# SCARP: 3D Shape Completion in ARbitrary Poses for Improved Grasping

Bipasha Sen∗1, Aditya Agarwal∗1, Gaurav Singh∗1, Brojeshwar B.2, Srinath
Sridhar3, and Madhava Krishna1 ∗Equal Contributions1Robotics Research Center,
IIIT-Hyderabad2TCS Research, India3Brown University

###### Abstract

Recovering full 3D shapes from partial observations is a challenging task that
has been extensively addressed in the computer vision community. Many deep
learning methods tackle this problem by training 3D shape generation networks
to learn a prior over the full 3D shapes. In this training regime, the methods
expect the inputs to be in a fixed canonical form, without which they fail to
learn a valid prior over the 3D shapes. We propose SCARP, a model that
performs Shape Completion in ARbitrary Poses. Given a partial pointcloud of an
object, SCARP learns a disentangled feature representation of pose and shape
by relying on rotationally equivariant pose features and geometric shape
features trained using a multi-tasking objective. Unlike existing methods that
depend on an external canonicalization, SCARP performs canonicalization, pose
estimation, and shape completion in a single network, improving the
performance by 45% over the existing baselines. In this work, we use SCARP for
improving grasp proposals on tabletop objects. By completing partial tabletop
objects directly in their observed poses, SCARP enables a SOTA grasp proposal
network improve their proposals by 71.2% on partial shapes. Project page:
https://bipashasen.github.io/scarp

##  I INTRODUCTION

Given a partial observation of an object, 3D shape completion aims to recover
the full 3D shape of the object. This has been widely addressed in computer
vision [1, 2, 3, 4, 5, 6, 7, 8] and has many diverse downstream applications
in robotics including visual servoing [9], manipulation [10, 11, 12, 13],
visual inspection [14], autonomous driving [15, 16, 17].

Many existing methods tackle shape completion by incorporating a training
scheme that learns a prior over the full 3D shapes. This is done by training
an autoencoder [18, 19, 1, 6] or a GAN [20] over many different instances of
full shapes. At inference, this learned prior space is conditionally queried
on the partial observations. These methods however, suffer from a major
limitation: they expect the partial input to be in a fixed canonical frame–a
common frame of reference that is shared between instances in that category
[21, 22]. A particular shape $X$ in two different poses $\\{R_{1},T_{1}\\}$
and $\\{R_{2},T_{2}\\}$ will have very different geometry. As a result, $X$ in
different poses appear as novel instances for these methods inhibiting them
from learning a valid prior over shapes.

Existing datasets like ShapeNet [23] have shapes that are manually aligned to
a canonical frame, but real shape observations (e.g., depth maps) do not
contain this information. A naive approach to tackling this challenge is to
_canonicalize_ , i.e., map a 3D (full or partial) shape to a category-level
canonical frame with [21] or without supervision [22, 24, 25]. A multi-stage
pipeline can be built involving the sequential steps of (1) canonicalization,
(2) shape completion, and (3) de-canonicalization (bringing the object back in
the original pose). In such a pipeline however, the performance of a shape
completion network directly depends on the output quality of the
canonicalization module. This can lead to errors propagating between these
modules leading to a sub-optimal completion.

Figure 1: SCARP performs Shape Completion in ARbitrary Poses (top-left). We
show an example of a real scene made of two tabletop objects. (a) The captured
scene is partial leaving out a portion of the objects. (b) This results in
grasp poses (in green) on the partial point cloud that directly collide (in
red) with the actual object leading to a collision between the object and the
Franka Panda’s gripper (grey). (c) SCARP improves grasp proposal by accurately
completing the partial pointcloud in the observed pose. (d) This enables the
grasp proposal network to propose grasp poses (shown in green) on the
completed pointcloud that do not collide with the actual object.

We propose SCARP, a method that performs Shape Completion in ARbitrary Poses.
Unlike existing methods that have to directly learn a prior over all possible
poses and shapes, we first disentangle the pose from the shape of a partial
pointcloud. We build a multi-task objective that: (1) generates a disentangled
feature representation of pose and shape by canonicalizing an object to a
fixed frame of reference, (2) estimates the exact pose of the object, and (3)
completes the shape of the object using the disentangled representation. This
multi-task objective allows our network to jointly understand the pose and
shape of the input. It does so by learning rotationally-equivariant and
translationally-invariant pose features using Tensor Field Networks [26], and
global geometric shape features using PointNet++ [27].

Application: Robotic grasp pose estimation [13, 12, 28, 29] is a challenging
area of research that often expects a faithful reconstruction of the scene in
3D. As shown in Fig. 1 (b), under a partial observation, [13] generates grasp
proposals that directly collide with the actual object in the scene (shown in
red). As a result, the manipulator is likely to collide with the object as it
attempts to grasp the objects using one of these predicted grasp poses. We use
SCARP to complete these partial shapes directly in their observed poses and
estimate grasp proposals on these completed shapes. We show that SCARP reduces
such invalid grasps by $71.2\%$ over predicting grasp poses directly on the
partial observations. To summarize, our contributions are:

  1. 1.

We propose SCARP, a novel architecture to perform shape completion from
partial pointclouds in arbitrary poses. To the best of our knowledge, this is
the first work to do so.

  2. 2.

We show for the first time how a multi-task objective can support: (1)
canonicalization, (2) 6D pose estimation, and (3) shape completion on partial
pointclouds.

  3. 3.

We demonstrate that SCARP outperforms the existing shape completion baselines
(with pre-canonicalization) by $45\%$ and improves grasp pose estimation by
reducing invalid grasp poses by $71\%$.

##  II Related Work

Partial Pointcloud Completion has been extensively addressed over the years
[7, 19, 2, 18, 30, 1, 31, 32, 6, 33, 20, 34]. Early 3D shape completion works
relied on intermediate voxel representation for representing the 3D objects
[35, 36, 37]. More recent works adopted an architecture similar to PointNet
[38] that aggregated point-wise embeddings to achieve a global pointcloud
feature. PCN [19] adopted such an architecture and pioneered learning-based
methods for pointcloud completion. More recent works [20, 1] adopted a
pointcloud generator network to learn a prior over the full pointcloud shapes.
[20] applied a differential degradation layer on the output of a pointcloud
generator [39] to obtain partial pointclouds from full outputs. Later, it
relied on conditional GAN inversion [40] to obtain the latent code of a
partial pointcloud’s full output. [1] used a similar approach by training a
quantized auto-encoder and adopting a second transformer based network to
learn the shape prior over the quantized space. Other lines of work aim to
generate high resolution pointclouds [7, 6], preserving high-details in the
output. All of these existing methods expect the partial pointcloud to be in a
fixed canonical frame. As a result, they depend on external canonicalization
methods [22, 24, 21] to bring the partial inputs to their fixed frame. We
explicitly disentangle the pointcloud’s pose and shape and train our network
to learn a prior over the pose and the shape separately. This allows us to
perform partial pointcloud completion directly in any arbitrary pose. [41, 2]
aim to complete pointclouds in arbitrary poses but need multiple observations
from multiple views for completing the pointcloud. Capturing a particular
object from multiple different views is not possible always, especially in a
moving frame of reference like moving cars. [42] performs pointcloud
completion with limited disturbance from the canonical pose and uses temporal
information for completion. Unlike them, we do not assume any restriction in
the pose and perform pointcloud completion from a single observation.

Figure 2: Overview of our proposed approach: The input to SCARP is a mean-
centered partial pointcloud $\hat{X_{p}}$ in an arbitrary orientation $R$. Our
feature extraction module (b) disentangles the partial pointcloud’s pose and
shape and is trained in a multi-tasking objective (a). In the first task,
SCARP combines Pointnet++ [27] and TFN [26] features to generate a shape
feature that is used by a pointcloud completion network, $G$, to generate
$X^{\prime}$. In the second task, the TFN pose feature is used to generate an
equivariant frame $\\{R^{\prime},T^{\prime}\\}$. Our loss functions enable the
overall network to learn a prior over the shape while understanding the pose
of the partial input.

Pointcloud Canonicalization canonicalizes an input pointcloud of a given
category to the category’s fixed canonical frame. This canonical frame is
implicitly defined by the network [22, 24]. Such a method can be clubbed with
the existing shape completion networks in a multi-stage pipeline: (1)
canonicalizing the partial pointcloud to a fixed implicit frame, (2) shape
completion in the fixed implicit frame, and (3) de-canonicalization. However,
the performance of the shape completion network is tightly coupled with the
canonicalization method. Morever, there is an additional training overhead in
such a method, where the existing dataset has to be first converted to the
implicit canonical frame before the shape completion model is trained. Our
model does not need an external canonicalization. We compare our model with
the multi-stage pipeline by clubbing [22] with the existing shape completion
networks [7, 20] and show substantial performance gain on shape completion
metrics.

##  III Background

Pointnet++ [27] is a 3D hierarchical network, $P$, that computes the geometric
shape feature, $p$, for a given pointcloud $X\in\mathbb{R}^{3\times K}$, where
$K$ is the user-defined number of points in the pointcloud. It computes a
global geometric feature by hierarchically aggregating local geometric
features of the points in a pointcloud. For a more detailed understanding,
please refer [38, 27]. These features are not rotation-aware. That is, for a
point cloud $X$ in any new orientation $\hat{R}(X)$, where $\hat{R}\in SO(3)$,
a unique $p$ is generated.

Tensor Field Networks [43, 26] is a 3D architecture, $\mathcal{X}$, that
computes rotationally equivariant and translationally invariant feature
matrix, $\hat{F}$. For a given pointcloud $X\in\mathbb{R}^{3\times K}$ and an
integer (aka type) $\ell\in\mathbb{N}$, a TFN produces global (type $\ell$)
feature vectors of dimension $2\ell+1$ stacked in a matrix
$\hat{F}^{\ell}\in\mathbb{R}^{(2\ell+1)\times N}$, where $N$ is user-defined
number of channel. $\hat{F}_{:,j}(X)$ satisfies the equivariance property
$\hat{F}_{:,j}(RX)=D(R)\hat{F}_{:,j}(X)$, where $D:SO(3)\rightarrow
SO(2\ell+1)$ is a Wigner matrix (of type $\ell$) [44, 45, 46]. Additional
details can be found in [26, 47, 48, 49, 43].

tree-GAN [39] is a GAN-based pointcloud generation network that adopts a graph
convolutional generator, $G$, and a discrimintor similar to r-GAN [50]. $G$
hierarchically upsamples a gaussian noise, $z\in\mathbb{R}^{E}$, sampled from
a standard normal distribution to a pointcloud
$X^{\prime}\in\mathbb{R}^{3\times K}$.

Density Aware Chamfer’s Distance [51] Given two pointclouds $X$ and
$Y\in\mathbb{R}^{3\times K}$ without known point-wise correspondences,
Chamfer’s Distance (CD)111https://pdal.io/en/stable/apps/chamfer.html can be
used to compute the distance $d_{CD}(X,Y)$ by considering the nearest neighbor
of the points $\\{x\in X,y\in Y\\}$ in $\\{Y,X\\}$ as their correspondence.
The distance is then given as:

$\sum_{x\in X}\min_{y\in Y}{||x-y||}_{2}^{2}+\sum_{y\in Y}\min_{x\in
X}{||x-y||}_{2}^{2}$ (1)

However, CD does not guarantee uniformity in the output density. Density-Aware
Chamfer’s Distance (DCD) [51] overcomes this issue by modifying CD as:

$\frac{1}{2}\left(\frac{1}{|X|}\sum_{x\in
X}\left(1-\frac{e^{\mathcal{Z}_{x}}}{n_{\hat{y}}}\right)+\frac{1}{|Y|}\sum_{y\in
Y}\left(1-\frac{e^{\mathcal{Z}_{y}}}{n_{\hat{x}}}\right)\right)$ (2)

Please refer to [51] for the exact notations.

##  IV SCARP: Shape Completion in ARbitrary Poses

Given a partial object pointcloud $\hat{X_{p}}$ at an unknown pose
$\\{R,T\\}$, we want to estimate this pose and the corresponding full object
pointcloud $\hat{X}$ in the same pose.

This is a challenging task as for a neural network, a pointcloud $X$ in two
different poses $\\{R_{1},T_{1}\\}$ and $\\{R_{2},T_{2}\\}$ are two completely
different pointclouds. Thus, we adopt a multi-tasking objective that
disentangles the pose and the shape of the input partial pointcloud
$\hat{X_{p}}$. The shape component allows us to understand that $\hat{X_{p}}$
is a partial observation of $X$ which is $\hat{X}$ in its canonical form. The
pose component is then used to estimate the pose transform $\\{R,T\\}$ between
$\hat{X}$ and $X$.

###  IV-A Multi-tasking Pipeline for disentangling Shape and Pose

Let $X_{p}$ and $X$ be a partial and its corresponding full pointcloud in a
fixed canonical frame. Then $\hat{X_{p}}$ and $\hat{X}$ are $X_{p}$ and $X$ in
an unknown arbitrary pose $\\{R,T\\}$ such that $\hat{X_{p}}=R(X_{p})+T$ and
$\hat{X}=R(X)+T$. The input to our network is $\hat{X_{p}}$ which is mean
centered at the origin. Our aim is to predict $\\{R,T\\}$ and the full
pointcloud $\hat{X}$ which is posed as:

$\\{R,T,\hat{X}\\}=\Phi(\hat{X_{p}})$ (3)

where $\Phi$ denotes our proposed network, SCARP.

Our multi-tasking objective is formulated to (1) complete the partial
pointcloud in a fixed canonical frame given by $X$ and (2) estimate the pose
transformation from the canonical frame to the original pose $\\{R,T\\}$. In
this pipeline, the two components (1) pose and (2) shape are predicted
separately using two different output heads as shown in Fig. 2.

####  IV-A1 Feature Extraction

To estimate the input’s shape, we compute global geometric shape features,
$p\in\mathbb{R}^{E}$, using Pointnet++ [27] as explained in Sec. III. To
estimate the pose of the input, we adapt TFN [26] as explained in Sec. III.
Our TFN computes a global equivariant feature, $F\in\mathbb{R}^{N\times E}$ by
max pooling over the types $\\{\ell\\}_{\ell=0}^{\ell=\ell_{max}}$, where $E$
is the dimension of the equivariant embeddings, $N$ and $\ell_{max}$ are user-
defined.

The input to our shape completion network is a non-linear combination of $p$
and a global invariant embedding, $F_{\mathcal{X}}\in\mathbb{R}^{E}$, computed
by max pooling $F$ over the channel dimension, $N$. Additionally, $F$ is used
to estimate an equivariant frame of reference,
$\\{R^{\prime}\in\mathbb{R}^{3\times 3},T^{\prime}\in\mathbb{R}^{3}\\}$ that
transforms the invariant embeddings to $X$’s original pose.

####  IV-A2 Task I: Shape Completion

Completing the shape of a partial input at any arbitrary orientation is
difficult. Therefore, we aim to first complete the shape at a fixed canonical
frame. To learn this canonical frame, the model needs to build an
understanding of the full shape of the partial input. To achieve this, we
train our model to predict a full canonicalized pointcloud $X^{\prime}$
directly from $\hat{X_{p}}$. Shape completion enables our model to learn a
prior over the global shape of a category (a typical chair would have four
legs and a backrest) enabling our network to directly canonicalize the partial
inputs accurately.

We adopt $G$, as explained in Sec. III, as our shape completion network where
(1) the input to $G$ is a semantically meaningful embedding generated from a
partial input $\hat{X_{p}}$ and (2) is trained using a distance loss against
the full pointcloud $X$ to learn a relationship between the partial input
$\hat{X_{p}}$ and the predicted full canonical pointcloud $X^{\prime}$. As
shown in Fig. 2 (right), the input to $G$ is a globally invariant feature
vector $f\in\mathbb{R}^{E}$ computed by combining $p=P(\hat{X_{p}})$ and
$F_{\mathcal{X}}=\mathcal{X}(\hat{X_{p}})$ non-linearly using a neural network
$\phi_{S}$ given as:

$X^{\prime}=G(f)\quad\textrm{and}\quad
f=\phi_{S}(\mathcal{X}(\hat{X_{p}})\oplus P(\hat{X_{p}})))$ (4)

####  IV-A3 Task II: Pose Estimation

Once $G$ predicts the full pointcloud $X^{\prime}$ in a canonical pose, it is
important to estimate the correct rotation $SO(3)$ matrix
$R\in\mathbb{R}^{3\times 3}$ and translation $T\in\mathbb{R}^{3}$ to register
$X^{\prime}$ back on $\hat{X_{p}}$. We predict $R^{\prime}$ and $T^{\prime}$
on the second head of our model using the rotationally equivariant TFN
features $F$ given as:

$R^{\prime}=\phi_{R}(F)\quad\textrm{and}\quad T^{\prime}=\phi_{T}(F)$ (5)

where $\phi_{R}$ and $\phi_{T}$ are multi-layered perceptrons.

###  IV-B Loss Functions for Multitask Training

####  IV-B1 Shape Completion in a fixed Canonical Frame

In the first task, we estimate the completed pointcloud in a fixed canonical
frame given by $X^{\prime}$. We use DCD (see Sec. III) to minimize the
distance between the predicted pointcloud $X^{\prime}$ and the ground truth
canonical pointcloud $X$ given by:

$\mathcal{L}_{\mathit{shape}}=d_{DCD}(X^{\prime},X)$ (6)

####  IV-B2 Estimating the pose of the object

Figure 3: Qualitative comparison of shape completion in arbitrary poses on
SCARP and the existing multi-stage baselines: Canonicalization using ConDor,
Shape Completion, and De-canonicalization. Pointr[7] is a SOTA pointcloud
completion network that generates high-resolution completed pointclouds. Shape
Inversion (SInv.) [20] is based on tree-GAN [39] that shares our generator
$G$.

Tabletop Off-Table gray!25 Bowl Bottle Can Mug Basket Plane Car Chair
Watercraft Average CD$\downarrow$ ConDor$+$SInv. 82.7 27.4 45.4 41.5 85.3 34.2
14.7 59.4 39.9 47.8 ConDor$+$Pointr 30.8 20.9 29.9 14.2 40.9 22.1 6.4 19.8 8.5
21.5 SCARP (Ours) 21.8 7.9 11.8 12.1 34.2 6.9 5.6 19.1 7.1 14.0 gray!25 MMD-
EMD$\downarrow$ Bowl Bottle Can Mug Basket Plane Car Chair Watercraft Average
ConDor$+$SInv. 27.3 17.2 20.1 19.9 29.2 19.6 11.3 22.2 18.9 20.6
ConDor$+$Pointr 21.6 13.6 14.8 12.6 18.8 14.4 8.1 13.5 9.1 14.1 SCARP (Ours)
9.6 6.3 8.8 8.4 10.6 5.0 5.6 8.4 6.0 7.6

TABLE I: Quantitative comparison of shape completion in arbitrary poses for
tabletop and off-tabletop objects. Most tabletop objects are symmetrical
whereas off-table objects have more variations in structure. Chamfer’s
Distance (CD) and Earth Movers Distance-Maximum Mean Discrepancy (MMD-EMD) are
explained in Sec. V and are scaled by $10^{3}$ and $10^{2}$.

To estimate the pose given by $\\{R,T\\}$, we use rotationally equivariant
pose features $F$ and pass it through $\\{\phi_{R},\phi_{T}\\}$. We constrain
this prediction against the canonical frame. To do so, we rotate the canonical
output $X^{\prime}$ to obtain $R^{\prime}(X^{\prime})$ and compare it against
the rotated ground truth $\hat{X}$. At this point however, the pointwise
correspondences between $X$ and $X^{\prime}$ are lost. Thus, a hard distance
loss such as Euclidean distance cannot be directly used. To tackle this, we
minimize permutation invariant CD objective as explained in Sec. III between
$X$ and $X^{\prime}$. However, CD only minimizes the distance between the
nearest neighbors of the points in the pointcloud. This results in local
minimas where the loss is minimal even when the actual correspondences are
far. As a result, the predicted pointcloud is often flipped about one of the
axes. To tackle this issue, we rotate the canonical ground truth $X$ using the
predicted $R^{\prime}$ and compare against $\hat{X}$ using $L2$ loss. The
overall loss is:

$\mathcal{L}_{\mathit{rot}}=\delta
d_{CD}(\hat{X},R^{\prime}(X^{\prime}))+\gamma{||\hat{X},R^{\prime}(X)||}_{2}$
(7)

$R^{\prime}(X^{\prime})$ is computed by detaching the forward computation
graph at the output of $G$. The gradients from the loss does not backpropogate
through $G$ at the first head.

For symmetrical objects such as bowls and glasses, multiple $R^{\prime}$
predictions can be correct. A hard $L2$ loss penalizes the network for correct
predictions even for correct $R^{\prime}$ if the correspondences do not
exactly match. Thus, for symmetrical objects, we keep $\delta\sim 1.0$ and
$\gamma\sim 0.0$ and for non-symmetrical objects we keep $\delta\sim 1.0$ and
$\gamma\sim 1.0$.

The input to our network is a mean-centered partial pointcloud $\hat{X_{p}}$.
At this point, we train our network to regress to $\hat{X_{p}}$’s centroid in
the full pointcloud $X$ given by $T^{\prime}$. We directly supervise
$T^{\prime}$ against the ground truth $T$ given as:

$\mathcal{L}_{\mathit{trans}}={||T^{\prime}-T||}_{2}$ (8)

The final output is obtained by rotating and translating our predicted
pointcloud $X^{\prime}$ by $R^{\prime}$ and $T^{\prime}$ respectively as:

$X_{o}=R^{\prime}(X^{\prime})+T^{\prime}$ (9)

Orthonormality Loss: The rotation $R^{\prime}$ predicted by our network is a
$3\times 3$ matrix in the $SO(3)$ space. However, the matrix predicted by Eqn.
5 is not guaranteed to be a valid $SO(3)$ matrix. We therefore, enforce
orthonormality on $R^{\prime}$ by minimizing its difference to its closest
orthonormal matrix. To do so, we compute the SVD decomposition of $R=U\Sigma
V^{T}$ and enforce unit eigenvalues as:

$\mathcal{L}_{\mathit{orth}}=||UV^{T}-R||_{2}$ (10)

####  IV-B3 Combined Loss

We train our network end-to-end by combining all the losses as:

$\mathcal{L}=\mathcal{L}_{\mathit{shape}}+\mathcal{L}_{\mathit{rot}}+\mathcal{L}_{\mathit{trans}}+\mathcal{L}_{\mathit{orth}}$
(11)

##  V Experiments

In this section, we evaluate SCARP on two tasks: (T1) Shape completion in
arbitrary poses and (T2) Improving grasp proposals by completing partial
pointclouds.

Baselines: As we are the first to perform the task T1, we modify the existing
shape completion networks by developing a multi-stage pipeline: (1) We use
ConDor[22] to first canonicalize the input partial pointclouds to a fixed
canonical frame defined implicitly by ConDor. (2) We train and test the
existing shape completion methods on ConDor’s canonical frame. (3) Bring the
completed pointcloud to the original orientation using a pose transform
predicted by ConDor. We compare against (1) ConDor$+$Pointr[7], a SOTA
pointcloud completion network that generates high-resolution completed
pointclouds and (2) ConDor$+$Shape Inversion (SInv.) [20] based on tree-GAN
[39] that shares our generator $G$.

Metrics:  We use Chamfer’s Distance (CD) as explained in Sec. III to compute
the distance between the ground truth pointcloud $\hat{X}$ and the predicted
pointcloud given as $R^{\prime}(X^{\prime})+T^{\prime}$ to evaluate the match
in shape.

Earth Movers Distance-Maximum Mean Discrepancy (MMD-EMD) [39, 50] is used to
evaluate for uniformity in the prediction by conducting bijective matching of
points between two pointclouds. As we only want to measure the output’s
uniformity, we compute this metric between the canonical ground truth $X$ and
the canonical prediction $X^{\prime}$.

We evaluate SCARP by measuring its impact in an important downstream task:
grasp pose estimation. In this, we measure the a) the number of grasp
proposals made on the partial object that collide with the actual object on
the table (shown in Fig. 1 and 4) denoted by $C$ and b) number of invalid
grasps that do not result in a valid grasping denoted by $I$. We then compute
the Grasping Error (GE) as:

$\frac{1}{\mathcal{D}}\sum_{i=1}^{\mathcal{D}}\frac{C+I}{\mathcal{N}}$ (12)

where $\mathcal{N}$ is the number of top grasp proposals and $\mathcal{D}$ is
the total number of pointcloud instances. In our case, $\mathcal{N}=30$.

Dataset: Our dataset is a subset of [23] derived from [22] and [19] made of 5
tabletop (Bowl, Bottle, Can, Mug, Basket) and 4 non-tabletop (Plane, Car,
Chair, Watercraft) categories. We evaluate $GE$ only on the tabletop objects.

Results: As shown in Table II, SCARP outperforms the existing multi-stage
baselines on all the categories on an average by $45\%$. The existing shape
completion methods rely on the output of an external canonicalization model
that suffer from their own inconsistencies as reported in their paper [22].
This results in an error propagation as the input to the shape completion
networks are not always in the exact canonical forms. The errors in the input
map to a larger error in the output of the networks. This is followed by an
error in the transform from the canonical form to the original pose. The
resulting output of the multi-stage pipeline suffer from high inconsitensies
and sub-optimal outputs. Unlike these networks, our model is trained jointly
on both tasks (canonicalization and shape-completion) using a multi-tasking
objective. As we show in the ablations, this objective plays a crucial role in
achieving a disentangled representation of shape and pose. Qualitative results
are shown in Fig. 3 that vividly show the closeness of SCARP’s output to the
ground truth when compared with others.

Figure 4: (left): Grasp proposals made by a SOTA grasp proposal network, [13],
on partial observations lead to collisions with the actual object. Partial is
a partial observation and Ground truth denotes the actual object. The
proposals are made on Partial (shown in green) but collide with the actual
object (shown in red). (right): We use SCARP to complete the partial
observations. Grasp proposals made on the completed objects align well with
the actual object on the table reducing such collisions by a large margin.

Improvement in Grasp Proposals: Generating grasp proposals for partial
pointclouds is a challenging task as a network may mistake a missing portion
of an object as a potential area to grasp (see Fig. 1 and Fig. 4). We apply
SCARP to complete these partial observations directly in the observed poses
and predict grasp poses on these completed pointclouds using a SOTA grasp
generation network Contact-Graspnet [13]. To evaluate the grasp proposals, we
compute GE on (1) partial observations, (2) completed observations by SCARP,
and (3) actual objects (ground truth). Actual objects are full pointclouds
with no missing portion. As we show in Table III, SCARP shows a relative
improvement of $71.2\%$ and an absolute improvement of $48.27\%$ over the
grasp proposals on the partial pointclouds. Moreover, there is only an
absolute degradation of $4.19\%$ vis-a-vis the ground truth. The ground truth
error in Table III is the datum error in the grasp proposals output by
Contact-Graspnet. Qualitative results are shown in Fig. 4. Green and red
proposals denote valid and colliding grasp proposals respectively. As can be
seen, the grasps proposed on partial observations collide with the actual
object (ground truth), whereas, the grasp proposals made on the completed
object by SCARP are valid.

w/o SC w/o $F_{\mathcal{X}}$ w/o $p$ Plane Ours 3.8 6.9 6.9 Modified 112.34
135.3 111.9 Bowl Ours 14.9 21.8 21.8 Modified 123.7 185.3 156.4 Mug Ours 10.79
12.1 12.1 Modified 47.6 45.5 44.3

TABLE II: Ablation Study: We show the affect of removing different components
of our network. In w/o SC, we train SCARP as an auto-encoding network to
verify if the model can still learns to canonicalize the input pointcloud. In
w/o $F_{\mathcal{X}}$ and w/o $p$, we remove TFN and pointnet features,
respectively, and evaluate the quality of shape completion. Each component
plays a crucial role in achieving SCARP as is evident by the drop in metrics.
Metrics are scaled by $10^{3}$.

Partial SCARP Ground Truth Bowl 62.18 21.14 16.86 Bottle 46.5 7.35 6.32 Can
81.33 22.33 16.0 Mug 71.33 25.0 23.5 Basket 77.33 21.5 13.66 Average 67.73
19.46 15.27

TABLE III: Quantitative metrics on GE (explained in Sec. V): % of grasp
proposals that are invalid or collide with the actual object on the table when
the proposals are made on (1) Partial Observations, (2) Shape Completed
objects by SCARP, and (3) Actual Objects on the table (Ground Truth). SCARP
reduces invalid and colliding grasp proposals by $71.2\%$ when only partial
observations are available by accurately completing the object in the observed
pose.

Ablation: SCARP is trained on a multi-tasking objective to achieve: (1)
canonicalization, (2) 6D pose estimation, and (3) shape completion. We
evaluate the contribution of the different components in our network in
achieving these tasks.

(A) Canonicalization without Shape Completion: Canonicalization involves
mapping an input $X$ to its category’s fixed canonical frame [21, 22, 25].
Learning a canonical frame for a partial input is challenging as a network may
struggle to understand the overall structure of the partial shape. In our
model, the structure of a category is correctly learned using the shape
completion task. Thus, we analyze if SCARP can canonicalize the partial inputs
without performing shape completion. To evaluate this, we modify SCARP by
training to auto-encode the partial input $\hat{X}_{p}$ while simultaneously
estimating its pose $\\{R,T\\}$. That is, our generator $G$ generates $X_{p}$
which is $\hat{X_{p}}$ in its canonical form and uses $R^{\prime}$ to rotate
$X_{p}$ back to $\hat{X_{p}}$. To measure the performance, we compute the CD
between $G^{\prime}s$ canonical output and the canonical ground truth. In case
of SCARP, this is given as $d_{CD}(X^{\prime},X)$, and in case of ablation,
this is given as $d_{CD}(X_{p}^{\prime},X_{p})$. As shown in Table. II (w/o
SC), on average $d_{CD}$ on SCARP is 9.83 whereas when the shape completion
aspect is removed, the average distance is 94.54. This indicates that the
network does not learn anything meaningful if the task of shape completion is
removed from the formulation.

(B) Shape Completion without pose and shape features: As shown in Fig. 2, $G$
expects a disentangled feature embedding that is a non-linear combination of
the pose $F_{\mathcal{X}}$ and shape embeddings $p$. We remove these features
one by one and observe their impact on shape completion. We measure the
performance of shape completion as $d_{CD}(\hat{X},R^{\prime}(X^{\prime}))$.
As shown in Table II (w/o $F_{\mathcal{X}}$ and w/o $p$), SCARP fails to
converge without either of the features. In both cases, the model fails to
learn a correct transformation between the canonical and the original pose.
Without pointnet (w/o $p$), the model collapses to a few different shapes
across the instances missing out on per-instance details. Without TFN (w/o
$F_{\mathcal{X}}$), completion in the canonical frame is more accurate but TFN
fails to estimate the correct pose transform. In summary, as the shape
completion in the canonical form suffers, the pose transform is also
inaccurate thus indicating the importance of the multi-task objective.

##  VI Conclusion

Existing shape completion works assume the partial inputs to be in a fixed
canonical frame. This is difficult to achieve in a robotics setting where the
objects are observed in arbitrary poses thus needing pre-canonicalization.
This leads to an error propagation resulting in a sub-optimal shape
completion. We propose SCARP, a novel architecture that performs Shape
Completion in ARbitrary Poses. SCARP is trained using a multi-task objective
to perform (1) canonicalization, (2) 6D pose estimation, and (3) shape
completion. SCARP outperforms the existing multi-stage baselines by $45\%$ and
showcases its potential in improving grasp proposals on tabletop objects,
reducing colliding grasps by more than $70\%$. SCARP has a huge potential in
many more robotics applications like collision avoidance in trajectory
planning or differential simulators for model-based RL planners.

Acknowledgement: We would like to thank Karthik Desingh, who is an Assistant
Professor at the University of Minnesota, and Adrien Poulenard, who is a
Postdoctoral Fellow at the Stanford University, for their valuable feedback.

## References

  * [1] X. Yan, L. Lin, N. J. Mitra, D. Lischinski, D. Cohen-Or, and H. Huang, “Shapeformer: Transformer-based shape completion via sparse representation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 6239–6249.
  * [2] L. Pan, T. Wu, Z. Cai, Z. Liu, X. Yu, Y. Rao, J. Lu, J. Zhou, M. Xu, X. Luo _et al._ , “Multi-view partial (mvp) point cloud challenge 2021 on completion and registration: Methods and results,” _arXiv preprint arXiv:2112.12053_ , 2021.
  * [3] P. Mittal, Y.-C. Cheng, M. Singh, and S. Tulsiani, “Autosdf: Shape priors for 3d completion, reconstruction and generation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 306–315.
  * [4] J. J. Park, P. Florence, J. Straub, R. Newcombe, and S. Lovegrove, “Deepsdf: Learning continuous signed distance functions for shape representation,” in _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2019, pp. 165–174.
  * [5] V. Sitzmann, E. Chan, R. Tucker, N. Snavely, and G. Wetzstein, “Metasdf: Meta-learning signed distance functions,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 10 136–10 147, 2020.
  * [6] H. Zhou, Y. Cao, W. Chu, J. Zhu, T. Lu, Y. Tai, and C. Wang, “Seedformer: Patch seeds based point cloud completion with upsample transformer,” 2022.
  * [7] X. Yu, Y. Rao, Z. Wang, Z. Liu, J. Lu, and J. Zhou, “Pointr: Diverse point cloud completion with geometry-aware transformers,” in _Proceedings of the IEEE/CVF international conference on computer vision_ , 2021, pp. 12 498–12 507.
  * [8] A. Avetisyan, A. Dai, and M. Nießner, “End-to-end cad model retrieval and 9dof alignment in 3d scans,” in _Proceedings of the IEEE/CVF International Conference on computer vision_ , 2019, pp. 2551–2560.
  * [9] G. Kumar, H. Pandya, A. Gaud, and K. M. Krishna, “Pose induction for visual servoing to a novel object instance.” IEEE Press, 2017, p. 2953–2959. [Online]. Available: https://doi.org/10.1109/IROS.2017.8206130
  * [10] C. C. Kemp, A. Edsinger, and E. Torres-Jara, “Challenges for robot manipulation in human environments [grand challenges of robotics],” _IEEE Robotics & Automation Magazine_, vol. 14, no. 1, pp. 20–29, 2007\.
  * [11] A. Agarwal, B. Sen, S. Narayanan, V. R. Mandadi, B. Bhowmick, and K. M. Krishna, “Approaches and challenges in robotic perception for table-top rearrangement and planning,” 2022. [Online]. Available: https://arxiv.org/abs/2205.04090
  * [12] A. Mousavian, C. Eppner, and D. Fox, “6-dof graspnet: Variational grasp generation for object manipulation,” in _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 2019, pp. 2901–2910.
  * [13] M. Sundermeyer, A. Mousavian, R. Triebel, and D. Fox, “Contact-graspnet: Efficient 6-dof grasp generation in cluttered scenes,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 13 438–13 444.
  * [14] H. Pandya, A. Gaud, G. Kumar, and M. Krishna, “Instance invariant visual servoing framework for part-aware autonomous vehicle inspection using mavs: Pandya et al.” _Journal of Field Robotics_ , vol. 36, 01 2019.
  * [15] M. Campbell, M. Egerstedt, J. How, and R. Murray, “Autonomous driving in urban environments: Approaches, lessons and challenges,” _Philosophical transactions. Series A, Mathematical, physical, and engineering sciences_ , vol. 368, pp. 4649–72, 10 2010.
  * [16] W. Zeng, S. Wang, R. Liao, Y. Chen, B. Yang, and R. Urtasun, “Dsdnet: Deep structured self-driving network,” in _European conference on computer vision_. Springer, 2020, pp. 156–172.
  * [17] S. Thrun, M. Montemerlo, H. Dahlkamp, D. Stavens, A. Aron, J. Diebel, P. Fong, J. Gale, M. Halpenny, G. Hoffmann, K. Lau, C. Oakley, M. Palatucci, V. Pratt, P. Stang, S. Strohband, C. Dupont, L.-E. Jendrossek, C. Koelen, and P. Mahoney, “Stanley: The robot that won the darpa grand challenge.” _J. Field Robotics_ , vol. 23, pp. 661–692, 01 2006.
  * [18] X. Wang, M. H. Ang Jr, and G. H. Lee, “Cascaded refinement network for point cloud completion,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 790–799.
  * [19] W. Yuan, T. Khot, D. Held, C. Mertz, and M. Hebert, “Pcn: Point completion network,” in _2018 International Conference on 3D Vision (3DV)_. IEEE, 2018, pp. 728–737.
  * [20] J. Zhang, X. Chen, Z. Cai, L. Pan, H. Zhao, S. Yi, C. K. Yeo, B. Dai, and C. C. Loy, “Unsupervised 3d shape completion through gan inversion,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 1768–1777.
  * [21] H. Wang, S. Sridhar, J. Huang, J. Valentin, S. Song, and L. J. Guibas, “Normalized object coordinate space for category-level 6d object pose and size estimation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2019, pp. 2642–2651.
  * [22] R. Sajnani, A. Poulenard, J. Jain, R. Dua, L. J. Guibas, and S. Sridhar, “Condor: Self-supervised canonicalization of 3d pose for partial shapes,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 16 969–16 979.
  * [23] A. X. Chang, T. Funkhouser, L. Guibas, P. Hanrahan, Q. Huang, Z. Li, S. Savarese, M. Savva, S. Song, H. Su _et al._ , “Shapenet: An information-rich 3d model repository,” _arXiv preprint arXiv:1512.03012_ , 2015.
  * [24] W. Sun, A. Tagliasacchi, B. Deng, S. Sabour, S. Yazdani, G. E. Hinton, and K. M. Yi, “Canonical capsules: Self-supervised capsules in canonical pose,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 24 993–25 005, 2021.
  * [25] R. Spezialetti, F. Stella, M. Marcon, L. Silva, S. Salti, and L. Di Stefano, “Learning to orient surfaces by self-supervised spherical cnns,” _Advances in Neural Information Processing Systems_ , vol. 33, pp. 5381–5392, 2020.
  * [26] A. Poulenard and L. J. Guibas, “A functional approach to rotation equivariant non-linearities for tensor field networks,” in _2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2021, pp. 13 169–13 178.
  * [27] C. R. Qi, L. Yi, H. Su, and L. J. Guibas, “Pointnet++: Deep hierarchical feature learning on point sets in a metric space,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [28] J. Mahler, J. Liang, S. Niyaz, M. Laskey, R. Doan, X. Liu, J. A. Ojea, and K. Goldberg, “Dex-net 2.0: Deep learning to plan robust grasps with synthetic point clouds and analytic grasp metrics,” _Robotics: Science and Systems_ , 2017.
  * [29] A. Murali, W. Liu, K. Marino, S. Chernova, and A. Gupta, “Same object, different grasps: Data and semantic knowledge for task-oriented grasping,” _Conference on Robot Learning_ , 2020.
  * [30] H. Xie, H. Yao, S. Zhou, J. Mao, S. Zhang, and W. Sun, “Grnet: Gridding residual network for dense point cloud completion,” in _European Conference on Computer Vision_. Springer, 2020, pp. 365–381.
  * [31] X. Wang, M. H. Ang, and G. H. Lee, “Point cloud completion by learning shape priors,” in _2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2020, pp. 10 719–10 726.
  * [32] Y. Wang, D. J. Tan, N. Navab, and F. Tombari, “Softpoolnet: Shape descriptor for point cloud completion and classification,” in _European Conference on Computer Vision_. Springer, 2020, pp. 70–85.
  * [33] X. Wang, M. H. Ang, and G. Lee, “Cascaded refinement network for point cloud completion with self-supervision,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2021.
  * [34] S. Song, F. Yu, A. Zeng, A. X. Chang, M. Savva, and T. Funkhouser, “Semantic scene completion from a single depth image,” _arXiv preprint arXiv:1611.08974_ , 2016.
  * [35] D. Stutz and A. Geiger, “Learning 3d shape completion from laser scan data with weak supervision,” in _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 1955–1964.
  * [36] X. Han, Z. Li, H. Huang, E. Kalogerakis, and Y. Yu, “High-resolution shape completion using deep neural networks for global structure and local geometry inference,” in _Proceedings of the IEEE international conference on computer vision_ , 2017, pp. 85–93.
  * [37] A. Dai, C. R. Qi, and M. Nießner, “Shape completion using 3d-encoder-predictor cnns and shape synthesis,” in _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 6545–6554.
  * [38] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, “Pointnet: Deep learning on point sets for 3d classification and segmentation,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 652–660.
  * [39] D. W. Shu, S. W. Park, and J. Kwon, “3d point cloud generative adversarial network based on tree structured graph convolutions,” in _Proceedings of the IEEE/CVF international conference on computer vision_ , 2019, pp. 3859–3868.
  * [40] W. Xia, Y. Zhang, Y. Yang, J.-H. Xue, B. Zhou, and M.-H. Yang, “Gan inversion: A survey,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2022.
  * [41] J. Gu, W.-C. Ma, S. Manivasagam, W. Zeng, Z. Wang, Y. Xiong, H. Su, and R. Urtasun, “Weakly-supervised 3d shape completion in the wild,” in _European Conference on Computer Vision_. Springer, 2020, pp. 283–299.
  * [42] J. Shi, L. Xu, P. Li, X. Chen, and S. Shen, “Temporal point cloud completion with pose disturbance,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 4165–4172, 2022.
  * [43] N. Thomas, T. Smidt, S. Kearnes, L. Yang, L. Li, K. Kohlhoff, and P. Riley, “Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds,” _arXiv preprint arXiv:1802.08219_ , 2018.
  * [44] L. Lang and M. Weiler, “A wigner-eckart theorem for group equivariant convolution kernels,” in _ICLR_ , 2021. [Online]. Available: https://openreview.net/forum?id=ajOrOhQOsYx
  * [45] G. Chirikjian and A. Kyatkin, “Engineering applications of noncommutative harmonic analysis: With emphasis on rotation and motion groups (1st ed.).” in _CRC Press_ , 2000. [Online]. Available: https://doi.org/10.1201/9781420041767
  * [46] A. W. Knapp, _Representation Theory of Semisimple Groups_. Princeton: Princeton University Press, 2016. [Online]. Available: https://doi.org/10.1515/9781400883974
  * [47] M. Weiler, M. Geiger, M. Welling, W. Boomsma, and T. S. Cohen, “3d steerable cnns: Learning rotationally equivariant features in volumetric data,” _Advances in Neural Information Processing Systems_ , vol. 31, 2018.
  * [48] J. Shotton, A. Fitzgibbon, M. Cook, T. Sharp, M. Finocchio, R. Moore, A. Kipman, and A. Blake, “Real-time human pose recognition in parts from single depth images,” in _CVPR 2011_ , 2011, pp. 1297–1304.
  * [49] B. Anderson, T. S. Hy, and R. Kondor, “Cormorant: Covariant molecular neural networks,” _Advances in neural information processing systems_ , vol. 32, 2019.
  * [50] P. Achlioptas, O. Diamanti, I. Mitliagkas, and L. Guibas, “Learning representations and generative models for 3d point clouds,” in _International conference on machine learning_. PMLR, 2018, pp. 40–49.
  * [51] T. Wu, L. Pan, J. Zhang, T. Wang, Z. Liu, and D. Lin, “Density-aware chamfer distance as a comprehensive metric for point cloud completion,” _Advances in Neural Information Processing Systems_ , 2021.


# On a Serrin-type overdetermined problem

Celentano A., Nitsch C., Trombetti C

###### Abstract

In this paper, we prove a Serrin-type result for an elliptic system of
equations, overdetermined with both Dirichlet and a generalized Neumann
conditions. With this tool, we characterize the critical shapes under volume
constraint of some domain functionals.

MSC 2020: 35B06; 35J57; 35N25.

Keywords:  Overdetermined problem; radial symmetry; first eigenvalue;
torsional rigidity; domain derivative.

E-mail address, A. Celentano<EMAIL_ADDRESS>
E-mail address, C. Nitsch<EMAIL_ADDRESS>
E-mail address, C. Trombetti<EMAIL_ADDRESS>
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

##  1 Introduction

In the study of overdetermined elliptic problems, Serrin’s pioneering work
[18] marked a turning point. He proved the radial symmetry property of the
solution to torsion problem, overdetermined with a Neumann boundary condition.
More precisely, main result of [18] states the following:

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a connected, bounded and open set whose
boundary is of class $C^{2}$. If there exists a solution $u\in
C^{2}(\overline{\Omega})$ satisfying

$\begin{cases}-\Delta u=1&\mbox{in }\Omega,\\\ u=0&\mbox{on
}\partial\Omega,\\\ \frac{\partial u}{\partial\nu}=c&\mbox{on
}\partial\Omega,\\\ \end{cases}$ (1)

where $\nu$ is the inward normal to $\partial\Omega$. Then
$\Omega=B_{R}(x_{0})$ is a ball of radius $R$, centered at $x_{0}$ and
$u(x)=\frac{R^{2}-|x-x_{0}|^{2}}{2n}$.

As remarked in [18], Theorem 1.1 still holds if Poisson equation in (1) is
replaced by a non linear elliptic equation, whose structure satisfies some
suitable conditions, provided the solution $u$ has constant sign in $\Omega$.
For instance, Poisson semilinear equation $-\Delta u=f(u)$ can be taken into
account.

The proof relies on two main tools: the moving planes method and a refinement
of the Hopf Boundary Lemma [18, Lemma 1]. The former was initially introduced
by Alexandrov in [1] to prove that the only compact, embedded
$(n-1)$-dimensional $C^{2}$ hypersurfaces in $\mathbb{R}^{n}$ with constant
mean curvature are the spheres. Serrin recovered this technique and adapted it
to the PDEs framework. Since then, many generalization of this method have
been made. We mention just a few of these. In [12], it was extended and
applied to show symmetry and monotonicity properties of positive solutions to
a class of elliptic problems with Dirichlet boundary condition. Afterwards, it
was improved in order to treat more general domains with weaker regularity
assumptions on the solution, see [3, 10]. The moving planes method can be
applied successfully also in the case of $p$-Laplace operator [8, 9]. For a
deep analysis involving also a quantitative approach, we refer to [7].

We point out that many different proofs of Theorem 1.1 and further extensions
are currently available in the literature, see [17] for a survey on this
topic. Weinberger provided an alternative short proof in [19], based on the
use of a suitable auxiliary function, called $P$-function, and Pohožaev
identity. These techniques were extended to handle a general quasilinear
operator in [11]. An other approach, which combines Pohožaev identity together
with Newton inequalities, is shown in [4]. With these tools, authors are also
able to deal with a class of fully non linear equations, namely Hessian
equations. We mention also the work [6], where a deep connection between shape
optimization and overdetermined problems is emphasized. Authors construct a
shape functional minimized by solutions to Serrin’s problem. They infer the
uniqueness of minimizer and then, expressing the optimality condition by means
of the domain derivative, they show that the mean curvature of the boundary of
the minimizer must be constant. Hence, it must be a ball.

In this paper, we deal with an extension of the Serrin’s problem to system of
elliptic equations with Dirichlet datus, overdetermined with an other boundary
condition which involves the normal derivatives of solutions and generalizes
the Neumann one. Constraints of this type arise from shape optimization
context, by imposing the optimality condition computed via domain derivative,
see for instance [5]. The main result of this work is the following:

######  Theorem 1.2.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be a bounded, open, connected
set of class $C^{2}$. For $i=1,...,m$, let $u_{i}\in
C^{2}(\overline{\Omega})$, $u_{i}>0$ in $\Omega$, be solutions to

$\begin{cases}-\Delta u_{i}=f_{i}(u_{i})&\mbox{in }\Omega,\\\ u_{i}=0&\mbox{on
}\partial\Omega,\\\ F\left(\frac{\partial
u_{1}}{\partial\nu},...,\frac{\partial u_{m}}{\partial\nu}\right)=c&\mbox{on
}\partial\Omega,\\\ \end{cases}$ (2)

where $\nu$ is the inner unit normal to $\partial\Omega$ and $c\in\mathbb{R}$.
Let us assume that $f_{i}$ and $F$ satisfy the following assumptions:

  1. (i)

$f_{i}=g_{i}+h_{i}$, with $g_{i}$ locally Lipschitz in
$\operatorname{\mathbb{R}}$ and $h_{i}$ non decreasing,

  2. (ii)

$F:[0,+\infty)^{m}\rightarrow\mathbb{R}$ and $F\in
C^{1}\left((0,+\infty)^{m}\right)$,

  3. (iii)

$\forall\,i\in\\{1,...,m\\}$, $\forall\,(x_{1},...,x_{m})\in[0,+\infty)^{m}$,
$\forall h>0$, it holds

$F(x_{1},...,x_{i},...,x_{m})\leq F(x_{1},...,x_{i}+h,...,x_{m})$

  4. (iv)

$\exists\,j\in\\{1,..,m\\}$ such that $\frac{\partial F}{\partial x_{j}}>0$ in
$(0,+\infty)^{m}$.

Then $\Omega$ is a ball of radius $R$ and the solutions $u_{i}$ are radially
symmetric. Moreover,

$\dfrac{\partial u_{i}}{\partial r}<0\quad\text{for
}\\{0<r<R\\},\,\forall\,i\in\\{1,...,m\\}.$

The proof is based on moving planes method and the arguments used are inspired
by those of Serrin. In the end, we provide an application of the previous
result to the study of critical shapes of some domain functionals involving
the torsional rigidity $T(\Omega)$ and the first eigenvalue
$\lambda_{1}(\Omega)$ of Dirichlet-Laplacian.

######  Corollary 1.3.

Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded, connected set of class
$C^{2,\gamma}$ for $\gamma\in(0,1)$. If $\Omega$ is a critical shape under
volume constraint for the functional
$J(\Omega)=T^{\alpha}(\Omega)\lambda_{1}^{\beta}(\Omega)$, with
$\alpha,\beta\in\mathbb{R}$ not both zero and such that $\alpha\beta\leq 0$,
then it is a ball.

The paper is organized as follows. In Section 2.1, we provide a detailed
exposure of the moving planes method, recalling also some useful results
proved in [12]. In Section 2.2, we give introductory notions and tools on the
torsional rigidity, the first eigenvalue of Laplacian with Dirichlet boundary
condition and domain derivative of shape functionals. Finally, in Section 3 we
prove Theorem 1.2 and Corollary 1.3.

##  2 Notions and preliminaries

###  2.1 The moving planes method

We provide a geometric description of the procedure of moving up planes
perpendicular to a fixed direction, until a critical position is reached. Let
$\Omega\subset\operatorname{\mathbb{R}}^{n}$ be a bounded domain with smooth
boundary. Assigned a unit vector $\gamma$ in $\operatorname{\mathbb{R}}^{n}$
and a real value $\lambda$, we define the hyperplane
$T_{\lambda}=\\{x\in\operatorname{\mathbb{R}}^{n}\,|\,\gamma\cdot
x=\lambda\\}$, that is the boundary of the open halfspace
$H_{\lambda}=\\{x\in\operatorname{\mathbb{R}}^{n}\,|\,\gamma\cdot
x>\lambda\\}$. If $\lambda$ is large enough, then
$T_{\lambda}\cap\overline{\Omega}=\emptyset$. Decreasing $\lambda$, the
hyperplane $T_{\lambda}$ moves continuously toward $\Omega$ along the
direction $-\gamma$ and it begins to intersect $\overline{\Omega}$ for
$\lambda=\lambda_{0}$. If $\lambda<\lambda_{0}$, the open cap
$\Sigma(\lambda)=H_{\lambda}\cap\Omega$ is nonempty. We denote respectively by
$\Sigma^{\prime}(\lambda)$ and $x^{\lambda}$, the reflection of the open cap
$\Sigma(\lambda)$ and of the point $x$ in the hyperplane $T_{\lambda}$. At the
beginning, $\Sigma^{\prime}(\lambda)$ will be in $\Omega$ and as $\lambda$
decreases, the reflected cap $\Sigma^{\prime}(\lambda)$ will remain in
$\Omega$, at least until one of the following two events occurs:

  1. (i)

$\Sigma^{\prime}(\lambda)$ becomes internally tangent to $\partial\Omega$ at
some point $\overline{x}$ not on $T_{\lambda}$,

  2. (ii)

$T_{\lambda}$ reaches a position such that it is orthogonal to
$\partial\Omega$ at some point $\overline{y}$.

We denote by $T_{\lambda_{c}}$ the hyperplane $T_{\lambda}$ when it first
reaches one of these positions. Now, we recall in a special form a result
proved by Gidas, Ni and Nirenberg in [12, Theorem 2.1].

######  Theorem 2.1.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be an open, connected set of
class $C^{2}$. Let $u\in C^{2}(\overline{\Omega})$, $u>0$ in $\Omega$, be a
solution to

$\begin{cases}-\Delta u=f(u)&\mbox{in }\Omega,\\\ u=0&\mbox{on
}\partial\Omega,\\\ \end{cases}$ (3)

with $f=f_{1}+f_{2}$, where $f_{1}$ is locally Lipschitz in
$\operatorname{\mathbb{R}}$ and $f_{2}$ is non decreasing. Then

$\nabla u(x)\cdot\gamma<0,\quad
u(x)<u(x^{\lambda})\quad\forall\,x\in\Sigma(\lambda),\quad\forall\,\lambda\in(\lambda_{0},\lambda_{c}).$
(4)

Thus $\nabla u\cdot\gamma<0$ in $\Sigma(\lambda_{c})$. Furthermore if $\nabla
u\cdot\gamma=0$ at some point on $\Omega\cap T_{\lambda_{c}}$ then necessarily
$u$ and $\Omega$ are symmetric in the hyperplane $T_{\lambda_{c}}$.

######  Remark 2.1.

As specified in [12], Theorem 2.1 is verified for a class of nonlinear
elliptic second order equations. With $\Omega$ as before, let $u\in
C^{2}(\overline{\Omega})$ be a solution to the following equation

${F(x,u,u_{x_{1}},...,u_{x_{n}},u_{x_{1}x_{1}},...,u_{x_{n}x_{n}})=0\quad\text{in
}\Omega,}$ (5)

which is elliptic in $\Omega$ if there exist positive constants $m,M$ such
that

${m|\xi|^{2}\leq\sum_{i,j=1}^{n}F_{u_{ij}}(x)\xi_{i}\xi_{j}\leq
M|\xi|^{2}\quad\forall\,\xi\in\mathbb{R}^{n},\quad\forall\,x\in\Omega.}$

The function $F(x,z,p_{k},r_{i,j})$ satisfies the following conditions:

  1. a)

$F$ is continuous and it has continuous first derivative with respect to
$z,p_{k}$ and $r_{i,j}$ for all values of these arguments, for every
$x\in\Omega$;

  2. b)

the function $g(x)=F(x,0,...,0)$ verifies

${g(x)\geq
0\quad\forall\,x\in\partial\Omega\cap\\{x_{1}>\lambda_{c}\\},\quad\text{or}\quad
g(x)<0\quad\forall x\in\partial\Omega\cap\\{x_{1}>\lambda_{c}\\};}$

  3. c)

for every $\lambda$ such that $\lambda_{c}\leq\lambda<\lambda_{0}$, for
$x\in\Sigma(\lambda)$ and all values of the arguments $z,p_{k},r_{i,j}$ with
$z>0$ e $p_{1}<0$, it results

${F(x^{\lambda},z,-p_{1},p_{2},...,p_{n},r_{1,1},-r_{1,\alpha},...,r_{\beta,\gamma})\geq
F(x,z,p_{1},p_{2},...,p_{n},r_{1,1},r_{i,j}),}$

where $i,j=1,...n$ and $\alpha,\beta,\gamma=2,...n$.

As a consequence of Theorem 2.1, the following symmetry result holds [12,
Theorem 1].

######  Theorem 2.2.

Let $f:\mathbb{R}\to\mathbb{R}$ be such that $f=f_{1}+f_{2}$, where $f_{1}$ is
a locally Lipschitz function and $f_{2}$ is non decreasing. Then, any positive
solution $u\in C^{2}(\overline{B})$ to the problem

$\begin{cases}-\Delta u=f(u)&\text{in }B,\\\ u=0&\text{on }\partial
B,\end{cases}$

where $B$ is a ball of $\mathbb{R}^{n}$ of radius $R$, has to be radial and

$\dfrac{\partial u}{\partial r}<0,\quad\text{for }\,0<r<R.$

In [12], Theorem 2.1 is also extended to some domains with corners, by proving
a refinement of the Hopf Boundary Lemma [12, Lemma S] reported below, which is
a generalization of a result due to Serrin in [18, Lemma 1].

######  Lemma 2.3.

Let $\Omega\subset\operatorname{\mathbb{R}}^{n}$ be an open, connected set
with the origin $O$ on its boundary. Assume that near $O$ the boundary
consists of two trasversally intersecting $C^{2}$ hypersurfaces $\rho=0$ and
$\sigma=0$. Suppose $\rho,\sigma<0$ in $\Omega$. Let $w\in
C^{2}(\overline{\Omega})$, with $w<0$ in $\Omega$, $w(O)=0$, satisfying the
differential inequality

$Lu=\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial
x_{j}}(x)+\sum_{i=1}^{n}b_{i}(x)\frac{\partial u}{\partial
x_{i}}(x)+c(x)u(x)\geq 0\quad\text{in }\Omega,$

where $L$ is a uniformly elliptic operator and its coefficients are uniformly
bounded. Assume

$\sum_{i,j=1}^{n}a_{ij}\rho_{x_{i}}\sigma_{x_{j}}\geq 0\quad\text{at O}.$ (6)

If this is $0$, assume furthermore that $a_{ij}\in C^{2}$ in
$\overline{\Omega}$ near $O$, and that

$\sum_{i,j=1}^{n}D(a_{ij}\rho_{x_{i}}\sigma_{x_{j}})=0\quad\text{at O},$

for any first order derivative $D$ at $O$ tangent to the submanifold
$\\{\rho=0\\}\cap\\{\sigma=0\\}$. Then, for any direction $s$ at $O$ which
enters $\Omega$ trasversally to each hypersurface

$\frac{\partial w}{\partial s}<0\quad\text{at O in case of strict inequality
in \eqref{cond1}},$ $\frac{\partial w}{\partial
s}<0\quad\text{or}\quad\frac{\partial^{2}w}{\partial s^{2}}<0\quad\text{at O
in case of equality in \eqref{cond1}}.$

###  2.2 The domain derivative

Fixed a bounded and open set $\Omega\subset\mathbb{R}^{n}$, the torsional
rigidity of $\Omega$ is

$\displaystyle T(\Omega)=\int_{\Omega}u_{\Omega}\,dx,$

where $u_{\Omega}$ is the torsion function of $\Omega$, that is the unique
solution of the boundary value problem

$\begin{cases}-\Delta u=1&\mbox{in }\Omega,\\\ u\in W^{1,2}_{0}(\Omega).\\\
\end{cases}$

Let $\lambda_{1}(\Omega)$ be the first eigenvalue of the Laplacian with
Dirichlet boundary conditions. It is known that $\lambda_{1}(\Omega)>0$, we
assume it is simple and we indicate by $v_{\Omega}$ the normalized
corresponding eigenfunction, which is the solution to

$\begin{cases}-\Delta v=\lambda_{1}(\Omega)v&\mbox{in }\Omega,\\\ v\in
W^{1,2}_{0}(\Omega),\\\ \end{cases}$

such that $\|v_{\Omega}\|_{L^{2}(\Omega)}=1$.

Next, we recall the first order shape derivative of the two functionals
$T(\Omega),\lambda_{1}(\Omega)$.
Let $\theta\in W^{1,\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ be a Lipschitz
continuous function. For given $t_{0}>0$, we denote by
$\\{\Omega_{t}\\}_{0\leq t<t_{0}}$ a family of perturbations of the domain
$\Omega$ of the form

$\Omega_{t}=(I+t\theta)(\Omega),$

where $I$ is the identity in $\mathbb{R}^{n}$.

######  Remark 2.2.

If $t_{0}$ is small enough, then $\Phi(t,x)=I(x)+t\theta(x)$ is a bi-Lipschitz
homeomorphism for every $0\leq t<t_{0}$. Accordingly, $\Omega_{t}$ is
measurable and open if and only if $\Omega$ is measurable and open.

######  Definition 2.1.

The derivative of a given functional $\mathcal{J}$ at $\Omega$ in the
direction $\theta$ is defined as

$d\mathcal{J}(\Omega,\theta):=\lim_{t\to
0^{+}}\frac{\mathcal{J}(\Omega_{t})-\mathcal{J}(\Omega)}{t},$ (7)

if the limit exists.

For the next two results, we refer to [15, Corollary 5.3.8, Theorem 5.7.1].

######  Theorem 2.4.

Let $\Omega$ be an open, bounded and of class $C^{1}$ set. Then

$dT(\Omega,\theta)=\int_{\partial\Omega}|\nabla u_{\Omega}|^{2}\theta\cdot
n\,d\,\mathcal{H}^{n-1},$

where $n$ denotes the outer unit normal to $\partial\Omega$.

######  Theorem 2.5.

Let $\Omega$ be an open, bounded and of class $C^{2}$ set. If
$\lambda_{1}(\Omega)$ is a simple eigenvalue, then

$d\lambda_{1}(\Omega,\theta)=-\int_{\partial\Omega}|\nabla
v_{\Omega}|^{2}\theta\cdot n\,d\,\mathcal{H}^{n-1}.$

Now, we consider the volume functional $V(\Omega)=|\Omega|$, which gives the
Lebesgue measure of $\Omega$. For a perturbation $\Phi$ of the type above, it
results [2, Section 2.3]

$dV(\Omega,\theta)=\int_{\Omega}\text{div}\,\theta\,dx.$

######  Definition 2.2.

The perturbation $\Phi(t,x)$ is called volume preserving if

$\int_{\Omega}\text{div}\,\theta\,dx=0.$ (8)

######  Definition 2.3.

A set $\Omega$ is said a critical shape under volume constraint for a
functional $\mathcal{J}$ if (7) vanishes for every choice of $\theta$
satisfying (8).

##  3 Proof of the main theorem

###### Proof of Theorem 1.2.

Let fix a direction $\gamma$ and let apply the procedure described before,
until the hyperplane $T_{\lambda}$ reaches a critical position. The goal is to
prove that $\Omega$ is symmetric with respect to the hyperplane
$T_{\lambda_{c}}$. Once this fact is proved then the statement follows, since
for every direction $\gamma$, $\Omega$ would be symmetric with respect to the
hyperplane normal to $\gamma$. Moreover, according to the construction,
$\Omega$ would also be simply connected, then it has to be a ball. In the end,
an application of Theorem 2.2 shows that $u_{i}$ is radially symmetric and non
increasing, for all $i=1,...,m$.
We set $u_{i}^{\lambda_{c}}$ the maps defined in
$\Sigma^{\prime}(\lambda_{c})$ by

$u_{i}^{\lambda_{c}}(x)=u_{i}(x^{\lambda_{c}})\quad
x\in\Sigma^{\prime}(\lambda_{c}),\quad\forall\,i\in\\{1,...,m\\}.$

The functions $u_{i}^{\lambda_{c}}$ satisfy

$\begin{cases}-\Delta u_{i}^{\lambda_{c}}=f_{i}(u_{i}^{\lambda_{c}})&\mbox{in
}\Sigma^{\prime}(\lambda_{c}),\\\ u_{i}=u_{i}^{\lambda_{c}}&\mbox{on
}\partial\Sigma^{\prime}(\lambda_{c})\cap\partial H_{\lambda_{c}},\\\
u_{i}^{\lambda_{c}}=0&\mbox{on
}\partial\Sigma^{\prime}(\lambda_{c})\setminus\partial H_{\lambda_{c}},\\\
F\left(\frac{\partial u_{1}^{\lambda_{c}}}{\partial\nu},...,\frac{\partial
u_{m}^{\lambda_{c}}}{\partial\nu}\right)=c&\mbox{on
}\partial\Sigma^{\prime}(\lambda_{c})\setminus\partial
H_{\lambda_{c}}.\end{cases}$

Since $\Sigma^{\prime}(\lambda_{c})$ is contained in $\Omega$, it’s possible
to consider $u_{i}-u_{i}^{\lambda_{c}}$. As a consequence of Theorem 2.1,
$u_{i}-u_{i}^{\lambda_{c}}\geq 0$ and it solves a linear differential
inequality

$0\geq
h_{i}(u_{i}^{\lambda_{c}})-h_{i}(u_{i})=\Delta(u_{i}-u_{\lambda_{c}})+g_{i}(u_{i})-g_{i}(u_{i}^{\lambda_{c}})=\Delta(u_{i}-u_{i}^{\lambda_{c}})+c_{i}(u_{i}-u_{i}^{\lambda_{c}}),$

where

${c_{i}(x)=\begin{cases}\dfrac{g_{i}(u_{i}(x))-g_{i}(u_{i}^{\lambda_{c}}(x))}{u_{i}(x)-u_{i}^{\lambda_{c}}(x)}&\mbox{if
}u(x)\neq u_{i}^{\lambda_{c}}(x),\\\ 0&\mbox{if
}u(x)=u_{i}^{\lambda_{c}}(x).\end{cases}}$

An application of the strong maximum principle gives either

$u_{i}-u_{i}^{\lambda_{c}}>0\quad\text{or}\quad
u_{i}-u_{i}^{\lambda_{c}}\equiv 0\quad\text{in }\Sigma^{\prime}(\lambda_{c}).$

The latter case would imply that $\Omega$ is symmetric about $T(\lambda_{c})$.
Suppose that case $i)$ occurs, that is $\Sigma^{\prime}(\lambda_{c})$ is
internally tangent to $\partial\Omega$ at a point $\overline{x}$ not belonging
to $T_{\lambda_{c}}$ and assume by contradiction that
$u_{i}-u_{i}^{\lambda_{c}}>0$ in $\Sigma^{\prime}(\lambda_{c})$ for all
$i=1,...,m$. Then Hopf Lemma ensures that

$\frac{\partial(u_{i}-u_{i}^{\lambda_{c}})}{\partial\nu(\overline{x})}>0.$

Hence, we get

$\frac{\partial u_{i}}{\partial\nu(\overline{x})}>\frac{\partial
u_{i}^{\lambda_{c}}}{\partial\nu(\overline{x})}\geq 0.$

Using the hypothesis on the function $F$, we get a contradiction

$c=F\left(\frac{\partial
u_{1}^{\lambda_{c}}}{\partial\nu(\overline{x})},...,\frac{\partial
u_{m}^{\lambda_{c}}}{\partial\nu(\overline{x})}\right)<F\left(\frac{\partial
u_{1}}{\partial\nu(\overline{x})},...,\frac{\partial
u_{m}}{\partial\nu(\overline{x})}\right)=c.$

In the case $ii)$, the proof makes use of Lemma 2.3. The goal is to prove that
$u_{i}-u_{i}^{\lambda_{c}}$ have in $\overline{y}$ a second order zero, for
some $i\in\\{1,...,m\\}$. Now, we fix a coordinate system with the origin at
$\overline{y}$, the $x_{n}$ axis in the direction of the inward normal to
$\partial\Omega$ at $\overline{y}$ and the $x_{1}$ axis in the direction of
$\gamma$, that is normal to $T_{\lambda_{c}}$. In this coordinates system, the
boundary of $\Omega$ is locally described by

$x_{n}=\phi(x_{1},x_{2},...,x_{n-1}),\quad\phi\in C^{2}.$

By construction, the function $\phi$ satisfies the following properties

$\nabla\phi(0)=0,\quad\frac{\partial^{2}\phi}{\partial x_{1}\partial
x_{j}}(0)=0\quad\text{for }j=2,...,n-1.$ (9)

Indeed, $\nu(0)=(-\nabla\phi(0),1)=(0,...,0,1)$ which gives the former in (9).
Moreover, $\frac{\partial\phi}{\partial x_{1}}$ has an extremum point at $0$
with respect to all but the first coordinates directions, since
$\Sigma^{\prime}(\lambda_{c})\subseteq\Omega$. Let us set

$u_{i}^{\lambda_{c}}(x_{1},x_{2},...,x_{n})=u_{i}(-x_{1},x_{2},...,x_{n}).$

In order to prove that all the first and second derivatives of $u_{i}$ and
$u_{i}^{\lambda_{c}}$ coincide at $0$, it is enough to show

$\frac{\partial u_{i}}{\partial
x_{1}}(0)=0,\quad\frac{\partial^{2}u_{i}}{\partial x_{1}\partial
x_{j}}(0)=0\quad\text{for }j=2,...,n.$

Since $u_{i}\in C^{2}$, the boundary condition $u_{i}=0$ on $\partial\Omega$
can be written as

$u_{i}(x_{1},x_{2},...,x_{n-1},\phi)=0\quad\text{for }i=1,...,m.$ (10)

Differentiating (10) with respect to $x_{j}$ for $j=1,...,n-1$, we have

$\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{i}}{\partial
x_{n}}\frac{\partial\phi}{\partial x_{j}}=0.$ (11)

Evaluating (11) at $0$ and for $j=1,...,n-1$, it results $\frac{\partial
u_{i}}{\partial x_{j}}(0)=0$. Differentiating (11) with respect to $x_{k}$ for
$k=1,...,n-1$, we get

$\frac{\partial^{2}u_{i}}{\partial x_{j}\partial
x_{k}}+\frac{\partial^{2}u_{i}}{\partial x_{j}\partial
x_{n}}\frac{\partial\phi}{\partial x_{k}}+\frac{\partial^{2}u_{i}}{\partial
x_{n}\partial x_{k}}\frac{\partial\phi}{\partial
x_{j}}+\frac{\partial^{2}u_{i}}{\partial x_{n}\partial
x_{n}}\frac{\partial\phi}{\partial x_{j}}\frac{\partial\phi}{\partial
x_{k}}+\frac{\partial u_{i}}{\partial x_{n}}\frac{\partial^{2}\phi}{\partial
x_{j}\partial x_{k}}=0.$ (12)

Evaluating (12) at $0$, for $j=1$ and $k=2,...,n-1$, it results
$\frac{\partial^{2}u_{i}}{\partial x_{1}\partial x_{k}}(0)=0$. If it exists
$j$ such that $\frac{\partial u_{j}}{\partial x_{n}}(0)=0$, then the normal
derivative $\frac{\partial u_{j}}{\partial\nu}$ has a minimum in $0$. Since

$\frac{\partial u_{j}}{\partial\nu}(x_{1},x_{2},...,x_{n-1},\phi)=\nabla
u_{j}(x_{1},...,x_{n-1},\phi)\cdot\frac{(-\nabla\phi(x_{1},...,x_{n-1}),1)}{\sqrt{1+|\nabla\phi(x_{1},...,x_{n-1})|^{2}}},$

a straightforward calculation shows that

$\displaystyle\frac{\partial}{\partial x_{1}}\left(\frac{\partial
u_{j}}{\partial\nu}(x_{1},...,x_{n-1},\phi)\right)=\Biggl{(}\,\frac{\partial^{2}u_{j}}{\partial
x_{n}\partial x_{1}}+\frac{\partial^{2}u_{j}}{\partial x_{n}\partial
x_{n}}\frac{\partial\phi}{\partial
x_{1}}-\sum_{k=1}^{n-1}\frac{\partial^{2}u_{j}}{\partial x_{k}\partial
x_{1}}\frac{\partial\phi}{\partial
x_{k}}-\sum_{k=1}^{n-1}\frac{\partial^{2}u_{j}}{\partial x_{k}\partial
x_{n}}\frac{\partial\phi}{\partial x_{1}}\frac{\partial\phi}{\partial x_{k}}+$
(13) $\displaystyle-\sum_{k=1}^{n-1}\frac{\partial u_{j}}{\partial
x_{k}}\frac{\partial^{2}\phi}{\partial x_{k}\partial
x_{1}}\Biggr{)}\frac{1}{\sqrt{1+|\nabla\phi|^{2}}}-\left(\frac{\partial
u_{j}}{\partial x_{n}}-\sum_{k=1}^{n-1}\frac{\partial u_{j}}{\partial
x_{k}}\frac{\partial\phi}{\partial
x_{k}}\right)\sum_{k=1}^{n-1}\frac{\partial\phi}{\partial
x_{k}}\frac{\partial^{2}\phi}{\partial x_{k}\partial
x_{1}}\left(1+|\nabla\phi|^{2}\right)^{-\frac{3}{2}}.$

Evaluating (13) in $0$, it follows $\frac{\partial^{2}u_{j}}{\partial
x_{n}\partial x_{1}}(0)=0$. In conclusion, the first and second derivatives of
$u_{j}$ and $u_{j}^{\lambda_{c}}$ agree at $0$. We can apply Lemma (2.3) to
the function $w=u_{j}^{\lambda_{c}}-u_{j}$ in $\Sigma^{\prime}(\lambda_{c})$
at $0$. This yields

$\frac{\partial w}{\partial
s}(0)<0\quad\text{or}\quad\frac{\partial^{2}w}{\partial s^{2}}(0)<0,$

contradicting the fact that both $u_{j}$ and $u_{j}^{\lambda_{c}}$ have the
same first and second partial derivatives at $\overline{y}$. If
$\frac{\partial u_{i}}{\partial x_{n}}(0)>0$ for $1\leq i\leq m$, in order to
study the values $\frac{\partial^{2}u_{i}}{\partial x_{1}\partial x_{n}}(0)$,
we take into account the second boundary condition in (2), that is

$F\left(\frac{\partial u_{1}}{\partial\nu(x)},...,\frac{\partial
u_{m}}{\partial\nu(x)}\right)=c.$ (14)

Differentiating (14) with respect to $x_{1}$ and evaluating in $0$, we obtain

$\displaystyle\sum_{i=1}^{m}\frac{\partial F}{\partial
x_{i}}\left(\frac{\partial u_{1}}{\partial x_{n}}(0),...,\frac{\partial
u_{m}}{\partial x_{n}}(0)\right)\frac{\partial^{2}u_{i}}{\partial
x_{n}\partial x_{1}}(0)=0.$ (15)

The sign of each of the terms on the left hand side of (15) is known. The
partial derivatives $\frac{\partial u_{i}}{\partial x_{1}}$ restricted to
$T_{\lambda_{c}}\cap\overline{\Omega}$ have a maximum at $0$, because of the
first inequality in (4). From Schwarz’s Theorem, it follows that
$\frac{\partial^{2}u_{i}}{\partial x_{n}\partial x_{1}}(0)\leq 0$. Therefore,
equation (15) is equivalent to the following system of equations

$\frac{\partial F}{\partial x_{i}}\left(\frac{\partial u_{1}}{\partial
x_{n}}(0),...,\frac{\partial u_{m}}{\partial
x_{n}}(0)\right)\frac{\partial^{2}u_{i}}{\partial x_{n}\partial
x_{1}}(0)=0\quad\forall i=1,...,m.$

By hypothesis, there exists an index $j$ such that $\frac{\partial F}{\partial
x_{j}}>0$ in $(0,+\infty)^{m}$. Hence, $\frac{\partial^{2}u_{j}}{\partial
x_{n}\partial x_{1}}(0)=0$ and then the first and second derivatives of
$u_{j}$ and $u_{j}^{\lambda_{c}}$ agree at $0$. We obtain a contradiction
arguing as above. ∎

######  Remark 3.1.

The previous result still holds if functions $u_{i}$ are solutions to
equations of the type described in Remark 2.1, provided the function
$F(x,z,p_{k},r_{i,j})$ is independent of variables $r_{1,\alpha}$ for
$\alpha>1$. This additional condition is necessary in order to verify
hypothesis (6) in Lemma 2.3.

######  Remark 3.2.

The monotonicity assumptions on function $F$ cannot be dropped as the
following examples show.
By removing hypothesis $iii)$, we can set
$F(x,y)=\begin{cases}\frac{x}{y}&\mbox{if }y>0,\\\ 0&\mbox{if }y=0,\\\
\end{cases}$. Let $u_{\Omega}$ be the torsion function of $\Omega$ and
$v=cu_{\Omega}$, for some constant $c$. Then the following system is satisfied

$\begin{cases}-\Delta u_{\Omega}=1&\mbox{in }\Omega,\\\ -\Delta v=c&\mbox{in
}\Omega,\\\ u_{\Omega}=v=0&\mbox{on }\partial\Omega,\\\ F\left(\frac{\partial
v}{\partial\nu},\frac{\partial u_{\Omega}}{\partial\nu}\right)=c&\mbox{on
}\partial\Omega,\\\ \end{cases}$

for every smooth and bounded open set $\Omega\subset\mathbb{R}^{n}$.
Similarly, we can set $u_{\Omega}$ and $v$ as above with $c\not=0$ and
$F(x,y)=x^{2}-c^{2}y^{2}$, then we have

$\begin{cases}-\Delta u_{\Omega}=1&\mbox{in }\Omega,\\\ -\Delta v=c&\mbox{in
}\Omega,\\\ u=v=0&\mbox{on }\partial\Omega,\\\ F\left(\frac{\partial
v}{\partial\nu},\frac{\partial u_{\Omega}}{\partial\nu}\right)=0&\mbox{on
}\partial\Omega,\\\ \end{cases}$

for every smooth and bounded open set $\Omega\subset\mathbb{R}^{n}$.
By removing hypothesis $iv)$, we can consider $F\equiv c$ identically
constant, then the boundary value problem (2) is not overdetermined anymore
and the thesis becomes false in general.

In the end, we prove Corollary 1.3 by means of Theorem 1.2 .

###### Proof of Corollary 1.3.

Standard regularity results [13, Theorem 6.8], [14, Theorem 1.2.12] ensures
that $u_{\Omega},v_{\Omega}\in C^{2,\gamma}(\overline{\Omega})$. Since
$\Omega$ is connected, [16, Theorem 11.5.4] assures that the first eigenvalue
$\lambda_{1}(\Omega)$ is simple and $v_{\Omega}$ can be chosen positive in
$\Omega$. An application of the strong maximum principle shows that
$u_{\Omega}>0$ in $\Omega$. By applying Theorems 2.4, 2.5 for $\theta\in
W^{1,\infty}(\mathbb{R}^{n},\mathbb{R}^{n})$ satisfying (8), we get

$dJ(\Omega,\theta)=\int_{\partial\Omega}\left[\alpha
T(\Omega)^{\alpha-1}|\nabla
u_{\Omega}|^{2}-\beta\lambda_{1}^{\beta-1}(\Omega)|\nabla
v_{\Omega}|^{2}\right]\theta\cdot n\,d\,\mathcal{H}^{n-1}=0,$ (16)

Since $\partial\Omega$ is smooth, condition (8) can be rewritten equivalently
as

$\int_{\Omega}\text{div }\theta\,dx=\int_{\partial\Omega}\theta\cdot
n\,d\,\mathcal{H}^{n-1}.$

Therefore, (16) implies

$\alpha T(\Omega)^{\alpha-1}|\nabla
u_{\Omega}|^{2}-\beta\lambda_{1}^{\beta-1}(\Omega)|\nabla
v_{\Omega}|^{2}=c\quad\text{on }\partial\Omega.$

If we set $F(x,y)=\alpha
T(\Omega)^{\alpha-1}x^{2}-\beta\lambda_{1}^{\beta-1}(\Omega)y^{2}$, then
Theorem 1.2 can be applied and the thesis follows. ∎

### Acknowledgements

The authors were partially supported by Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto
Nazionale di Alta Matematica (INdAM).

## References

  * [1] A. D. Alexandrov.  Uniqueness theorems for surfaces in the large. V.  Amer. Math. Soc. Transl. (2), 21:412–416, 1962.
  * [2] C. Bandle and A. Wagner.  Shape Optimization: Variations of Domains and Applications.  Frontiers in Mathematics. De Gruyter, 2023.
  * [3] H. Berestycki and L. Nirenberg.  On the method of moving planes and the sliding method.  Bol. Soc. Brasil. Mat. (N.S.), 22(1):1–37, 1991.
  * [4] B. Brandolini, C. Nitsch, P. Salani, and C. Trombetti.  Serrin-Type Overdetermined Problems: an Alternative Proof.  Arch. Rational Mech. Anal., 190:267–280, 2008.
  * [5] G. Buttazzo, F. P. Maiale, D. Mazzoleni, G. Tortone, and B. Velichkov.  Regularity of the optimal sets for a class of integral shape functionals.  ArXiv preprint, 2022.
  * [6] M. Choulli and A. Henrot.  Use of the Domain Derivative to Prove Symmetry Results in Partial Differential Equations.  Mathematische Nachrichten, 192:91–103, 1998.
  * [7] G. Ciraolo and A. Roncoroni.  The method of moving planes: a quantitative approach.  Bruno Pini Math. Anal. Semin., 9:41–77, 2018.
  * [8] L. Damascelli and F. Pacella.  Monotonicity and symmetry results for $p$-Laplace equations and applications.  Adv. Differential Equations, 5(7-9):1179–1200, 2000.
  * [9] L. Damascelli and B. Sciunzi.  Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations.  J. Differential Equations, 206(2):483–515, 2004.
  * [10] E.N. Dancer.  Some notes on the method of moving planes.  Bulletin of the Australian Mathematical Society., 46:425–434, 1992\.
  * [11] A. Farina and B. Kawohl.  Remarks on an overdetermined boundary value problem.  Calc. Var., 31:351–357, 2008.
  * [12] B. Gidas, W. M. Ni, and L. Nirenberg.  Symmetry and related properties via the maximum principle.  Comm. Math. Phys., 68(3):209–243, 1979.
  * [13] D. Gilbarg and N.S. Trudinger.  Elliptic Partial Differential Equations of Second Order.  Classics in Mathematics. Springer Berlin Heidelberg, 2001.
  * [14] A. Henrot.  Extremum Problems for Eigenvalues of Elliptic Operators.  Frontiers in Mathematics. Birkhäuser Basel, 2006.
  * [15] A. Henrot and M. Pierre.  Shape Variation and Optimization: A Geometrical Analysis.  EMS Tracts in Mathematics. European Mathematical Society, 2018.
  * [16] J. Jost.  Partial Differential Equations.  Graduate Texts in Mathematics. Springer New York, NY, 2013.
  * [17] C. Nitsch and C. Trombetti.  The classical overdetermined serrin problem.  Complex Variables and Elliptic Equations, 63(7-8):1107–1122, 2018\.
  * [18] J. Serrin.  A symmetry problem in potential theory.  Arch. Rational Mech. Anal., 43:304–318, 1971.
  * [19] H.F. Weinberger.  Remark on the preceding paper of Serrin.  Arch. Rational Mech. Anal., 43:319–320, 1971.