Hopf algebra of labeled simple graphs arising from super-shuffle product

1.   Introduction

The main purpose of the present paper is to derive a sharp quantitative stability estimate for the rigidity of the spherical configuration of $\Omega$ under non-radial perturbations of the boundary data $g$, where a bounded domain $\Omega\subset\mathbb{R}^n$ with $n\geq 2$ admits a solution $u$ to the overdetermined problem

Here, $\nu$ is the unit outer normal vector to $\partial\Omega$, and $f, g$ are prescribed functions of the form

where $\alpha, \beta\in\mathbb{R}$, $g_0$ is a function defined on the unit sphere $\mathbb{S}$ and the coefficient $n+\alpha$ is only for the normalization. In order to clarify the sense in which (1.1) is to be understood, we shall additionally assume $\alpha > -n$, so that

is a solution to (1.1) in the sense of distributions when $\Omega$ is the unit ball $\mathbb{B}$, $g_0 = 1$ and $\beta\in\mathbb{R}$. Equation (1.1) arises as the Euler-Lagrange equation of a variational problem for a weighted torsional rigidity (see Section 2). The particular case $\alpha = \beta = 0$ has been extensively studied in the literature and is sometimes referred to as Serrin's overdetermined problem.

In the case where $f, g$ are positive constants (i.e., $\alpha = \beta = 0$ and $g_0 > 0$ a constant), it is well-known that $\Omega$ must be a ball if (1.1) has a solution $u\in C^2(\overline{\Omega})$. This rigidity result was proved in a seminal paper ^[34] by Serrin, with an innovative argument called the method of moving planes based on Alexandrov's reflection principle and a refined boundary point lemma for corners, which in fact applies to nonlinear equations. Weinberger ^[37] provided an alternative proof based on the observation that, if $u$ satisfies (1.1), the Cauchy-Schwarz deficit

becomes identically zero and thus $u$ is a quadratic function as (1.3) (see ^[22,23,31] for refined arguments). Another interesting proof was introduced by Brandolini, Nitsch, Salani and Trombetti ^[7] using an integral quantity related to Newton's inequalities involving elementary symmetric functions of the eigenvalues of the Hessian matrix $D^2 u$.

There have been numerous studies on the stability of the spherical configuration of $\Omega$ when $f = n$ (i.e., $\alpha = 0$) and $g$ is slightly perturbed from a constant. Here we recall a few relevant results (without mentioning technical assumptions) from a methodological point of view, but not in chronological order. In order to describe the results in a unified manner, let $\Omega_\rho$ be the bounded star-shaped domain enclosed by

for $\rho\in C^{2+\gamma}(\mathbb{S})$ with $-1 < \rho(\xi) < \infty$ and $0 < \gamma < 1$, and let $u_\rho$ denote a unique solution to the Dirichlet problem consisting in the first two equations in (1.1) for $\Omega = \Omega_\rho$. Aftalion, Busca and Reichel ^[3] initiated the stability analysis of (1.1) by developing a quantitative version of the method of moving planes and proved that, up to translation,

holds if the quantity $\|\partial_\nu u_\rho+1\|_{C^1(\partial\Omega_\rho)}$ is sufficiently small. This inequality shows that the deviation $\rho$ of domain $\Omega_\rho$ from the unit ball $\Omega_0 = \mathbb{B}$ can be controlled by that of the Neumann data $g$ from the constant $c = 1$. This method was further developed by Ciraolo, Magnanini and Vespri ^[13] using a quantitative Harnack's inequality, and the logarithmic estimate (1.5) was sharpened to a power-type estimate. In fact, these two results apply to general nonlinear equations. For the particular case $f = n$, in a series of papers ^{[22,23,24,25]}, Magnanini and Poggesi improved (1.5) by establishing an integral identity that relates $d(u)$ to the deviation $\partial_\nu u+1$ and estimating both sides of the identity. The resulting estimate is

where $\tau_2 = 1$, $\tau_3 = 1-\varepsilon$ for any $\varepsilon > 0$, and $\tau_n = 4/(n+1)$ for $n\geq 4$ (see ^[25] for a sharper estimate in the case $n = 3$). In particular, this estimate is optimal for $n = 2$, and almost optimal for $n = 3$, as one can confirm by choosing $\Omega_\rho$ as ellipsoids that linear estimates (i.e., $\tau_n = 1$) are sharpest. Optimal linear estimates for any spatial dimensions $n\geq2$ have been established either when the norm of the left hand side is weakened or when the norm of the right hand side is strengthened. Indeed, Feldman ^[16] proved

by refining an argument of Brandolini, Nitsch, Salani and Trombetti ^[8], in which a power-type stability estimate was obtained for the first time by exploiting their own proof of the symmetry in ^[7]. Another optimal estimate obtained by Gilsbach and the present author ^[18] states that

This estimate is a consequence of the detailed linear analysis of (1.1) for $\alpha = \beta = 0$ based on a new implicit function theorem for triplets of Banach spaces, and it also provides the existence and local uniqueness of $\Omega_\rho$ for small perturbations of $g$ from $c = 1$.

For non-constant $f, g$, the overdetermined problem (1.1) or other variants were treated in ^{[1,2,4,5,9,19,20,28,29,30,35]}. Bianchini, Henrot and Salani ^[6] studied the existence, uniqueness and geometric properties of $\Omega$ for (1.1) with $\alpha = 0$, $\beta > 0$ and $\beta\neq 1$ by a variational method and the maximum principle. In particular, for $\alpha = 0$ and $\beta > 1$, they proved the stability estimate

or equivalently

The restriction $\beta > 1$ hinges on the availability of the comparison principle for domains $\Omega$ with different values of $g$, and indeed (1.7) was proved by a comparison of $\Omega_\rho$ with radial domains. Note that a priori estimates for $\alpha = \beta = 0$ such as (1.6) cannot yield the corresponding estimates for general $\alpha, \beta$ by a direct use of the triangle inequality.

Our purpose in this paper is to prove an optimal quantitative stability estimate of the radial configuration of $\Omega$ for non-radial perturbations of $g$ in any spatial dimensions $n\geq 2$ by linearization approach as in ^[18]. Since this approach relies only on the non-degeneracy of the linearized problem, we can treat general $\alpha, \beta$ unless

In fact, our linear analysis suggests that a symmetry-breaking bifurcation should occur at these exceptional values of $\alpha, \beta$. We refer to ^{[10,11,12,14,15,21,26,27,33,36]} for the linearization approach to bifurcation phenomena in overdetermined problems. We also emphasize that our approach yields the existence and local uniqueness of $\Omega$ for given perturbations $g_0$ even if $\beta\leq0$ for which the variational method in general fails (see ^[6]).

To state our main result, for $k\in\mathbb{N}$ and $0 < \gamma < 1$, we set $h^{k+\gamma}(\overline{\Omega})$, called the little Hölder space, to be a closed proper subspace of $C^{k+\gamma}(\overline{\Omega})$ defined as the closure of $C^\infty(\overline{\Omega})$ in $C^{k+\gamma}(\overline{\Omega})$, and similarly we define $h^{k+\gamma}(\Gamma)$ for a hypersurface $\Gamma$. The little Hölder space $h^{k+\gamma}(\mathbb{S})$ is suitable for our linearization approach, since the set of spherical harmonics spans a dense subspace of $h^{k+\gamma}(\mathbb{S})$.

Theorem 1.1. Let $\alpha > -n$, $\beta\in\mathbb{R}$ and $0 < \gamma < 1$ satisfy

Then, there are $\delta, \varepsilon > 0$ such that, for any $g_0\in h^{2+\gamma}(\mathbb{S})$ with $\|g_0-1\|_{h^{2+\gamma}(\mathbb{S})} < \delta$, there exists a unique $\rho\in h^{3+\gamma}(\mathbb{S})$ with $\|\rho\|_{h^{3+\gamma}(\mathbb{S})} < \varepsilon$ such that $(1.1)$ is solvable in $\Omega = \Omega_{\rho}$ with $f, g$ defined by $(1.2)$. Moreover, $\rho = \rho(g_0)$ satisfies the following:

(i) If $g_0\to 1$ in $h^{2+\gamma}(\mathbb{S})$, then $\rho\to 0$ in $h^{3+\gamma}(\mathbb{S})$.

(ii) There is a constant $C > 0$ such that

holds for any $g_0$ with $\|g_0-1\|_{h^{2+\gamma}(\mathbb{S})} < \delta$.

In the case where $\alpha-\beta+1 < 0$, the uniqueness in fact holds among all bounded domains $\Omega$ having $C^1$-boundary with $0\in\Omega$.

Remark 1.2. Theorem 1.1 contains, as a special case, the radial symmetry of $\Omega$ for $g_0 = 1$. In fact, the global rigidity/symmetry of $\Omega$ for $\alpha-\beta+1 < 0$ has its counterpart for the endpoint case $\alpha-\beta+1 = 0$ (see Proposition 2.2). Moreover, we prove a global uniqueness result in this special case $\alpha-\beta+1 = 0$, where the solvability of (1.1) is invariant under rescaling of $\Omega$ (see Proposition 5.2). $

Remark 1.3. The stability estimate $(1.8)$ still holds for $h^{k+\gamma}(\mathbb{S})$ with arbitrary $k\geq 2$ in both sides. This can be easily verified, as all the succeeding arguments equally proceed with $h^{k-1+\gamma}, h^{k+\gamma}, h^{k+1+\gamma}$ instead of $h^{1+\gamma}, h^{2+\gamma}, h^{3+\gamma}$.

The structure of the present paper is as follows. In Section 2, we shall discuss the radial symmetry of $\Omega$ in the case $g_0 = 1$ with various different techniques. In Section 3, we introduce a functional analytic formulation of (1.1) and derive the linearized problem. A detailed linear analysis is carried out by spherical harmonics. In Section 4, we derive the stability estimate (1.8) as well as the existence and local uniqueness of $\Omega$ by an implicit function theorem in ^[18]. Lastly, in Section 5, we study the global uniqueness of $\Omega$ when $\alpha-\beta+1\leq 0$.

2.   Radial symmetry of domains

This section concerns the radial symmetry of $\Omega$ when it admits a solution $u$ to (1.1) for radial data $f, g$. Although some of the symmetry results presented in this section are well-known or easily deduced from existing methods, we briefly discuss them so as to compare the well-known arguments with ours.

Let us begin with a variational structure of (1.1). Indeed, (1.1) is derived as the Euler-Lagrange equation of the minimization problem of the generalized torsion functional

among all sets $\Omega$ of equal weighted volume

In the case where $g_0$ is a positive constant and

we can show that $J(\Omega)$ is minimized when $\Omega$ is a ball centered at the origin by a rearrangement inequality as in Pólya ^[32]. Indeed, Sobolev's inequality implies that $J(\Omega)$ is attained by a nonnegative function $u_\Omega\in H_0^1(\Omega)$ for $\alpha > -(n+2)/2$ and thus

If we denote by $\Omega^\ast$ the ball centered at the origin having the same volume as $\Omega$, and by $u_\Omega^\ast$ the symmetric decreasing rearrangement of $u_\Omega$, we see that

Thus, choosing a larger ball $B\supset \Omega^\ast$ with $V(B) = V(\Omega)$, we have

with equality only if $\Omega$ is a ball. We emphasize that this symmetry result only holds for the minimizer $\Omega$, but not for every critical point $\Omega$ that admits a solution $u$ to (1.1).

The method of moving planes can be used to deduce the radial symmetry of any bounded domain $\Omega$ having $C^2$-boundary in which (1.1) has a solution $u$ with

Indeed, the method is based on the comparison between the solution $u$ and its reflection $\tilde{u}(x): = u(x')$ in the hyperplane $x_1 = \lambda$ in a maximal cap

where $\lambda\geq 0$ is chosen to be the smallest number so that the reflected caps

are all contained in $\Omega$. Since $f$ is non-increasing and $g$ is non-decreasing in the radial direction if (2.1) holds, the difference $u-\tilde{u}$ is subharmonic in $\Omega_\lambda$, and Hopf's boundary lemma or its refined version by Serrin ^[34] derives a contradiction at a boundary point $x\in\partial\Omega_\lambda$ unless $\lambda = 0$ or $u$ is symmetric with respect to the hyperplane $x_1 = \lambda$. The same argument with $x_1 = \lambda$ moved from the opposite side, i.e., from $\lambda = -\infty$ toward $\lambda = 0$, deduces the symmetry of $\Omega$ with respect to $x_1$. Hence, choosing the moving plane in every direction, we can obtain the radial symmetry of $\Omega$.

The aforementioned arguments by Weinberger ^[37] and Brandolini, Nitsch, Salani and Trombetti ^[7] using integral quantities and algebraic inequalities apparently work only for $\alpha = \beta = 0$.

Our approach here is based on the existence of a spherical foliation of $\mathbb{R}^n\setminus\{0\}$ consisting of the boundaries of parametrized solutions $\Omega(t)$ (see ^[29,30], where a similar argument was used for a different overdetermined problem). This argument only relies on the structure of spherical solutions and is irrelevant to the monotonicity of $f, g$; and thus our result applies even to the case $\alpha > 0$ or $\beta < 0$. Moreover, the result holds under a minimal regularity assumption on $\partial\Omega$. However, we point out that the result does not fully cover the case (2.1).

Proposition 2.1. Let $\alpha > -n$, $\beta\in\mathbb{R}$ satisfy

and let $\Omega$ be a bounded domain having $C^1$-boundary and $0\in\Omega$. If $(1.1)$ has a solution $u\in C^1(\overline{\Omega}\setminus\{0\})\cap C^2(\Omega\setminus\{0\})$ for $f, g$ defined by $(1.2)$ with $g_0 = 1$, then $\Omega$ must be the unit ball $\mathbb{B}$.

Proof. For $0 < t < \infty$, let us consider the parametrized overdetermined problem

It is easy to check that $\Omega(t) = \mathbb{B}_{t}$, the ball centered at the origin with radius $t$, has a solution $u = u_t$ to (2.2) given by

Now let us suppose that there is a bounded domain $\Omega$ admitting a solution $u\in C^2(\overline{\Omega}\setminus\{0\})$ to (1.1) with $g_0 = 1$ and $0\in\Omega$. We choose the largest number $t_\ast > 0$ and the smallest number $t^\ast > 0$ such that

We will show by contradiction that $t_\ast\geq 1$ and $t^\ast\leq 1$; and thus $\Omega = \mathbb{B}$. Let us suppose $t_\ast < 1$ and take a point $x_0\in\partial\Omega\cap\partial\mathbb{B}_{t_\ast}$. Then, $w: = u-u_{t_\ast}$ satisfies

Hence we arrive at a contradiction as

Similarly, $t^\ast > 1$ leads to a contradiction by considering $w: = u-u_{t^\ast}$ in $\Omega$.

We also obtain a symmetry result in the endpoint case $\alpha-\beta+1 = 0$. In this particular case, $\mathbb{B}_t$ for arbitrary radius $t > 0$ allows the solvability of (1.1) for $g_0 = 1$. Indeed, $u_t$ solves (2.2) in $\Omega(t) = \mathbb{B}_t$ with $t^{\alpha-\beta+1} = 1$ for any $t > 0$.

Proposition 2.2. Let $\alpha > -n$, $\beta\in\mathbb{R}$ satisfy

and let $\Omega$ be a bounded domain having $C^1$-boundary and $0\in\Omega$. If $(1.1)$ has a solution $u\in C^1(\overline{\Omega}\setminus\{0\})\cap C^2(\Omega\setminus\{0\})$ for $f, g$ defined by $(1.2)$ with $g_0 = 1$, then $\Omega$ must be a ball centered at the origin.

Proof. The proof proceeds similarly as before, except that in the inclusion (2.3) we will only prove that $t_\ast = t^\ast$. If this is not true, then $\mathbb{B}_{t_\ast}\subsetneq\Omega$ and $w: = u-u_{t_\ast}$ satisfies (2.4) and $w > 0$ in $\mathbb{B}_{t_\ast}$ by the strong maximum principle. Hence by Hopf's lemma (used in $\mathbb{B}_{t_\ast}$) we arrive at a contradiction as

Thus $B_{t_\ast} = \Omega$ as desired.

In the proofs above, the crucial step for the radial symmetry of $\Omega$ is the construction of a foliated family of domains $\Omega(t)$. This technique will be used in Section 5 to prove the uniqueness of $\Omega$ with a different foliation.

3.   Linearized problem

In this section, we will first derive a functional analytic formulation of the problem of finding $\Omega$ for a prescribed $g$ such that (1.1) is solvable, as presented in ^[18]. For this purpose, throughout this section we fix $\alpha > -n$, $\beta\in\mathbb{R}$ and $0 < \gamma < 1$, and set

for $\delta > 0$ and $k\in\mathbb{N}$. If $\delta > 0$ is sufficiently small, for $\rho\in\mathcal{U}_\delta^{2+\gamma}$, we may define the domain $\Omega_\rho$ by (1.4) and a diffeomorphism $\theta_\rho\in h^{2+\alpha}(\overline{\mathbb{B}}, \overline{\Omega_\rho})$ by

where $\eta\in C^\infty(\mathbb{R})$ is a cut-off function satisfying $0\leq \eta\leq 1$, $|\eta'|\leq 4$, $\eta(r) = 1$ for $|r|\leq 1/4$, and $\eta(r) = 0$ for $|r|\geq 3/4$. This induces the pullback and pushforward isomorphisms $\theta_\rho^\ast\in{\rm Isom}(h^{k+\gamma}(\overline{\Omega_\rho}), h^{k+\gamma}(\overline{\mathbb{B}}))$, $\theta^\rho_\ast\in{\rm Isom}(h^{k+\gamma}(\overline{\mathbb{B}}), h^{k+\gamma}(\overline{\Omega_\rho}))$ for $0\leq k\leq 2$, as well as the corresponding boundary isomorphisms, defined by

For each $\rho\in\mathcal{U}^{2+\gamma}_\delta$, the Dirichlet problem consisting in the first two equations in (1.1) has a unique solution $u_\rho\in h^{2+\gamma}(\overline{\Omega_\rho})$ by the Schauder theory. Consequently, we can define the mapping $F\in C(\mathcal{U}_\delta^{2+\gamma}\times h^{1+\gamma}(\mathbb{S}), h^{1+\gamma}(\mathbb{S}))$ by

where $\nu_\rho\in h^{1+\gamma}(\partial\Omega_\rho, \mathbb{R}^n)$ is the unit outer normal vector field on $\partial\Omega_\rho$. Thus, for a given $g_0\in h^{1+\gamma}(\mathbb{S})$, our problem reduces to finding a solution $\rho\in \mathcal{U}_\delta^{2+\gamma}$ to

Indeed, for such a $\rho$, $u_\rho$ additionally satisfies the Neumann boundary condition in (1.1). In terms of these notations, the spherical solution $\Omega = \mathbb{B}$ for $g_0 = 1$ corresponds to $F(0, 1) = 0$.

In order to construct a solution $\rho$ for $g_0\neq 1$ by an implicit function theorem, we will differentiate (3.1) with respect to $\rho$. At this point, we encounter a regularity issue, that is, we need to impose the higher regularity assumption $\rho\in h^{3+\gamma}(\mathbb{S})$ for the differentiability of $F$ as stated in the following lemma. Note that $u_\rho\in h^{3+\gamma}(\overline{\Omega_\rho})$ under this assumption. Here, we shall use the notation

by which $\nu_\rho$ and the normal and tangential components $\mu_\rho, \tau_\rho\in h^{2+\gamma}(\partial\Omega_\rho, \mathbb{R}^n)$ of the vector field $\theta^\rho_\ast\nu_0$ on $\partial\Omega_\rho$ are represented by

where we have used $\theta^\rho_\ast\nu_0(x) = \nu_0(\xi) = \xi = x/|x|$ for $x = \xi+\rho(\xi)\xi\in\partial\Omega_\rho$.

Lemma 3.1. For sufficiently small $\delta > 0$, we have

and the following hold:

(i) The Fréchet derivative of $F$ with respect to $\rho$ is given by, for $\tilde{\rho}\in h^{3+\gamma}(\mathbb{S})$,

where $p\in h^{2+\gamma}(\overline{\Omega_\rho})$ is the unique solution to

and $H_{\partial\Omega_\rho}\in h^{1+\gamma}(\partial\Omega_\rho)$ is the mean curvature of $\partial\Omega_\rho$ normalized in such a way that $H_{\partial\mathbb{B}} = n-1$.

(ii) $\partial_\rho F(\rho, g_0)$ has a continuous extension in $\mathcal{L}(h^{2+\gamma}(\mathbb{S}), h^{1+\gamma}(\mathbb{S}))$ and

Proof. We give a sketch of proof and refer to [18,Lemma 2.1] for details. Let us first derive the formula (3.4) in the simplest case where $\rho = 0$. To this end, for $\tilde{\rho}\in h^{3+\alpha}(\mathbb{S})$ and small $\varepsilon\in\mathbb{R}$, we substitute $\rho = \varepsilon \tilde{\rho}$ and the formal expansion $u_\rho = u_0+\varepsilon p+o(\varepsilon)$ into

and use the corresponding equation for $u_0$ to obtain

Thus, letting $\varepsilon\to0$, we see that $p$ satisfies (3.5) for $\rho = 0$. Moreover, for $x\in\mathbb{S}$,

where we used the fact that $\nu_\rho$ and $\partial^2 u_0/\partial \nu_0^2$ can be represented by a tangent vector $\tau$ to $\mathbb{S}$ and the Laplace-Beltrami operator $\Delta_{\mathbb{S}}$ on $\mathbb{S}$ as

This shows (i) for $\rho = 0$ by letting $\varepsilon\to0$. For general $\rho\neq 0$, we use the same argument as above with the reference domain $\Omega_\rho$ instead of $\Omega_0 = \mathbb{B}$. In particular, every occurrence of $\tilde{\rho}$ must now be replaced by $\theta_\ast^\rho\tilde{\rho}/|\nabla N_\rho|$ and the extra term $\varepsilon(\theta_\ast^\rho\tilde{\rho})\partial_{\tau_\rho}\partial_{\nu_\rho}u_\rho$ appears in the expansion of $F(\rho+\varepsilon\tilde{\rho}\nu_0, g)$, since for $x\in\mathbb{S}$

For (ii), we observe that the formula (3.4) with (3.5) still makes sense for $\rho\in\mathcal{U}_\delta^{3+\gamma}$ and $\tilde{\rho}\in h^{2+\gamma}(\mathbb{S})$; and the extension in (ii) is thus defined. Finally, the analyticity of $\partial_\rho F$ follows from that of $\mathcal{U}_{\delta}^{3+\gamma}\ni\rho\mapsto\theta_\rho^\ast H_{\partial\Omega_\rho}\in h^{1+\gamma}(\mathbb{S})$, $\theta_\rho^\ast u_\rho\in h^{3+\gamma}(\overline{\mathbb{B}})$ and $\theta_\rho^\ast p\in h^{2+\gamma}(\overline{\mathbb{B}})$.

Remark 3.2. The required higher regularity $\rho\in \mathcal{U}_\delta^{3+\gamma}$ is adequate in view of the formula in (i), since, if $\rho\in h^{k+\gamma}(\mathbb{S})$, then at most $H_{\partial\Omega_\rho}\in h^{k-2+\gamma}(\partial\Omega_\rho)$, $u_\rho\in h^{k+\gamma}(\overline{\Omega_\rho})$ and $p\in h^{k-1+\gamma}(\overline{\Omega_\rho})$.

As stated in the next lemma, the representation formula (3.4) of the Fréchet derivative of $F$ in Lemma 3.1 yields a characterization of the invertibility of the extended operator $\partial_\rho F(\rho, g_0)$ in terms of the elliptic boundary value problem

Note that $F(\rho, g_0) = 0$ implies that $g = -\partial_{\nu_\rho}u_\rho\in h^{2+\gamma}(\partial\Omega_\rho)$ and $g > 0$ on $\partial\Omega_\rho$, where the latter follows from the maximum principle.

Lemma 3.3. Suppose that $\rho\in\mathcal{U}_\delta^{3+\gamma}$ and $g_0\in h^{2+\gamma}(\mathbb{S})$ satisfy $F(\rho, g_0) = 0$. Then, the inverse

exists if and only if $(3.6)$ has a unique solution $p\in h^{2+\gamma}(\overline{\Omega_\rho})$ for $\varphi\in h^{1+\gamma}(\partial\Omega_\rho)$. Furthermore, the inverse is then given by

Proof. By assumption, $-\partial_{\nu_\rho}u_\rho = g$ on $\partial\Omega_\rho$. Moreover, in view of (1.2) and (3.3),

Hence, the boundary condition in (3.5) becomes

and the remaining condition in (3.5) and (3.4) are

Since $\theta_\rho^\ast, \theta_\ast^\rho$ are isomorphisms and (3.8) yields a one-to-one correspondence between $\tilde{\rho}$ and $p$, the invertibility of $\partial_\rho F(\rho, g_0)\in\mathcal{L}(h^{2+\gamma}(\mathbb{S}), h^{1+\gamma}(\mathbb{S}))$ is equivalent to the unique existence of a solution $p\in h^{2+\gamma}(\overline{\Omega_\rho})$ to (3.6) for any given boundary data $\varphi\in h^{1+\gamma}(\partial\Omega_\rho)$. The formula (3.7) follows from the above equations.

A well-known sufficient condition (see Gilbarg and Trudinger [17,Theorem 6.31]) for the unique solvability of (3.6) is

In particular, for $\rho = 0$ and $g_0 = 1$, this positivity condition is nothing but

In fact, we can classify all the values of $\alpha, \beta$ for which (3.6) is uniquely solvable by virtue of spherical harmonics. To this end, we recall some basic properties of spherical harmonics. Let us denote by $H_l$ the vector space of all homogeneous harmonic polynomials of degree $l\in\mathbb{N}\cup\{0\}$ on $\mathbb{R}^n$. The dimension $d_l^{(n)}$ of $H_l$ is given by $d_0^{(n)} = 1$, $d_1^{(n)} = n$ and

If we regard $H_l$ as a subspace of $L^2(\mathbb{S})$ and choose an orthonormal basis

then it is known that

forms a complete orthonormal system of $L^2(\mathbb{S})$. In particular, $u\in h^{k+\gamma}(\mathbb{S})$ can be expressed by its Fourier series in $L^2(\mathbb{S})$ as

Moreover, if $u\in C^\infty(\mathbb{S})$, the coefficients $\hat{u}_{l, m}$ are rapidly decaying so that the series on the right hand side converges in the norm in $C^{k+\gamma}(\mathbb{S})$. Thus, the linear subspace spanned by $\mathcal{B}$ is dense in $h^{k+\gamma}(\mathbb{S})$. We also note that $h_{l, m}$ satisfies

In other words, the Dirichlet-to-Neumann operator $\mathcal{N}\in\mathcal{L}(h^{k+\gamma}(\mathbb{S}), h^{k-1+\gamma}(\mathbb{S}))$, for $k\geq 2$, defined by

where $v\in h^{k+\gamma}(\overline{\mathbb{B}})$ is the unique solution to the Dirichlet problem

satisfies

Lemma 3.4. Let $\rho = 0$, $g_0 = 1$, $\alpha > -n$ and $\beta\in\mathbb{R}$. The boundary value problem $(3.6)$ has a unique solution $p\in h^{2+\gamma}(\overline{\mathbb{B}})$ for any $\varphi\in h^{1+\gamma}(\mathbb{S})$ if and only if

Proof. The boundary condition in (3.6) can be written as

Hence, (3.6) is uniquely solvable if and only if

is invertible. As remarked earlier, the latter holds if $-1-\alpha+\beta > 0$; and in particular $(I+\mathcal{N})^{-1}\in\mathcal{L}(h^{1+\gamma}(\mathbb{S}), h^{2+\gamma}(\mathbb{S}))$ exists. Thus, by the Fredholm theory (see e.g., [17, Theorem 5.3]) applied to

where $(I+\mathcal{N})^{-1}$ is compact as a mapping from $h^{1+\gamma}(\mathbb{S})$ to itself, the range of (3.10) is closed and the invertibility follows from the surjectivity of (3.10). Now, since

the condition (3.9) implies that the range of (3.10) contains the linear span of $\mathcal{B}$ and hence its closure $h^{1+\gamma}(\mathbb{S})$. On the other hand, if there is an $l\in\mathbb{N}\cup\{0\}$ such that $\alpha-\beta+1 = l$, then obviously (3.10) is not injective from the above computation.

4.   Quantitative stability estimates

Our goal in this section is to derive the existence, uniqueness and regularity of a solution $\rho$ to the nonlinear equation $F(\rho, g_0) = 0$, based on the linear analysis in the previous section. Lemmas 3.3 and 3.4 show that the linearized operator $\partial_\rho F(0, 1)$ has the inverse

as long as (3.9) holds. This indicates that $F(\rho, g_0) = 0$ can be locally solved and the solution map $g_0\mapsto \rho$ is differentiable.

However, a classical perturbation method generally fails for our nonlinear equation defined by (3.1). Indeed, in contrast to $F(\cdot, g_0)\in C^1(\mathcal{U}_\delta^{3+\gamma}, h^{1+\gamma}(\mathbb{S}))$, the inverse (4.1) only recovers a partial regularity that is not sufficient for a successive approximation of the form

to converge, since $\rho_j\in h^{3+\gamma}(\mathbb{S})$ only results in $\rho_{j+1}\in h^{2+\gamma}(\mathbb{S})$. This regularity deficit called the loss of derivatives can be circumvented by the following implicit function theorem introduced by Gilsbach and the author [18,Theorem 4.2]. It requires some additional regularity assumptions on $F$ at the single point $(\rho, g_0) = (0, 1)$, but provides the existence, local uniqueness and differentiability of $g_0\mapsto \rho$.

Proposition 4.1. Let $X_2\subset X_1\subset X_0$, $Z_2\subset Z_1\subset Z_0$ and $Y$ be Banach spaces with inclusions being continuous embeddings, and let $D_j$ be a neighborhood of a point $(x_0, y_0)\in X_2\times Y$ in $X_j\times Y$ with $D_2\subset D_1\subset D_0$ for $j = 0, 1, 2$. Suppose that

satisfies the following conditions:

(a) $F(x_0, y_0) = 0$;

(b) $\partial_xF(x, y)\in\mathcal{L}(X_2, Z_1)\cap\mathcal{L}(X_1, Z_0)$ has an extension with

(c) $F$ is differentiable at $(x_0, y_0)$ as a mapping from $D_j$ to $Z_j$ for $j = 1, 2$;

(d) $\partial_xF(x_0, y_0)^{-1}\in\mathcal{L}(Z_j, X_j)$ exists for $j = 0, 1, 2$.

Then, there are a neighborhood $U_1$ of $x_0\in X_1$, a neighborhood $V$ of $y_0\in Y$ and a mapping $\upsilon: V\to U_1$ satisfying

(i) $F(\upsilon(y), y) = 0$ for all $y\in V$;

(ii) $\upsilon(y_0) = x_0$ and $\upsilon(y)\to x_0$ in $X_1$ as $y\to y_0$ in $Y$;

(iii) $F(x, y) = 0$, $x\in U_1$ and $y\in V$ imply that $x = \upsilon(y)$;

(iv) $\upsilon\in C^1(V, X_0)$ and

Remark 4.2. The continuity in (ii) is not explicitly stated in [18,Theorem 4.2]. But it is clear from the proofs of [18,Theorems 4.1 and 4.2]: choose an arbitrarily small neighborhood $U_1'\subset U_1$ of $x_0$ and then take a small $V'\subset V$ accordingly and use the contraction mapping principle to get $\upsilon(y)\in U_1'$ for $y\in V'$. $

This proposition enables us to handle nonlinear problems having a particular type of loss of derivatives specified in the conditions (b) and (d). The assumption (c) can be regarded as no loss of derivatives occurring at the single point $(x_0, y_0)$.

In order to apply Proposition 4.1 to our problem with $F$ defined by (3.1), we set, for $j = 0, 1, 2$,

As in Lemma 3.1, we have

This implies that $F$ meets the regularity assumption (4.2). Similarly, (a), (b) are easily confirmed with $(x_0, y_0) = (0, 1)$. Now we use Lemma 3.4 and its variant in higher regular spaces to conclude that the non-degeneracy condition (d) is satisfied if (3.9) holds. For the remaining condition (c), we recall that the loss of derivatives is caused by the regularity of the mean curvature $H_{\partial\Omega_\rho}$ (see Remark 3.2) and by several other terms in (3.4). However, at $(\rho, g_0) = (0, 1)$ where $\partial\Omega_0 = \mathbb{S}$, non-smooth terms vanish and we have

and $p$ is as smooth as $\tilde{\rho}$ by (3.5). Thus one can check that $F\in C(D_j, Z_j)$, with the image space having the stronger topology, is still differentiable at $(\rho, g_0) = (0, 1)$ for $j = 1, 2$. As a conclusion, we obtain open sets $U_1\subset h^{3+\gamma}(\mathbb{S})$ and $V\subset h^{2+\gamma}(\mathbb{S})$ with $(0, 1)\in U_1\times V$ and a solution map $\rho:V\to U_1$ such that

(i) $F(\rho(g_0), g_0) = 0$ for all $g_0\in h^{2+\gamma}(\mathbb{S})$;

(ii) $\rho(1) = 0$ and $\rho(g_0)\to 0$ in $h^{3+\gamma}(\mathbb{S})$ as $g_0\to 1$ in $h^{2+\gamma}(\mathbb{S})$;

(iii) $F(\tilde{\rho}, g_0) = 0$, $\tilde{\rho}\in U_1$ and $g_0\in V$ imply that $\tilde{\rho} = \rho(g_0)$.

Moreover, $\rho\in C^1(V, h^{2+\gamma}(\mathbb{S}))$ and hence the linear stability estimate

holds for any $g_0\in V$ in a small neighborhood of $1\in h^{2+\gamma}(\mathbb{S})$. This concludes the proof of Theorem 1.1, except the last assertion on the global uniqueness of $\Omega$.

5.   Global uniqueness by foliations

Our remaining task is to prove that $\Omega_\rho$ constructed in the previous section is the only possible bounded domain having a solution $u$ to (1.1) in the case where

The technique we employ is based on the construction of a foliation of $\mathbb{R}^n\setminus\{0\}$ by the boundaries of particular solutions $\Omega(t)$ to (1.1), as in the proof of Propositions 2.1 and 2.2. However, for non-constant $g_0$, the spherical foliation is no longer suitable and we need to construct a non-spherical one. We rely on an argument used by Bianchini, Henrot and Salani [6,Theorem 3.4], where the uniqueness is proved for $\alpha = 0$ by constructing a non-spherical foliation as the boundaries of the rescaled family

The following proposition is a generalization of [6,Theorem 3.4] to the case of arbitrary $\alpha > -n$, which completes the proof of Theorem 1.1.

Proposition 5.1. Let $\alpha > -n$, $\beta\in\mathbb{R}$ and $g_0\in C(\mathbb{S})$ satisfy

and let $\Omega$ and $\tilde{\Omega}$ be bounded domains having $C^1$-boundaries and $0\in\Omega\cap\tilde{\Omega}$. Suppose that $(1.1)$ in $\Omega$ and $\tilde{\Omega}$ respectively admit solutions

with $f, g$ defined by $(1.2)$. Then $\Omega = \tilde{\Omega}$.

Proof. The proof is similar to that of Proposition 2.1 with $\Omega(t) = t\Omega$ for $0 < t < \infty$. It is easy to see that the parametrized overdetermined problem

has a solution

As in the proof of Proposition 2.1, we can choose the largest number $t_\ast > 0$ and the smallest number $t^\ast > 0$ such that

and prove that $t_\ast\geq 1$ and $t^\ast\leq 1$ and hence $\tilde{\Omega} = \Omega(1) = \Omega$ by contradiction. Indeed, if $t_\ast < 1$, we take a point $x_0\in\partial\tilde{\Omega}\cap\partial\Omega(t_\ast)$ and observe that $w: = \tilde{u}-u_{t_\ast}$ satisfies

Hence we arrive at a contradiction as

Similarly, $t^\ast > 1$ leads to a contradiction by considering $w: = \tilde{u}-u_{t^\ast}$ in $\tilde{\Omega}$.

In the endpoint case $\alpha-\beta+1 = 0$, the existence of $\Omega$ for $g_0\neq 1$ is not guaranteed due to the degeneracy of $\partial_\rho F(0, 1)$. However, we can prove the uniqueness of $\Omega$ up to dilation as in Proposition 2.2.

We say that $\Omega$ satisfies the interior sphere condition if for any point $x\in\partial\Omega$ there is a ball $B\subset\Omega$ such that $x\in \partial B$. In particular, this condition is fulfilled if $\partial\Omega$ is of class $C^2$. The interior sphere condition allows us to use Hopf's lemma.

Proposition 5.2. Let $\alpha > -n$, $\beta\in\mathbb{R}$ and $g_0\in C(\mathbb{S})$ satisfy

and let $\Omega$ and $\tilde{\Omega}$ be bounded domains having $C^1$-boundaries and $0\in\Omega\cap\tilde{\Omega}$, and moreover suppose that $\Omega$ or $\tilde{\Omega}$ satisfies the interior sphere condition. If $(1.1)$ in $\Omega$ and $\tilde{\Omega}$ respectively admit solutions

with $f, g$ defined by $(1.2)$, then $\tilde{\Omega} = t\Omega$ for some $t > 0$.

Proof. We may suppose that $\Omega$ satisfies the interior sphere condition. The same argument as in the proof of Proposition 5.1 yields $\Omega(t_\ast)\subset\tilde{\Omega}$ with a common boundary point $x_0\in\partial\tilde{\Omega}\cap\partial\Omega(t_\ast)$. If $\Omega(t_\ast)\neq\tilde{\Omega}$, then $w: = \tilde{u}-u_{t_\ast}$ satisfies (5.1) and $w > 0$ in $\Omega(t_\ast)$ by the strong maximum principle. Hence Hopf's lemma derives a contradiction as

Thus $\Omega(t_\ast) = \tilde{\Omega}$ as desired.

Acknowledgments

The author was supported in part by the Grant-in-Aid for Scientific Research (C) 20K03673, Japan Society for the Promotion of Science.

Conflict of interest

The author declares no conflict of interest.