z$. In addition to the solution $\rho(g)$ being locally unique, the mapping $g\mapsto\rho(g)\in X$ is in $C^{1}$. In the current setting, the Neumann boundary condition (1.1b) yields such a mapping $F$, and the linearized equation $\partial_{\rho}F(\rho_{0},g_{0})[\tilde{\rho}]=\varphi$ is reflected by (1.6). However, the linearized equation (1.6) has a regularity defect called loss of derivatives, i.e. $\partial_{\rho}F(\rho_{0},g_{0})^{-1}\not\in\mathscr{L}(Z,X)$. Since solutions $\tilde{\rho}$ to (1.6) are less regular than $\rho_{0}$, and hence the typical iterative scheme in the classical implicit function theorem fails.

One method to overcome this regularity issue is the Nash-Moser theorem, a generalization of the classical implicit function theorem introduced by Nash in [16] and generalized by Moser in [15]. The introduction of a smoothing operator combined with Newton’s method for improved convergence was there shown to be a mean to overcome the regularity deficit. For the Nash-Moser theorem to work,

1. (i)

   regularity properties are required for $F\colon X_{i}\times Y\to Z_{i}$, where $(X_{i},Z_{i})_{i}$ is a family of pairs of Banach spaces such that $X_{i}\subset X_{i-1}$, $Z_{i}\subset Z_{i-1}$. Furthermore,

2. (ii)

   a (right) inverse of $\partial_{\rho}F(\rho,g)$ has to exist for $(\rho,g)$ in a neighbourhood of $(\rho_{0},g_{0})$.

In this setting, for every $g$ in a neighbourhood of $g_{0}$, the existence of $\rho(g)$ in $X_{0}$ is then given. Note that there are various versions of the Nash-Moser theorem, also referred to as Nash-Moser-Hörmander theorem. We refer as an example to the work of Baldi and Haus [3] and the references therein.

Instead of applying the Nash-Moser theorem, we introduce a new modified version of the classical implicit function theorem, which has the constraint that a loss of derivatives may take place except at the point $(\rho_{0},g_{0})$. We require for a pair of Banach triplets $X_{2}\subset X_{1}\subset X_{0}$ and $Z_{2}\subset Z_{1}\subset Z_{0}$ that

1. (i)

   for $j=1,2$, $F$ is continuous in a neighbourhood of $(\rho_{0},g_{0})$ from $X_{j-1}\times Y$ to $Z_{j-1}$, and that it is in $C^{1}$ in a neighbourhood of $(\rho_{0},g_{0})$ from $X_{j}\times Y$ to $Z_{j-1}$, For $(\rho,g)$ in a neighbourhood of $(\rho_{0},g_{0})\in X_{j}\times Y$, we have $\partial_{\rho}F(\rho,g)\in\mathscr{L}(Z_{j-1},X_{j-1})$. Further,

2. (ii)

   $F\colon X_{j}\times Y\to Z_{j}$ is Fréchet-differentiable at $(\rho_{0},g_{0})$ for $j=1,2$ and $\partial_{\rho}F(\rho_{0},g_{0})\in\mathscr{L}(X_{j},Z_{j})$ is invertible for $j=0,1,2$.

The first point reflects the loss of regularity, the second point reflects that it does not occur at the point under consideration. Under these assumptions, we derive a modified implicit function theorem that yields local uniqueness of a solution $\rho(g)\in X_{1}$ for all $g$ in a neighbourhood of $g_{0}$, and the mapping $g\mapsto\rho(g)\in X_{0}$ is in $C^{1}$. Note that in the setting of (1.1) and (1.6), the loss of derivatives does indeed not occur in the case of the solution of (1.1) for constant $f$, as then $\Gamma$ and $u$ are smooth.

However, a second obstacle apart from the loss of derivatives arises. Due to the translational invariance of (1.1), the linearized equation (1.6a) for $g=0$ and $\Omega_{\rho_{0}}=\mathbb{B}$ is not solvable for arbitrary $\varphi\in h^{2+\alpha}(\mathbb{S})$, and for $\varphi=0$ it has an $n$-dimensional space of solutions. This implies that the partial derivative of $F$ at $(0,0)$ is not invertible, which is necessary also for the modified implicit function theorem. We will remove this degeneracy by imposing an additional condition

(1.7)

$$\int_{\Omega}x_{j}\,dx=0\quad(j=1,\ldots,n),$$

so that the barycenter of $\Omega$ is fixed to be the origin, and by decomposing the space $h^{k+\alpha}(\mathbb{S})$ into $h^{k+\alpha}(\mathbb{S})=X_{l}\oplus K$, where

(1.8)

$\displaystyle\begin{aligned} X_{l}&:=\left\{\rho\in h^{l+\alpha}(\mathbb{S})\mid\langle\rho,x_{j}\rangle_{L^{2}(\mathbb{S})}=0,\quad j=1,\ldots,n\right\},\\ K&:=\text{span}\left\{x_{1},\ldots,x_{n}\right\}.\end{aligned}$

This allows for a decomposition of a domain perturbation $\rho\in h^{l+\alpha}(\mathbb{S})$ into $\rho_{1}\in X_{l}$, $\rho_{2}\in K$, as well as a decomposition of the function $g$ in (1.2) likewise, and we examine $F(\rho_{1}+\rho_{2},g_{1}+g_{2})=0$.

With these preparations, we present the main result of this work.

While the existence of $\rho$ is guaranteed in $h^{3+\alpha}(\mathbb{S})$, only the weaker norm, i.e. the $h^{2+\alpha}(\mathbb{S})$ norm, of $\rho$ is estimated in (1.9). This is due to the fact that the linear stability estimate requires $C^{1}$-regularity of the mapping $g_{1}\mapsto\rho$, and this regularity is expected only when the image space is $h^{2+\alpha}(\mathbb{S})$ due to the loss of derivatives.

The paper is organised as follows. In Section 2, we introduce the perturbed problem as well as derive in detail the formulation via the linearized equation (1.6). We motivate the application of an implicit function theorem to a mapping $F$ that is derived from the Neumann boundary condition. This application is obstructed by the degeneracy of the derivative of $F$ as well as the loss of derivatives. The degeneracy of the derivative of $F$ stemming from the inherent symmetry of (1.1) will be addressed in Section 3. There, also the decomposition for the little Hölder spaces is motivated as well as the necessity of using the setting of the little Hölder spaces. In Section 4, we will revisit the implicit function theorem and establish a modified version fitting our setting. This is then applied to the perturbed problem in Section 5 to prove Theorem 1.1.

We formally set up the perturbed problem defined in (1.1). We want to know whether for a perturbation $g$ there exists an open bounded domain $\Omega=\Omega(g)$ admitting a solution $u_{\Omega}$ to (1.1) with (1.2), i.e. $f=\frac{1}{n}+g$.

We restrict the domain $\Omega$ to be in such a way that it may be modelled as a deviation of $\mathbb{B}$, the domain admitting a solution to (1.1) with $f=\frac{1}{n}$. For this reference domain $\Omega_{0}=\mathbb{B}$, with $\partial\Omega_{0}=\Gamma_{0}=\mathbb{S}$, we define the perturbed domain $\Omega_{\rho}$ by its $h^{m+\alpha}$-boundary $\Gamma_{\rho}=\partial\Omega_{\rho}$ in the following way. We set for $m\in\mathbb{N}$

$$U_{\gamma,m}:=\left\{v\in h^{m+\alpha}(\mathbb{S})\,\big{|}\,\left\|v\right\|_{h^{m+\alpha}(\mathbb{S})}<\gamma\right\},$$

with $\gamma\leq 1$ sufficiently small. Next, we define

$$\theta:\,\mathbb{S}\times(-1,\infty)\to\theta\left(\mathbb{S}\times(-1,\infty)\right),\quad\theta(\zeta,r):=\zeta+r\nu_{0}(\zeta)=\zeta+r\zeta.$$

In general, $\nu_{\rho}$ denotes the outer unit normal vector of $\Gamma_{\rho}$; for $\rho=0$ we have $\nu_{0}(x)=x$. Then we set

$$\Gamma_{\rho}=\left\{\zeta+\rho(\zeta)\nu_{0}(\zeta)\in\mathbb{R}^{n}\,\big{|}\,\zeta\in\Gamma_{0}\right\}=\left\{\zeta+\rho(\zeta)\zeta\in\mathbb{R}^{n}\,\big{|}\,\zeta\in\mathbb{S}\right\},$$

and $\rho\in U_{\gamma,m}$. $\rho$ models the velocity of the boundary, and will be used to measure how much $\Gamma_{\rho}$ deviates from $\Gamma_{0}$. Using this, we define the diffeomorphism

$$\theta_{\rho}(x):=\begin{cases}x+\varphi\left(|x|-1\right)\rho\left(\frac{x}{|x|}\right)\frac{x}{|x|},\quad&\text{for }x\neq 0,\\ 0\quad&\text{for }x=0,\end{cases}$$

from $\Omega_{0}=\mathbb{B}$ to $\Omega_{\rho}$, where $\varphi\colon\mathbb{R}\to\mathbb{R}$ is a smooth cut-off function with $0\leq\varphi(r)\leq 1$, $\varphi(r)=1$ for $|r|\leq\frac{1}{4}$ and $\varphi(r)=0$ for $|r|\geq\frac{3}{4}$, as well as $\left|\frac{d\varphi}{dr}(r)\right|\leq 4$. The diffeomorphism $\theta_{\rho}$ induces pullback and pushforward operators

	$\displaystyle\theta_{\rho}^{*}u$	$\displaystyle:=u\circ\theta_{\rho},\quad$	$\displaystyle\theta_{\rho}^{*}\colon h^{k+\alpha}(\Gamma_{\rho})\to h^{k+\alpha}(\mathbb{S}),$
	$\displaystyle\theta_{*}^{\rho}v$	$\displaystyle:=v\circ\theta_{\rho}^{-1},\quad$	$\displaystyle\theta_{*}^{\rho}\colon h^{k+\alpha}(\mathbb{S})\to h^{k+\alpha}(\Gamma_{\rho}),$

with $k\in\mathbb{N}\cup\{0\}$.

Our problem now becomes the following:

By elliptic regularity theory, we have, for given $\rho\in U_{\gamma,2}$, the existence and uniqueness of a solution $u_{\rho}\in h^{2+\alpha}(\overline{\Omega}_{\rho})$ when only considering (2.1a). Therefore, for the examination of this problem, it is sufficient to focus on the perturbation, i.e. (2.1b).

We define $F\in C(U_{\gamma,2}\times h^{1+\alpha}(\mathbb{S}),h^{1+\alpha}(\mathbb{S}))$ by

(2.2)

$$F(\rho,g):=\theta_{\rho}^{*}\left(\frac{\partial u_{\rho}}{\partial\nu_{\rho}}\right)+\frac{1}{n}+g,$$

where $u_{\rho}$ is the unique solution of (2.1a). Then $\Omega_{\rho}$ admits a solution to (2.1) for given $g\in h^{1+\alpha}(\mathbb{S})$ if and only if $F(\rho,g)=0$.

This structure tempts to use the implicit function theorem to arrive at solutions in a neighbourhood of $(0,0)$. However, we shall arrive at two obstacles. The first is the derivative $\partial_{\rho}F(0,0)$ not being bijective due to the inherent translational invariance, an observation that will be treated in Section 3. The second is the loss of derivatives, a regularity issue of the $\rho$-derivative of $F$ that will be discussed in the following.

In view of this, note that for $g\in h^{2+\alpha}(\mathbb{S})$ and for $m=2,3$, we have

(2.3)

$$F\in C(U_{\gamma,m}\times h^{2+\alpha}(\mathbb{S}),h^{m-1+\alpha}(\mathbb{S})).$$