# A Pendulum-Driven Legless Rolling Jumping Robot

Jake Buzhardt†1, Prashanth Chivkula†1 and Phanindra Tallapragada1 † Indicates
equal contribution.View the supplemental video at
https://youtu.be/9hKQilCpeaw1The authors are with the Department of Mechanical
Engineering, Clemson University, Clemson, SC, USA,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a novel rolling, jumping robot. The robot consists
of a driven pendulum mounted to a wheel in a compact, lightweight, 3D printed
design. We show that by driving the pendulum to shift the robot’s weight
distribution, the robot is able to obtain significant rolling speed, achieve
jumps of up to 2.5 body lengths vertically, and clear horizontal distances of
over 6 body lengths. The robot’s dynamic model is derived and simulation
results indicate that it is consistent with the rolling motion and jumping
observed on the robot. The ability to both roll and jump effectively using a
minimalistic design makes this robot unique and could inspire the use of
similar mechanisms on robots intended for applications in which agile
locomotion on unstructured terrain is necessary, such as disaster response or
planetary exploration.

##  I Introduction

The mobility of a robot in unstructured terrain can greatly improve with the
ability to execute small hops or jumps repeatedly with little relaxation. Even
in structured settings, such added agility can allow mobile robots to clear
obstacles or climb up stairs. In recent years, many jumping robot designs have
been proposed, with most of them achieving a jump using a spring-like
mechanism to store potential energy [1, 2, 3, 4]. Such a mechanism can then be
rapidly released to convert the potential energy to kinetic energy of a jump.

In this paper, we take an alternative approach by proposing a rolling, jumping
wheel robot which achieves both of these modes of motion by shifting its
internal mass distribution without the aid of any elastic element. The design
is inspired by the complex dynamics of a rolling hoop with an unbalanced mass,
the so-called Littlewood hoop [5], which has been the subject of many
theoretical and experimental investigations [6, 7, 8, 9, 10]. Just as the
original works on passive walking [11, 12] inspired the development of many
actuated walking mechanisms [13], here we let the passive rolling jumping
mechanism of Littlewood’s hopping hoop serve as inspiration for an actuated
rolling, jumping robot. The specific design used in this paper is a wheel that
contains an actuated internal pendulum. This wheel mechanism can roll due to a
torque exerted via the internal pendulum and a high speed swing of the
pendulum produces a discontinuity in the normal reaction on the robot from the
ground leading to a jump. With such a framework, even in the first design
iteration, the robot is capable of a vertical jump height of over two body
lengths and a horizontal jump distance of over 6 body lengths. Furthermore the
robot can execute repeated jumps without a long relaxation time. The repeated
jumping motion takes advantage of the slip motion of the wheel upon impact and
the resulting friction force.

The proposed legless rolling-jumping robot is distinct in design and mechanics
from the existing few jumping robots that can also roll or tip over
repeatedly. Past rolling-jumping robots such as [14, 15, 16] still relied on
elastic components or shape memory to store and release energy. For instance
the jollbot in [14] is an elastic spheroid which uses a pendulum mechanism for
rolling, but can also compress and change shape significantly to store elastic
potential energy that can be released for a jump of about $0.6$ body lengths.
The two wheeled robot in [16] has an elastic chassis that can snap through and
buckle and produce a jump that is $0.25$ body lengths in height and $1$ body
lengths in width. Despite the lack of any elastic components, the novel
legless rolling-jumping robot presented in this paper can jump significantly
higher and farther.

Another class of related robots are those which employ momentum-based
strategies for rolling, walking, or jumping. For example, the robot in [17] is
able to jump up and down stairs by swinging a pair of pendulum arms. Several
other works have recently proposed mechanisms which utilize the transfer of
angular momentum that occurs from suddenly braking a rapidly spinning momentum
wheel to initiate a jumping, tipping, or standing motion [18, 19, 20, 21, 22].
However, these robots are either unable to locomote aside from their jumping
behavior or rely on other mechanisms for more efficient movement. Many such
jumping robots do not take advantage of other efficient modes of locomotion
that may be possible on different terrains. Similarly, several robots have
used pendulum-based mechanisms for rolling or walking, such as pendulum-driven
rolling spheres [23] and torso-actuated walking mechanisms [24, 25], but these
applications focus only on ground-based locomotion without consideration of
how such a mechanism could be used for jumping. In this paper we propose a
legless robot design that is capable of two modes of motion – rolling and
jumping. Even more importantly this paper presents a fresh look at the
theoretical models and mechanics that can lead to the design of robots with
superior mobility.

##  II Robot design and system description

The jumping mechanism in consideration here requires a wheel body with an
approximately uniform weight distribution with an offset mass attached to an
arm which can be driven by a motor from the center of the wheel. Such a robot
was constructed from two 3D printed hoops with a pendulum driven by a
brushless DC motor affixed to the back hoop. This setup is shown in Fig. 1 and
the key electronic components are summarized in Table I.

d

bce

a

Figure 1: Top left: pendulum-driven jumping wheel robot. Top right: side view
of disassembled robot. Bottom right: Front view of pendulum mounted to a
brushless DC motor affixed to the back hoop. Bottom left: rear view of robot
showing electronics. Labels correspond to labels in Table I.

The total mass of the robot with all components attached is approximately 600
g and the offset mass of 125 g is attached to a pendulum which rotates at a
radial distance of 51 mm from the geometric center of the hoop. The diameter
of the hoop is 152 mm. The length of the two hoops together is 106mm, with the
front and rear hoops having lengths 75 mm and 31 mm. These relative lengths
are chosen so that the weight distribution of the robot is approximately
centered along the axis passing through the centers of the hoops when the
pendulum and motor are mounted to the back hoop.

TABLE I: Component specifications Component | Specification | Label
---|---|---
Motor | T-motor MN4006 KV380 | (a)
Single Board Computer | Raspberry Pi Zero W | (b)
Motor Driver | Tinymovr R5.1 | (c)
IMU | Adafruit LSM9DS0 9 DOF | (d)
CAN-USB Adapter | CANine USB-CAN adapter | (e)
Batteries | 11.1 V (3S), 450 mAh |
| 7.4 V (2S), 400 mAh |

The electronics and controls configuration of the robot consists of a T-motor
MN4006 380 Kv brushless DC motor (BLDC) driven by a Tinymovr R5.1 motor
driver. Being designed for use in drones, this BLDC has a very compact and
lightweight design, with a diameter of 44mm and weight of 68g, while being
capable of delivering a torque of up to 0.376Nm when used with the Tinymovr
R5.1. The Tinymovr driver uses field oriented control (FOC) to drive the BLDC
using a built-in magnetic encoder to measure motor rotation and speed. In our
application, the Tinymovr is used in velocity-control mode, in which a PI
controller feeds forward a motor current value to the FOC control loop based
on errors from a motor velocity reference. These control loops run at 20 kHz.
The Tinymovr communicates with a Raspberry Pi Zero W over a CAN connection via
the CANine CAN-USB adapter. The Raspberry Pi allows for higher level
controllers to be implemented, as it delivers velocity setpoint values to the
velocity control loop running on the Tinymovr. However, in this work, the
velocity setpoint values are fed forward in an open-loop manner, as higher
level control strategies for this robot are still the subject of ongoing work.

##  III Legless jumping - concept and model

The mechanical system in consideration consists of a thin hoop of mass $m_{o}$
and radius $R$ with a mass moment of inertia $I_{o}$ about it’s geometric
center. Attached to the hoop from the center is a pendulum of mass $m_{p}$ and
length $l_{p}$. The arm of the pendulum is taken to be massless and the
pendulum is driven relative to the hoop by a torque applied by a motor
connecting the pendulum to the hoop.

We express the motion of the system in terms of the generalized coordinates
$\mathbf{q}=[\phi,\theta,x,y]^{T}$ depicted in Fig. 2(a). The angles $\phi$
and $\theta$ denote the angles of the hoop and the pendulum, respectively,
both relative to a spatially fixed frame of reference. The $\phi=\theta=0$
configuration corresponds to the pendulum in the vertically downward position
with no relative angular displacement between the pendulum and the hoop. The
$x$ and $y$ coordinates refer to the horizontal and vertical displacement of
the geometric center of the hoop, respectively, relative to a spatially fixed
frame. The $y=0$ position is defined as the configuration in which the hoop is
in contact with the ground.

(a)

(b)

Figure 2: (a) Pendulum-driven wheel robot system with generalized coordinates
$\phi$, $\theta$, $x$, $y$. (b) Free body diagram of detached wheel,
indicating the effects of the pendulum acceleration on the wheel.

The kinetic energy of the system is

$\mathcal{T}=\frac{1}{2}m_{o}|\mathbf{v}_{o}|^{2}+\frac{1}{2}I_{o}\dot{\phi}^{2}+\frac{1}{2}m_{p}|\mathbf{v}_{p}|^{2}$
(1)

where the velocities of the center of the hoop and the pendulum are
$\mathbf{v}_{o}=\dot{x}\,\mathbf{i}+\dot{y}\,\mathbf{j}$ and
$\mathbf{v}_{p}=\left(\dot{x}-l_{p}\dot{\theta}\cos\theta\right)\mathbf{i}+\left(\dot{y}+l_{p}\dot{\theta}\sin\theta\right)\mathbf{j}$,
where $\mathbf{i}$ and $\mathbf{j}$ are the unit vectors associated with the
spatially fixed reference frame. The potential energy of the system is

$\mathcal{V}=m_{o}gy+m_{p}g(y-l_{p}\cos\theta)$ (2)

Then, taking the Lagrangian as $\mathcal{L}=\mathcal{T}-\mathcal{V}$, we can
write the equations of motion from the Euler-Lagrange equations:

$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}_{i}}\right)-\frac{\partial\mathcal{L}}{\partial
q_{i}}=\sum_{j}\lambda_{j}\frac{\partial f^{\text{con}}_{j}}{\partial
q_{i}}+f^{\text{ext}}_{i}$ (3)

for $i=1,\dots,4$ where $f^{\text{con}}_{j}$ ($j=1,2$) are the holonomic
constraints acting on the system, $\lambda_{j}$ are the constraint forces
required to satisfy these constraints, and $f^{\text{ext}}_{i}$ are the non-
conservative generalized forces acting on the system. The holonomic
constraints associated with this system are those of pure rolling,
$f^{\text{con}}_{1}=\phi R-x=0$, and contact with the flat surface,
$f^{\text{con}}_{2}=y=0$. The external generalized forces are those associated
with the torque, $\tau$, applied by the motor to the pendulum and the hoop.

###  III-A Rolling phase

During the rolling phase of motion, the wheel remains in contact with the
ground and rolls without slip. Applying the Euler-Lagrange equation with both
constraints active yield the following Equations of motion during the rolling
phase.

$\displaystyle(I_{o}+mR^{2})\ddot{\phi}+m_{p}l_{p}R(-\ddot{\theta}\cos\theta+\dot{\theta}^{2}\sin\theta)$
$\displaystyle=-\tau$ (4a) $\displaystyle
m_{p}l_{p}^{2}\ddot{\theta}+m_{p}l_{p}(-R\ddot{\phi}\cos\theta+g\sin\theta)$
$\displaystyle=\tau$ (4b)

The expressions for the friction, $\lambda_{1}$, required to prevent slipping
and the normal reaction, $\lambda_{2}$, during rolling are given as follows.

$\displaystyle\lambda_{1}$
$\displaystyle=m_{p}l_{p}\left(-\dot{\theta}^{2}\sin\theta+\ddot{\theta}\cos\theta\right)-mR\ddot{\phi}$
(5a) $\displaystyle\lambda_{2}$
$\displaystyle=m_{p}l_{p}\left(\dot{\theta}^{2}\cos\theta+\ddot{\theta}\sin\theta\right)+mg$
(5b)

The transition from a rolling phase to a flight phase occurs when the normal
force vanishes and the velocity of the center of mass has a positive upward
component. It is also possible for an intermediate phase to occur in which the
friction provided by the rolling surface is insufficient to maintain the pure
rolling constraint. In this case, the hoop could undergo slipping, skidding,
or gliding motions [26, 9]. In this work, we will assume that such effects are
negligible.

###  III-B Flight phase

During the flight phase of motion, neither the contact nor rolling constraints
are active, and thus the corresponding constraint forces can be set to zero,
$\lambda_{1}=\lambda_{2}=0$, to give the following equations of motion.

$\displaystyle I_{o}\ddot{\phi}=-\tau$ (6a) $\displaystyle
m_{p}l_{p}^{2}\ddot{\theta}+m_{p}l_{p}\left(-\ddot{x}\cos\theta+\ddot{y}\sin\theta+g\sin\theta\right)=\tau$
(6b) $\displaystyle
m\ddot{x}+m_{p}l_{p}\left(\dot{\theta}^{2}\sin\theta-\ddot{\theta}\cos\theta\right)=0$
(6c) $\displaystyle
m\ddot{y}+m_{p}l_{p}\left(\dot{\theta}^{2}\cos\theta+\ddot{\theta}\sin\theta\right)+mg=0$
(6d)

The center of mass of the system will undergo projectile motion during the
flight phase, and thus will follow a parabolic trajectory, but this full
system of equations is needed to describe the internal rotations of the system
and the trajectory of the geometric center of the hoop.

##  IV Control strategy and numerical simulation

To numerically simulate this system, we implement the equations of motion
described in Sec. III in MATLAB. To demonstrate the rolling, jumping motion,
we consider the case where the robot begins at rest with the pendulum in the
vertically downward position and the robot will be controlled using a
proportional controller to choose the torque $\tau$ based on the error from a
reference signal for the angular velocity of the pendulum relative to the
wheel. We denote this angular velocity as
$\dot{\psi}=\dot{\theta}-\dot{\phi}$. Such a simulation is shown in Fig. 3 and
is explained in more detail in Sec. IV-C, but first we will discuss the
control strategies implemented to achieve rolling and jumping motions.

###  IV-A Rolling strategy

To achieve a smooth rolling motion from rest, a torque is applied to lift the
pendulum in the desired direction of motion. The resulting offset of the
pendulum will cause the wheel to roll to restore the pendulum to the stable,
downward equilibrium, thus leading the wheel to accelerate in the direction of
the pendulum. Once the wheel is in rolling motion, it’s velocity can be
maintained by driving the pendulum so that it remains in the downward
position. This is done by driving the pendulum so that its angular velocity
relative to the wheel offsets the velocity of the wheel; that is,
$\dot{\psi}=-\dot{\phi}$ so that $\dot{\theta}=0$. This has the effect of
holding the pendulum in the downward position in the spatial frame, but
requires continuous actuation to move the pendulum relative to the rolling
wheel. Following a similar reasoning, the wheel speed can be increased in a
given direction by driving the pendulum relative to the wheel in such a way
that the pendulum is displaced in that direction in the spatially fixed frame.

(a)

(b)

(c)

(d)

(e)

(f)

$\lambda_{2}=0$ Jump occurs (g)

Figure 3: Numerical simulation of the jumping wheel system. Dashed lines
indicate the rolling phase while solid lines indicate the flight phase. (a)
Spatial trajectory of the system center of mass and geometric center. (b)-(c)
trajectories of these points over time. (d) relative angular velocity between
the pendulum and the wheel (e) angular velocity of the wheel(f) torque values
$\tau$ and corresponding motor currents $I_{q}$. (g) normal reaction
$\lambda_{2}$ during rolling.

###  IV-B Jumping strategy

As mentioned previously in Sec. III, jumping motion will tend to occur when
the pendulum has large angular velocity (relative to the spatially fixed
frame) and the pendulum nears the upward configuration ($\theta\approx n\pi$
and $\dot{\theta}^{2}\gg 1)$, which causes the
$m_{p}l_{p}\dot{\theta}^{2}\cos\theta$ term in Eq. 5b to become large, which
leads the normal reaction $\lambda_{2}$ to vanish with the center of mass
velocity having a positive vertical component. Therefore, to achieve a jump,
the pendulum should be driven at high angular velocity toward the upward
configuration once the desired rolling velocity is reached.

Increasing the horizontal distance traversed during the jump requires
increasing the horizontal component of the velocity of the center of mass at
the instant that the loss of contact occurs, as from that point forward, the
center of mass behaves as a projectile, and thus its displacement is a
function of its initial position and velocity alone. The horizontal component
of the center of mass velocity during rolling is
$\dot{\phi}R-\frac{m_{p}l_{p}}{m_{o}+m_{p}}\dot{\theta}\cos\theta\,.$ Thus,
the horizontal displacement can be increased by increasing the rolling speed
of the wheel, $\dot{\phi}R$. However, at higher rolling speed there is less
time for the pendulum to accelerate from its downward position during rolling
to the upward configuration necessary for a jump. Therefore, better jumping
can be achieved by first initiating “swing-up” phase in which the pendulum is
brought to the upward position at a low angular velocity before actuating the
pendulum with maximum torque in the opposite direction so that the pendulum
has more time to accelerate and maximize $\dot{\theta}^{2}$.

###  IV-C Numerical simulation

As proof of concept, we implement a numerical simulation of the proposed
control strategy. In this simulation, the wheel is started from rest with the
pendulum in the vertically downward position. The torque is chosen by applying
a proportional controller to track a reference relative angular velocity,
$\dot{\psi}_{\text{ref}}$. This reference angular velocity is chosen to give a
5s rolling period in which the wheel accelerates from rest to an angular
velocity of approximately 10 rad/s. For this, the reference angular velocity
is chosen to be linear in time. Following this rolling phase, the swing-up
phase is initiated. Finally, the pendulum is actuated clockwise at a large
angular velocity in order to achieve a jump as the pendulum nears the upward
configuration. Specifically, the relative angular velocity reference command
is chosen as follows:

$\dot{\psi}_{\text{ref}}(t)=\begin{cases}-2t&t<5\\\ -40&5.0<t<5.2\\\
200&t>5.2\end{cases}$ (7)

where the times are given in seconds and the angular velocities are in rad/s.
All parameters used in this simulation are selected to be representative of
the robot described in Sec II.

The simulation is conducted by first simulating the rolling dynamics given by
Eqs. 4 until the normal reaction vanishes with a positive vertical component
of the center of mass velocity. This state is then taken as the initial
condition for the flight phase and the system is simulated from there using
the flight dynamics given by Eqs. 6. The results of this simulation are shown
in Fig. 3. It can be seen from Figs. 3(a)-3(c) that while in the air, the
center of the hoop follows a cycloidal trajectory, revolving about the
trajectory of the center of mass, which follows a parabolic trajectory. The
simulation is ended when the $y$ coordinate of the center of the wheel returns
to zero, indicating that the wheel has landed on the ground. Further
simulation beyond this point would require additional modelling of the
collision dynamics associated with the impact with the ground.

2.4 body lengths
(a)

(b)

(c)

(d)

(e)

Figure 4: Experimental results for vertical jumping. (a) Four still images
from a video of the jumping motion overlaid to illustrate the robot
trajectory. (b)-(c) Trajectory of a point on the front of the wheel over time.
(d) Relative angular velocity between pendulum and wheel. (e) Angular velocity
of the wheel.

(a)

(b)

(c)

(d)

(e)

Figure 5: Jumping while traversing a horizontal distance. Top row: a sequence
of frames from an overhead camera. Black floor markings indicate distances of
one body length. Red lines indicate the landing positions of the robot and the
distances traversed between them. The distance indicated in panel (c) is
approximately 4 body lengths and the distance indicated in panel (e) is
approximately 6 body lengths. Bottom row: corresponding images from the same
sequence, taken from a side view panning along with the robot.

##  V Experimental results

###  V-A Jumping experiment

Following the results of the numerical experiment described in the previous
section, the same experiment is conducted on the robot described in Sec. II.
In this experiment, the same reference velocity is provided to the motor as
given by Eq. 7. The result of this experiment is shown in Fig. 4. Video of the
experiment was recorded at 240 frames/second (fps) from a side view with a
stationary camera. Fig. 4(a) shows four frames from this recording overlaid
with one another to represent the path followed by the robot during the jump.
The video was post-processed using the software, Tracker
(https://physlets.org/tracker/) to obtain the trajectory followed by a point
located on the center of the front hoop. This trajectory is given by the curve
plotted over the image and is shown over time in Figs. 4(b), 4(c). The four
points on the trajectory that are overlaid in Fig. 4(a) are indicated the
plots in Fig. 4(b), 4(c) by the blue circles. It can be seen from this
trajectory that the motion of the robot agrees with the numerical simulation
of the previous section. The robot completes a small hop followed by a small
jump and then a large jump. The simulation of the previous section predicts
the small jump height well, both showing this height to be approximately
0.7-0.8 body lengths. In the experiment, following this first jump, the robot
rebounds into a second jump, achieving a height of approximately 2.4 body
lengths, as shown in Fig. 4(a). Fig. 4(d) shows the relative angular velocity
of the pendulum to the hoop during the experiment as measured from the onboard
encoder along with the corresponding reference signal in Eq. 7. Fig. 4(e)
shows the wheel angular velocity as measured from the onboard IMU.

###  V-B Horizontal jumping

In an additional experiment, we apply a different reference angular velocity
in an attempt to clear a larger horizontal distance during the jump. In
particular, we attempt to have the robot reach a larger horizontal velocity
before initiating the jump. This is done by driving the pendulum so that the
angular velocity of the wheel increases more rapidly, accelerating from rest
to an angular velocity of approximately 23 rad/s over 3.75s. At this point,
the jump is initiated by applying an angular velocity reference of 150 rad/s
in the same direction as the rotation of the pendulum for jumping. This
reference signal is summarized as follows.

$\dot{\psi}_{\text{ref}}(t)=\begin{cases}-6.22t&t<3.75\\\
-150&3.75<t<4.75\end{cases}$ (8)

The direction of rotation for initiating the jump for this test was chosen to
be the same direction as the rolling direction because it was seen in previous
experiments (including the experiment of Fig. 4) that by initiating the jump
by swinging the pendulum counter to the direction needed for rolling, while
leading to a larger vertical height of the jump, can have a braking effect on
the rolling motion, leading to a reduced horizontal velocity of the center of
mass at the loss of contact, and therefore a reduced horizontal distance
traversed while in flight.

(a)

(b)

Figure 6: Measured data from horizontal jumping experiment shown in Fig. 5.
(a) Relative angular velocity between pendulum and wheel. (b) Angular velocity
of the wheel.

Results from this test are shown in Fig. 5. The trial was recorded at 240 fps
from a side view taken from a panning camera and additionally from a
stationary camera mounted overhead, recording at 120 fps. Five still images
from each of these views are aligned and shown in Fig. 5. The top row of Fig.
5 shows views from the overhead camera, where the black markings on the ground
are evenly spaced ticks with spacings of one diameter separating each.

During this experiment, once the jump is initiated, the robot completes one
small hop, followed by two jumps in which a significant horizontal distance is
cleared. The red marking in the top panel of Fig. 5a indicates the landing
position after the first small hop. Fig. 5b shows the robot in mid-flight
after launching from the position shown in Fig. 5a. Fig. 5c shows the landing
position of the robot following the first significant jump. The top panel of
Fig. 5c shows a red marking indicating the landing position along with a line
indicating the distance traversed during this first jump, which spans
approximately 4 body lengths. Fig. 5d shows the robot in mid-flight after
launching from the position shown in Fig. 5c . Finally, Fig. 5e shows the
landing position of the robot during the second jump, with the top panel
indicating the distance traversed during this jump, a horizontal span of over
6 body lengths. Fig. 6 shows the measured data collected in this experiment
from the onboard encoder and IMU, respectively, along with the reference
signal for the relative angular velocity in Eq. 8.

##  VI Conclusion

We have presented a novel design for a pendulum-driven wheel robot that can
both roll and jump. In experiments, the robot was capable of jumping up to 2.4
body lengths vertically and, in separate tests, was able to traverse
horizontal lengths of over 6 body lengths in flight. Ongoing and future work
on this robot include the development and implementation of trajectory
planning and model-based control strategies, consideration of the full
dynamics including 3D rotations and collision effects, designs and control
strategies to allow for stabilization following a jump.

## References

  * [1] J. Zhao, J. Xu, B. Gao, F. Cintron, M. Mutka, and L. Xiao, “Msu jumper: A single-motor-actuated miniature steerable jumping robot,” IEEE Transactions on Robotics, vol. 29, no. 3, pp. 602–614, 2013.
  * [2] A. De and D. Koditschek, “Parallel composition of templates for tail-energized planar hopping,” in IEEE International Conference on Robotics and Automation, pp. 4562–4569, 2015.
  * [3] U. Scarfogliero, C. Stefanini, and P. Dario, “Design and development of the long-jumping “grillo” minirobot,,” in IEEE International Conference on Robotics and Automation, pp. 467–472, 2007.
  * [4] D. W. Haldane, M. M. Plecnik, J. K. Yim, and R. S. Fearing, “Robotic vertical jumping agility via series-elastic power modulation,” Science Robotics, vol. 1, no. 1, 2008.
  * [5] J. Littlewood, A Mathematicians Miscellany.  London: Methuen & Co. Ltd, 1953.
  * [6] T. F. Tokieda, “The Hopping Hoop,” American Mathematical Monthly, vol. 104, pp. 152–154, Feb. 1997.
  * [7] Y. Shimomura, M. Branicki, and H. Moffatt, “Dynamics of an axisymmetric body spinning on a horizontal surface. II. Self-induced jumping,” Proceedings of the Royal Society A, vol. 461, no. 2058, pp. 1753–1774, 2005\.
  * [8] A. Ivanov, “On detachment conditions in the problem on the motion of a rigid body on a rough plane,” Regular and Chaotic Dynamics, vol. 13, p. 355–368, 2008.
  * [9] A. Bronars and O. M. O’Reilly, “Gliding motions of a rigid body: the curious dynamics of littlewood’s rolling hoop,” Proceedings of the Royal Society A, vol. 475, no. 2231, p. 20190440, 2019.
  * [10] P. Tallapragada, J. Buzhardt, and R. Seney, “A passive jumping mechanism,” in Dynamic Systems and Control Conference, ASME, 2019.
  * [11] T. McGeer, “Passive dynamic walking,” International Journal of Robotics Research, vol. 9, no. 2, pp. 62–82, 1990.
  * [12] M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, “The Simplest Walking Model: Stability, Complexity, and Scaling,” Journal of Biomechanical Engineering, vol. 120, pp. 281–288, 04 1998.
  * [13] S. Collins, A. Ruina, R. Tedrake, and M. Wisse, “Efficient bipedal robots based on passive-dynamic walkers,” Science, vol. 307, no. 5712, pp. 1082–1085, 2005.
  * [14] R. Armour, K. Paskins, A. Bowyer, J. Vincent, and W. Megill, “Jumping robots: a biomimetic solution to locomotion across rough terrain,” Bioinspiration & Biomimetics, vol. 2, no. 3, p. S65, 2007.
  * [15] Y. Matsuyama and S. Hirai, “Analysis of circular robot jumping by body deformation,” in International Conference on Robotics and Automation (ICRA), IEEE, 2007.
  * [16] K. Misu, A. Yoshii, and H. Mochiyama, “A Compact Wheeled Robot that Can Jump while Rolling,” in International Conference on Intelligent Robots and Systems (IROS), IEEE, Oct. 2018.
  * [17] R. Hayashi and S. Tsujio, “High-performance jumping movements by pendulum-type jumping machines,” in International Conference on Intelligent Robots and Systems. (IROS), IEEE, 2001.
  * [18] B. J. Hockman, A. Frick, R. G. Reid, I. A. Nesnas, and M. Pavone, “Design, control, and experimentation of internally-actuated rovers for the exploration of low-gravity planetary bodies,” Journal of Field Robotics, vol. 34, no. 1, pp. 5–24, 2017.
  * [19] T.-M. Ho, V. Baturkin, C. Grimm, J. T. Grundmann, C. Hobbie, E. Ksenik, C. Lange, K. Sasaki, M. Schlotterer, M. Talapina, et al., “Mascot—the mobile asteroid surface scout onboard the hayabusa2 mission,” Space Science Reviews, vol. 208, pp. 339–374, 2017.
  * [20] A. R. Geist, J. Fiene, N. Tashiro, Z. Jia, and S. Trimpe, “The Wheelbot: A Jumping Reaction Wheel Unicycle,” IEEE Robotics and Automation Letters, vol. 7, pp. 9683–9690, Oct. 2022.
  * [21] J. W. Romanishin, K. Gilpin, S. Claici, and D. Rus, “3d m-blocks: Self-reconfiguring robots capable of locomotion via pivoting in three dimensions,” in International Conference on Robotics and Automation (ICRA), IEEE, 2015.
  * [22] M. Muehlebach and R. D’Andrea, “Nonlinear analysis and control of a reaction-wheel-based 3-d inverted pendulum,” IEEE Transactions on Control Systems Technology, vol. 25, no. 1, pp. 235–246, 2016.
  * [23] E. Kayacan, Z. Y. Bayraktaroglu, and W. Saeys, “Modeling and control of a spherical rolling robot: a decoupled dynamics approach,” Robotica, vol. 30, no. 4, pp. 671–680, 2012.
  * [24] G. W. Howell, Analysis and control of superarticulated biped robots.  PhD thesis, Boston University, 2000.
  * [25] S. Sanchez and P. A. Bhounsule, “A differential drive rimless wheel that can move straight and turn,” in International Conference on Advanced Intelligent Mechatronics (AIM), IEEE, 2020.
  * [26] A. Taylor and M. Fehrs, “The dynamics of an eccentrically loaded hoop,” American Journal of Physics, vol. 78, pp. 496–498, May 2010.


# A note on the Winterbottom shape

Leonard Kreutz  School of Computation, Information and Technology, Technical
University of Munich
Boltzmannstraße 3, 85748 Garching bei München, Germany<EMAIL_ADDRESS>and  Bernd Schmidt  Institut für Mathematik, Universität Augsburg,
Universitätsstr. 14, 86159 Augsburg, Germany<EMAIL_ADDRESS>augsburg.de

(Date: August 31, 2024)

###### Abstract.

In this short note we review results on equilibrium shapes of minimizers to
the sessile drop problem. More precisely, we study the Winterbottom problem
and prove that the Winterbottom shape is indeed optimal. The arguments
presented here are based on relaxation and the (anisotropic) isoperimetric
inequality.

###### Key words and phrases:

wetting, dewetting, Winterbottom shape, crystallization, interfacial energies,
capillarity problems

##  1\. Introduction

For solid crystals with sufficiently small grains, Herring [11] claims that
the bulk contribution to their configurational energy is negligible with
respect to the surface tension. Thus, in this setting, when determining the
equilibrium shape of a crystal, interfacial energies of the type

$\displaystyle
P_{\varphi}(E)=\int_{\partial^{*}E}\varphi(\nu_{E})\,\mathrm{d}\mathcal{H}^{d-1}\,$

play a fundamental role. Here, $\varphi\colon\mathbb{R}^{d}\to[0,+\infty)$ is
a convex and positively homogeneous function of degree one that describes the
possibly anisotropic surface tension of the crystal. The set $E$ is assumed to
be sufficiently regular (i.e. $E$ is a set of finite perimeter),
$\partial^{*}E$ denotes its (measure theoretic) boundary, and $\nu_{E}(x)$ the
outer normal to the set $E$ at the point $x\in\partial^{*}E$. The Wulff
problem consists in studying solutions of

$\displaystyle
E\in\mathrm{argmin}\left\\{P_{\varphi}(E)\colon|E|=v\right\\}\,,$ (1.1)

where $v>0$ is given. This problem is an anistropic generalization of the
classical isoperimetric problem. In the early 1900s Wulff [18] proposed a
geometric construction to (1.1) given by

$\displaystyle W_{\varphi}=\\{x\in\mathbb{R}^{d}\colon
x\cdot\nu\leq\varphi(\nu)\text{ for all }\nu\in\mathbb{S}^{d-1}\\}\,.$ (1.2)

This shape is nowadays known as the Wulff set (or Wulff crystal) of $\varphi$.
Dinghas proved formally in [5] that among convex polyhedra the Wulff set is
the shape having the least surface integral for the volume it contains. The
proof has later been rendered precise by Taylor [14, 15, 16] using arguments
from geometric measure theory.

The Wulff variational problem provides a description of an equilibrium crystal
shape deep inside a region in the gas phase. This leads to the natural
question if, likewise, the shape of a crystal growing on a substrate can be
determined that minimizes a suitable combination of its surface tension and
the interaction energy with the substrate. Such a situation may be described
as follows. For $\varphi$ be as above and $\lambda\in\mathbb{R}$ we set

$\displaystyle F_{\lambda,\varphi}(E)=\int_{\partial^{*}E\cap
H^{+}}\varphi(\nu_{E})\,\mathrm{d}\mathcal{H}^{d-1}+\lambda\mathcal{H}^{d-1}(\partial^{*}E\cap
H)\,,$ (1.3)

where $H^{+}=\\{x\in\mathbb{R}^{d}\colon x_{d}>0\\}$,
$H=\\{x\in\mathbb{R}^{d}\colon x_{d}=0\\}$ and $E$ is a set of finite
perimeter in $H^{+}$, i.e. $|E\setminus H^{+}|=0$, which we simply denote by
$E\subseteq H^{+}$, cf. Fig. 1.

$\partial^{*}E$$E$$\nu_{E}$$H$$H^{+}$$\partial^{*}E\cap H$ Figure 1. A generic
configuration admissible to (1.4). The substrate is illustrated in dark gray,
the region occupied by the crystal is illustrated in light gray, and the
contact surface is illustrated in bold.

Here, as in the Wulff problem, $\varphi$ represents the (anisotropic) surface
tension density between crystal and vapor and $\lambda\in\mathbb{R}$ is the
relative adhesion coefficient between the crystal and the substrate (i.e., the
difference of the crystal-substrate and the substrate-vapor interfacial energy
per unit surface area). In this setting, the Winterbottom problem [17]
consists in studying the existence and finding the solutions of

$\displaystyle E\in\mathrm{argmin}\left\\{F_{\lambda,\varphi}(E)\colon
E\subseteq H^{+},|E|=v\right\\}\,,$ (1.4)

where $v>0$. There are three interesting parameter regimes to consider (see
Remark 2.4):

  * (1)

If $\lambda\geq\varphi(-e_{d})$ it is energetically inconvenient for the
crystal to attach to the substrate and therefore the solution (1.4) coincides
(up to translation) with the solution of (1.1). This phenomenon is called
complete drying.

  * (2)

If $\lambda\in(-\varphi(e_{d}),\varphi(-e_{d}))$ a solution to the
Winterbottom exists, but its shape differs from the Wulff shape given in
(1.2). In fact, the solution shape is now the Winterbottom shape given by

$\displaystyle W_{\lambda,\varphi}=W_{\varphi}\cap\\{x\in\mathbb{R}^{d}\colon
x_{d}\geq-\lambda\\}$ (1.5)

suitably rescaled and placed in order to be in contact with the substrate. In
this case we speak of partial drying/wetting.

  * (3)

If $\lambda\leq-\varphi(e_{d})$ complete wetting occurs: It is energetically
favorable to create large surface area in common with the substrate. This
allows to create arbitrarily small energy (if $\lambda=-\varphi(e_{d})$) or
even energy diverging to $-\infty$ (if $\lambda<-\varphi(e_{d})$).

We rigorously study all the cases. From the point of view of the mathematical
analysis, the most interesting regime to consider is the partial wetting
regime (2). There, our first main result Theorem 2.3 proves that indeed (1.5)
solves (1.4).

In the case that $\varphi$ is smooth the solution given in (1.5) in particular
recovers Young’s law [19]. This law relates the contact angle of the boundary
of equilibrium configurations with the adhesion coefficient $\lambda$. More
precisely, if $\partial(\overline{E}\cap H)$ denotes the boundary of
$\overline{E}\cap H$ in $H$, then

$\displaystyle\nabla\varphi(\nu_{E}(x))\cdot(-e_{d})=\lambda\,,\quad\text{ for
all }x\in\partial(\overline{E}\cap H)\,$ (1.6)

(see Remark 2.4(4) below). In the isotropic case, i.e. $\varphi(\nu)=|\nu|$,
this reads (cf. Fig. 2)

$\displaystyle\nu_{E}(x)\cdot(-e_{d})=\lambda\,,\quad\text{ for all
}x\in\partial(\overline{E}\cap H)\,.$

The equilibrium condition (1.6) has been derived more generally for anistropic
capillarity problems in [12] and for epitaxially-strained thin films in [4].

$\nu_{E}$$-e_{d}$$x\in\partial(\overline{E}\cap H)$ Figure 2. Young’s law for
the contact angle. $\nu_{E}$ is the normal at the point
$x\in\partial(\overline{E}\cap H)$ and $-e_{d}$ is the normal vector of $H$
pointing outwards with respect to the region the crystal may occupy.

The proof that (1.5) is the solution shape to (1.4) provides a rigorous
justification of an ansatz that has been previously considered in [13]. It
essentially lies on the anistropic isoperimetric inequality [6] for the case
$\lambda>0$ and change of coordinates in the case that $\lambda\leq 0$. In
particular the quantitative isoperimetric inequality [3, 10, 7], implies the
stability of the Winterbottom shape. The stability of the Winterbottom shape
has already been proved in two dimensions in [13] with the use of the
(generalized) Bonnesen inequalities [2]. The stability of the Winterbottom
shape in any dimension is the content of our second main Theorem 2.5: For a
set $E\subset H^{+}$ we show that its squared asymmetry index (i.e., the
Lebesgue measure of the symmetric difference of $E$ with the optimally placed
Winterbottom shape of equal volume) is controlled by the isoperimetric
difference (the energy deficit of $E$ and the Winterbottom shape).

The paper is organized as follows. In Section 2, we introduce the problems,
discuss the different parameter regimes, and state the main theorems. In
Section 3, we prove the main theorem. Here, we would like to point out that,
due to the presence of substrate, we are working with discontinuous integrands
in general and thus, some of the results already present for continuous
integrands need to be proved again in our setting, cf. Proposition 2.2.

##  2\. Settings and Main results

### Notation and Preliminaries

For a measurable subset $B$ of $\mathbb{R}^{d}$ we denote by $|B|$ its
$d$-dimensional Lebesgue measure and by $\mathcal{H}^{k}(B)$ its
$k$-dimensional Hausdorff measure. Given $x,y\in\mathbb{R}^{d}$ we denote by
$x\cdot y$ their scalar product and by $\|x\|$ the Euclidean norm of $x$. For
$r>0$, $x\in\mathbb{R}^{d}$ we write
$B_{r}(x)=\\{y\in\mathbb{R}^{d}\colon\|y-x\|<r\\}$ and abbreviate
$\mathbb{S}^{d-1}=\partial B_{1}(0)$. We also introduce the half space
$H^{+}=\\{x\in\mathbb{R}^{d}\colon x_{d}>0\\}$ and the hyperplane
$H=\\{x\in\mathbb{R}^{d}\colon x_{d}=0\\}$. If $E\subset\mathbb{R}^{d}$ is a
set of finite perimeter (i.e., its characteristic function is of bounded
variation), we write $\partial^{*}E$ for the reduced boundary of $E$ and
denote by $\nu_{E}\colon\partial^{*}E\to\mathbb{S}^{d-1}$ the generalized
outer normal to $E$. We refer to [1] for the definition of these objects and
basic facts about sets of finite perimeter. In particular, if
$K\subseteq\mathbb{R}^{d}$ is a bounded convex set, then $K$ is of finite
perimeter and $\nu_{K}(x)\in\mathcal{N}_{K}(x)$ for all
$x\in\partial^{*}K\subseteq\partial K$, where, for each $x\in\partial K$,
$\mathcal{N}_{K}(x):=\\{\zeta\in\mathbb{R}^{d}\colon\zeta\cdot(x^{\prime}-x)\leq
0\leavevmode\nobreak\ \forall\,x^{\prime}\in K\\}$ denotes the normal cone to
$K$ at $x$.

### Energy

Let $\varphi\colon\mathbb{R}^{d}\to[0,+\infty)$ be a convex and positively
homogeneous function of degree one that is bounded from below, i.e. there
exists $c>0$ such that

$\displaystyle\varphi(\nu)\geq c\,\|\nu\|\text{ for all
}\nu\in\mathbb{R}^{d}\,.$ (2.1)

For a set of finite perimeter $E\subseteq H^{+}$ we define
$F_{\lambda,\varphi}(E)$ as in (1.3). Given $v>0$, we are interested in
studying the shape of the solutions to (1.4).

######  Remark 2.1 (Scaling).

It is obvious that $F_{\lambda,\varphi}(rE)=r^{d-1}F_{\lambda,\varphi}(E)$ for
any $r>0$ and, in particular, the minimal energy
$m_{\lambda,\varphi}(v)=\inf\left\\{F_{\lambda,\varphi}(E)\colon|E|=v\right\\}$
satisfies

$\displaystyle
m_{\lambda,\varphi}(v)=v^{\frac{d-1}{d}}m_{\lambda,\varphi}(1)\,.$

### Wulff shape

In order to construct solutions to (1.4) we first define the _Wulff set_ of
$\varphi$, yet in a more general set-up: For any positively $1$-homogeneous
Borel function $\varphi\colon\mathbb{R}^{d}\to[0,+\infty)$ which is bounded
from below (cf. (2.1)) but not necessarily continuous or even convex, we set

$\displaystyle W_{\varphi}:=\\{x\in\mathbb{R}^{d}\colon\nu\cdot
x\leq\varphi(\nu)\leavevmode\nobreak\ \forall\,\nu\in\mathbb{S}^{n-1}\\}\,.$

In fact, for continuous $\varphi$, $W_{\varphi}$ is the – up to translations
unique – minimizer among sets with equal volume to the problem
$E\mapsto\int_{\partial^{*}E}\varphi(\nu_{E})\,\mathrm{d}\mathcal{H}^{d-1}$
without substrate, cf. [16, 8, 9]. We recall some basic facts on Wulff sets.

######  Proposition 2.2.

  1. (1)

$W_{\varphi}$ is a bounded, convex and closed set with
$0\in\operatorname{int}W_{\varphi}$.

  2. (2)

The convex conjugate $\varphi^{*}$ is equal to the indicator function of
$W_{\varphi}$, i.e., $\varphi^{*}(x)=0$ if $x\in W_{\varphi}$ and
$\varphi^{*}(x)=\infty$ if $x\notin W_{\varphi}$.

  3. (3)

For $x\in\partial W_{\varphi}$ we have that
$\mathcal{N}_{W_{\varphi}}(x)=\partial\varphi^{*}(x)$, where
$\partial\varphi^{*}(x)$ is the subgradient of $\varphi^{*}$ at $x$.

  4. (4)

The Wulff shapes of $\varphi$ and its convex envelope $\varphi^{**}$ coincide:
$W_{\varphi}=W_{\varphi^{**}}$.

  5. (5)

If $x\in\partial W_{\varphi}$ and $\zeta$ lies in the the normal cone to
$W_{\varphi}$ at $x$, then $x\cdot\zeta=\varphi^{**}(\zeta)$.

###### Proof.

Convexity and closedness of $W_{\varphi}$ and $0\in B_{c}\subseteq
W_{\varphi}$ for $c>0$ given by (2.1) are immediate. Boundedness follows in
our set-up from $-\varphi(-e_{k})\leq x_{k}\leq\varphi(e_{k})$ for
$k=1,\ldots,n$ for any $x\in W_{\varphi}$. This proves (1). Now, (2)–(5) are
found in [8, Propositions 3.4 & 3.5]. A careful insepection of the proofs in
[8] reveals that the relevant assertions do not need continuity of $\varphi$
(which is assumed there). The statement in (5) follows since, by (3),
$\zeta\in\partial\varphi^{*}(x)$, whence the Fenchel-Young identity implies
$x\cdot\zeta=\varphi^{*}(x)+\varphi^{**}(\zeta)$ and so
$x\cdot\zeta=\varphi^{**}(\zeta)$ by (2). ∎

### Winterbottom shape

We now return to our convex, positively $1$-homogeneous and lower bounded
$\varphi$ and set

$\displaystyle W_{\lambda,\varphi}=W_{\varphi}\cap\\{x\in\mathbb{R}^{d}\colon
x_{d}\geq-\lambda\\}\,.$

If $|W_{\lambda,\varphi}|>0$, the _Winterbottom set_ with volume $v>0$ is then
defined as

$\displaystyle
W_{\lambda,\varphi}(v)=\left(\frac{v}{|W_{\lambda,\varphi}|}\right)^{\frac{1}{d}}\left(W_{\lambda,\varphi}+\lambda
e_{d}\right)\,.$ (2.2)

### Minimizers

We are describing quickly the different regimes for $\lambda$:

  1. (1)

_Complete drying:_ $\lambda\geq\varphi(-e_{d})$. In this case
$W_{\lambda,\varphi}=W_{\varphi}$. In Remark 2.4 below we will see that
minimizers to $F_{\lambda,\varphi}$ for a given volume are those of the
unconstrained system without substrate.

  2. (2)

_Partial drying/wetting:_
$\lambda\in\left(-\varphi(e_{d}),\varphi(-e_{d})\right)$. Here we have
$W_{\lambda,\varphi}\subsetneq W_{\varphi}$ as shown in Lemma 3.1 below.
Minimality of the Winterbottom shape will be established in Theorem 2.3 below.

  3. (3)

_Complete wetting:_ $\lambda\leq-\varphi(e_{d})$. Here
$W_{\lambda,\varphi}=\emptyset$. In Remark 2.4 below we will see that
solutions to (1.4) do not exist if $\lambda<-\varphi(e_{d})$ as “wetting”
configurations that intersect large parts of $H$ have arbitrarily small
energy. Generically there are no minimizers in the special case
$\lambda=-\varphi(e_{d})$ either, while here for particular $\varphi$ (even
non-unique) minimizers might exist.

The first main theorem is the following characterization of minimizers in the
partial drying/wetting regime.

######  Theorem 2.3.

Let $\lambda\in\left(-\varphi(e_{d}),\varphi(-e_{d})\right)$ and $v>0$. Then

$\displaystyle F_{\lambda,\varphi}(W_{\lambda,\varphi}(v))\leq
F_{\lambda,\varphi}(E)$

for all $E\subseteq H^{+}$ sets of finite perimeter such that $|E|=v$.
Moreover, equality holds if and only if there exists $\tau\in H$ such that
$|E\Delta\left(W_{\lambda,\varphi}(v)+\tau\right)|=0$.

######  Remark 2.4.

  1. (1)

_Complete drying:_ $\lambda\geq\varphi(-e_{d})$. Comparison to the
unconstrained case shows that $E$ is a solution to (1.4) if and only if
$|E\Delta\left(W_{\lambda,\varphi}(v)+\tau\right)|=0$ (recall definition (2.2)
of $W_{\lambda,\varphi}(v)$) for some $\tau\in H^{+}$ (if
$\lambda>\varphi(-e_{d})$ and $\mathcal{H}^{d-1}(\\{x\in
W_{\varphi}:x_{d}=-\varphi(-e_{d})\\})>0$), respectively, $\tau\in H^{+}\cup
H$ (if $\lambda=\varphi(-e_{d})$ or $\mathcal{H}^{d-1}(\\{x\in
W_{\varphi}:x_{d}=-\varphi(-e_{d})\\})=0$).

  2. (2)

_Complete drying and partial drying/partial wetting:_
$\lambda>-\varphi(e_{d})$. For every set of finite perimeter $E\subseteq
H^{+}$ one has

$\displaystyle\int_{\partial^{*}E\cap
H^{+}}\varphi(\nu_{E})\,\mathrm{d}\mathcal{H}^{d-1}\geq\varphi\Big{(}\int_{\partial^{*}E\cap
H^{+}}\nu_{E}\,\mathrm{d}\mathcal{H}^{d-1}\Big{)}=\mathcal{H}^{d-1}(\partial^{*}E\cap
H)\varphi(e_{d})\,,$ (2.3)

where we have used Jensen’s inequality, the homogeneity of $\varphi$ and the
fact that, by the Gauss-Green Theorem for sets of finite perimeter,
$\int_{\partial^{*}E}\nu_{E}\,\mathrm{d}\mathcal{H}^{d-1}=0$. This shows that
$m_{\lambda,\varphi}$ (cf. Remark 2.1) is positive if
$\lambda>-\varphi(e_{d})$.

  3. (3)

_Complete wetting:_ $\lambda\leq-\varphi(e_{d})$. In case
$\lambda<-\varphi(e_{d})$ one may consider cylindrical sets
$E_{R}=(0,R)^{d-1}\times(0,v/R^{d-1})$ with $R\to\infty$ to see that
$m_{\lambda,\varphi}=-\infty$. In particular, solutions to (1.4) do not exist.
In case $\lambda=-\varphi(e_{d})$ the trial configurations $E_{R}$ show that
$m_{\lambda,\varphi}\leq 0$. Together with (2.3) this implies
$m_{\lambda,\varphi}=0$. However, in this case both existence and non-
existence of minimizers might occur: If, e.g., $\varphi$ is strictly convex,
the above argument shows that $F_{\lambda,\varphi}(E)>0$ for $|E|>0$. If, by
way of contrast, $\varphi$ is affine near $e_{d}$ the above computation shows
that for a spherical cap $C_{\varepsilon}=\\{x\in B_{1}(0):x_{d}\geq
1-\varepsilon\\}$, $0<\varepsilon\ll 1$, the set
$E=\frac{v}{|C_{\varepsilon}|}(C_{\varepsilon}-(1-\varepsilon)e_{d})$ has
$F_{\lambda,\varphi}(E)=0$.

  4. (4)

_Young’s law:_ The relation (1.6) can be seen as follows. Let $x\in\partial
W_{\lambda,\varphi}$ with $x_{d}=-\lambda$ and
$\nu(x)\in\mathcal{N}_{W_{\varphi}}(x)$. Then Proposition 2.2(3) and the
Fenchel-Young identity imply
$x\in\partial\varphi^{**}(\nu(x))=\partial\varphi(\nu(x))$. For smooth
$\varphi$ this shows $\nabla\varphi(\nu(x))=x$ and, hence,

$\displaystyle\nabla\varphi(\nu(x))\cdot(-e_{d})=-x_{d}=\lambda\,.$

Furthermore, we are able to establish the stability of the Winterbottom shape
improving Theorem 2.3. This is formulated in our second main theorem.

######  Theorem 2.5.

Let $\lambda\in\left(-\varphi(e_{d}),\varphi(-e_{d})\right)$ and $v>0$. There
is a constant $C>0$ such that for every set $E\subseteq H^{+}$ of finite
perimeter with $|E|=v$ there exists $\tau\in H$ that verifies

$\displaystyle\frac{|E\triangle(W_{\lambda,\varphi}(v)+\tau)|^{2}}{v^{2}}\leq
C\left(\frac{F_{\lambda,\varphi}(E)-F_{\lambda,\varphi}(W_{\lambda,\varphi}(v))}{F_{\lambda,\varphi}(W_{\lambda,\varphi}(v))}\right)\,.$

##  3\. Proof of the main theorems

In this section we prove Theorem 2.3 and Theorem 2.5. In the following we will
make use of various unconstrained auxiliary functionals. If
$\psi:\mathbb{R}^{n}\to\mathbb{R}$ is a homogeneous function of degree one
(not necessarily convex or bounded from below), for a set $E\subseteq H^{+}$
of finite perimeter we define

$\displaystyle
P_{\psi}(E)=\int_{\partial^{*}E}\psi(\nu)\,\,\mathrm{d}\mathcal{H}^{d-1}\,.$

Now suppose $\varphi_{\lambda}\colon\mathbb{R}^{d}\to\mathbb{R}$ is given by

$\displaystyle\varphi_{\lambda}(\nu)=\begin{cases}\lambda t&\text{if
}\nu=-te_{d}\text{ with }t>0\,,\\\ \varphi(\nu)&\text{otherwise.}\end{cases}$

######  Lemma 3.1.

Let $-\varphi(e_{d})<\lambda<\varphi(-e_{d})$. Then,

$\displaystyle P_{\varphi_{\lambda}}(E)\leq F_{\lambda,\varphi}(E)$

for all sets of finite perimeter $E\subseteq H^{+}$ with equality if and only
if

$\displaystyle\mathcal{H}^{d-1}\left(\\{x\in\partial^{*}E\colon\nu_{E}(x)=-e_{d}\\}\setminus
H\right)=0\,.$

In particular, $W_{\lambda,\varphi}(1)\subsetneq W_{\varphi}(1)$ and

$\displaystyle
P_{\varphi_{\lambda}}(W_{\lambda,\varphi}(1))=F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))\,.$

###### Proof.

The first two statements are direct from the definition of
$P_{\varphi_{\lambda}}$. As $W_{\lambda,\varphi}(1)$ is convex and
$W_{\lambda,\varphi}(1)\subseteq\\{x_{d}\geq 0\\}$, it suffices to show
$W_{\lambda,\varphi}(1)\cap H\neq\emptyset$, or equivalently
$W_{\varphi}\cap\\{x_{d}=-\lambda\\}\neq\emptyset$. In fact, if there exists
$x\in\partial W_{\lambda,\varphi}(1)$ such that $x_{d}>0$ and
$-e_{d}=\nu(x)\in\mathcal{N}_{W_{\lambda,\varphi}(1)}(x)$, then

$\displaystyle-x_{d}=x\cdot(-e_{d})\geq y\cdot(-e_{d})=-y_{d}\quad\forall y\in
W_{\lambda,\varphi}(1)\iff y_{d}\geq x_{d}>0\quad\forall y\in
W_{\lambda,\varphi}(1)\,.$

This contradicts $W_{\lambda,\varphi}(1)\cap H\neq\emptyset$. In order to see
that $W_{\varphi}\cap\\{x_{d}=-\lambda\\}\neq\emptyset$ we choose
$x_{+},x_{-}\in W_{\varphi}$ such that $\pm
e_{d}\in\mathcal{N}_{W_{\varphi}}(x_{\pm})$. Then, by Proposition 2.2(5) we
have $(x_{\pm})_{d}=\pm x_{\pm}\cdot(\pm e_{d})=\pm\varphi(\pm e_{d})$. Thus
by convexity of $W_{\varphi}$ there exists $x\in W_{\varphi}$ such that
$x_{d}=-\lambda$.

∎

######  Lemma 3.2.

Let $0<\lambda<\varphi(-e_{d})$. Then

$\displaystyle
P_{\varphi_{\lambda}^{**}}(W_{\lambda,\varphi}(1))=P_{\varphi_{\lambda}}(W_{\lambda,\varphi}(1))\leq
P_{\varphi_{\lambda}}(E)$

for all sets of finite perimeter $E\subseteq H^{+}$ such that $|E|=1$.
Moreover, equality holds if and only if there exists $\tau\in H^{+}\cup H$
such that $|E\Delta(W_{\lambda,\varphi}(1)+\tau)|=0$.

###### Proof of Lemma 3.2.

First note that the argument in the preceding proof shows that for
$x\in\partial^{*}W_{\lambda,\varphi}$

$\displaystyle\nu_{W_{\lambda,\varphi}}(x)=\begin{cases}\nu_{W_{\varphi}}(x)\neq-
e_{d}&\mbox{if }x_{d}>-\lambda\,,\\\ -e_{d}&\mbox{if
}x_{d}=-\lambda\,.\end{cases}$ (3.1)

Since $0<\lambda<\varphi(-e_{d})$ we have that $\varphi_{\lambda}$ is bounded
from below and that $W_{\lambda,\varphi}=W_{\varphi_{\lambda}}$ is the Wulff
set of $\varphi_{\lambda}$. By Proposition 2.2(4) we therefore have

$\displaystyle W_{\lambda,\varphi}=W_{\varphi_{\lambda}^{**}}\,.$ (3.2)

Let
$x\in\partial^{*}W_{\lambda,\varphi}=\partial^{*}W_{\varphi_{\lambda}^{**}}$.
If $x_{d}>-\lambda$, we apply Proposition 2.2(5) to both
$\varphi_{\lambda}^{**}$ and $\varphi$ and get

$\varphi_{\lambda}^{**}(\nu_{W_{\lambda,\varphi}}(x))=x\cdot\nu_{W_{\lambda,\varphi}}(x)=\varphi(\nu_{W_{\lambda,\varphi}}(x))=\varphi_{\lambda}(\nu_{W_{\lambda,\varphi}}(x))\,,$

by (3.1). If $x_{d}=-\lambda$, we apply Proposition 2.2(5) to
$\varphi_{\lambda}^{**}$ and use (3.1) to obtain

$\varphi_{\lambda}^{**}(\nu_{W_{\lambda,\varphi}}(x))=x\cdot(-e_{d})=\lambda\,=\varphi_{\lambda}(\nu_{W_{\lambda,\varphi}}(x)),$

too. It follows that

$\displaystyle
P_{\varphi_{\lambda}^{**}}(W_{\lambda,\varphi})=P_{\varphi_{\lambda}}(W_{\lambda,\varphi})\,.$
(3.3)

We can now conclude by referring to known results [16, 8, 9] on the
unconstrained functional $P_{\varphi_{\lambda}^{**}}$ as, by (3.3) and (3.2),
for any set $E\subseteq H^{+}$ of finite perimeter with $|E|=1$ we have

$\displaystyle
P_{\varphi_{\lambda}}(W_{\lambda,\varphi}(1))=P_{\varphi^{**}_{\lambda}}(W_{\lambda,\varphi}(1))=P_{\varphi^{**}_{\lambda}}(|W_{\varphi_{\lambda}^{**}}|^{-1/d}W_{\varphi_{\lambda}^{**}})\leq
P_{\varphi^{**}_{\lambda}}(E)\leq P_{\varphi_{\lambda}}(E)$

with equality only if $E=W_{\lambda,\varphi}(1)+\hat{\tau}$ such that
$E\subseteq H^{+}$, i.e. $E=W_{\lambda,\varphi}(1)+\tau$ with $\tau\in
H^{+}\cup H$. ∎

The following change of coordinates allows to reduce to the case $\lambda>0$.
Given $x_{0}\in\mathrm{int}\,(W_{\varphi})$ we define

$\displaystyle\varphi_{x_{0}}(\nu)=\varphi(\nu)-x_{0}\cdot\nu\,.$

######  Lemma 3.3.

Let $-\varphi(e_{d})<\lambda<\varphi(-e_{d})$. For all
$x_{0}\in\mathrm{int}(W_{\varphi})$ there exists
$\varepsilon=\varepsilon(x_{0})>0$ such that

$\displaystyle\varphi_{x_{0}}(\nu)\geq\varepsilon\|\nu\|\quad\text{for all
}\nu\in\mathbb{R}^{d}\,.$ (3.4)

Furthermore, there exists $x_{0}\in\mathrm{int}(W_{\varphi})$ such that for
$\lambda^{\prime}=\lambda+x_{0}\cdot e_{d}$ we have that

$\displaystyle{\rm(i)}\leavevmode\nobreak\
0<\lambda^{\prime}<\varphi_{x_{0}}(-e_{d})\,\quad\text{and}\quad{\rm(ii)}\leavevmode\nobreak\
P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)=P_{\varphi_{\lambda}}(E)\,.$ (3.5)

In addition, we have

$\displaystyle{\rm(i)}\leavevmode\nobreak\ x\in W_{\varphi}\iff x-x_{0}\in
W_{{\varphi}_{x_{0}}}\quad\text{and}\quad{\rm(ii)}\leavevmode\nobreak\
W_{\lambda^{\prime},\varphi_{x_{0}}}(1)=W_{\lambda,\varphi}(1)+\tau$ (3.6)

for $\tau=|W_{\lambda,\varphi}|^{-1/d}(-x_{0}+(x_{0}\cdot e_{d})e_{d})\in H$.

###### Proof.

We first prove (3.4). As $\varphi_{x_{0}}$ is positively $1$-homogeneous, it
suffices to prove the claim for $\nu\in\mathbb{S}^{d-1}$. By construction of
$W_{\varphi}$ for all $x\in W_{\varphi}$ we have $x\cdot\nu\leq\varphi(\nu)$
and hence

$\displaystyle(x-x_{0})\cdot\nu\leq\varphi(\nu)-x_{0}\cdot\nu=\varphi_{x_{0}}(\nu)\quad\text{for
all }\nu\in\mathbb{S}^{d-1}\,.$

Since $x_{0}\in\mathrm{int}\,(W_{\varphi})$ there is $\varepsilon>0$ such that
$\overline{B}_{\varepsilon}(x_{0})\subseteq W_{\varphi}$ and thus choosing
$x_{\varepsilon,\nu}=x_{0}+\varepsilon\nu\in\overline{B}_{\varepsilon}(x_{0})$
we find

$\displaystyle\varepsilon=(x_{\varepsilon,\nu}-x_{0})\cdot\nu\leq\varphi_{x_{0}}(\nu)\quad\text{for
all }\nu\in\mathbb{S}^{d-1}\,.$

This shows (3.4). Next, we prove (3.5)(i). To see this, observe

$\displaystyle 0<\lambda^{\prime}<\varphi_{x_{0}}(-e_{d})\iff-x_{0}\cdot
e_{d}<\lambda<\varphi(-e_{d})\,.$

By assumption, $\lambda<\varphi(-e_{d})$. By choosing an element $\bar{x}$
with maximal last component in the compact set $W_{\varphi}$ and observing
that $e_{d}\in\mathcal{N}_{W_{\varphi}}(\bar{x})$, with the help of
Proposition 2.2(5) we also get

$\displaystyle-\max\\{x_{0}\cdot e_{d}\colon x_{0}\in
W_{\varphi}\\}=-\bar{x}\cdot e_{d}=-\varphi(e_{d})\,.$

Since $\lambda>-\varphi(e_{d})$, there exists
$x_{0}\in\mathrm{int}\,(W_{\varphi})$ such that $-x_{0}\cdot e_{d}<\lambda$,
too, as claimed. Next, we show (3.5)(ii). To this end, note that
$(\varphi_{x_{0}})_{\lambda^{\prime}}(\nu)=\varphi_{\lambda}(\nu)-x_{0}\cdot\nu$
for all $\nu\in\mathbb{S}^{d-1}$ since

$(\varphi_{x_{0}})_{\lambda^{\prime}}(\nu)=\left\\{\begin{aligned}
\varphi_{x_{0}}(\nu)&&=&&\varphi(\nu)-x_{0}\cdot\nu&&=&&\varphi_{\lambda}(\nu)-x_{0}\cdot\nu&&\leavevmode\nobreak\
\text{ if }\nu\neq-e_{d}\,,\\\ \lambda^{\prime}&&=&&\lambda-
x_{0}\cdot(-e_{d})&&=&&\varphi_{\lambda}(\nu)-x_{0}\cdot\nu&&\leavevmode\nobreak\
\text{ if }\nu=-e_{d}\,.\end{aligned}\right.$

Now, by the Gauss-Green Theorem for sets of finite perimeter, for $E$ a set of
finite perimeter we obtain

$\displaystyle P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)$
$\displaystyle=\int_{\partial^{*}E}(\varphi_{x_{0}})_{\lambda^{\prime}}(\nu)\,\mathrm{d}\mathcal{H}^{d-1}=\int_{\partial^{*}E}\varphi_{\lambda}(\nu)-x_{0}\cdot\nu\,\mathrm{d}\mathcal{H}^{d-1}$
$\displaystyle=\int_{\partial^{*}E}\varphi_{\lambda}(\nu)\,\mathrm{d}\mathcal{H}^{d-1}-\int_{E}\mathrm{div}(x_{0})\,\mathrm{d}x=\int_{\partial^{*}E}\varphi_{\lambda}(\nu)\,\mathrm{d}\mathcal{H}^{d-1}=P_{\varphi_{\lambda}}(E)\,,$

where we used that $\mathrm{div}(x_{0})=0$. To see (3.6)(i), it suffices to
note that

$\displaystyle x\in W_{\varphi}$ $\displaystyle\iff
x\cdot\nu\leq\varphi(\nu)\quad\forall\,\nu\in\mathbb{S}^{d-1}\iff(x-x_{0})\cdot\nu\leq\varphi(\nu)-x_{0}\cdot\nu\quad\forall\,\nu\in\mathbb{S}^{d-1}$
$\displaystyle\iff(x-x_{0})\cdot\nu\leq\varphi_{x_{0}}(\nu)\quad\forall\,\nu\in\mathbb{S}^{d-1}\iff
x-x_{0}\in W_{\varphi_{x_{0}}}\,.$

In order to prove (3.6)(ii), Note that

$\displaystyle W_{\lambda^{\prime},\varphi_{x_{0}}}$ $\displaystyle=\\{x\in
W_{\varphi_{x_{0}}}\colon x_{d}\geq-\lambda^{\prime}\\}=\\{x\in
W_{\varphi_{x_{0}}}\colon x_{d}\geq-\lambda-x_{0}\cdot e_{d}\\}$
$\displaystyle=\\{x\in
W_{\varphi}-x_{0}\colon(x+x_{0})_{d}\geq-\lambda\\}=\\{x\in W_{\varphi}\colon
x_{d}\geq-\lambda\\}-x_{0}=W_{\lambda,\varphi}-x_{0}\,.$

Therefore,

$\displaystyle W_{\lambda^{\prime},\varphi_{x_{0}}}(1)$
$\displaystyle=|W_{\lambda^{\prime},\varphi_{x_{0}}}|^{-1/d}(W_{\lambda^{\prime},\varphi_{x_{0}}}+\lambda^{\prime}e_{d})=|W_{\lambda,\varphi}|^{-1/d}(W_{\lambda,\varphi}-x_{0}+(\lambda+x_{0}\cdot
e_{d})e_{d})$
$\displaystyle=|W_{\lambda,\varphi}|^{-1/d}(W_{\lambda,\varphi}+\lambda\,e_{d}-x_{0}+(x_{0}\cdot
e_{d})e_{d})=W_{\lambda,\varphi}(1)+\tau\,,$

where $\tau=|W_{\lambda,\varphi}|^{-1/d}(-x_{0}+(x_{0}\cdot e_{d})e_{d})\in
H$. ∎

We are now in position to prove the main Theorems.

###### Proof of Theorem 2.3.

By Remark 2.1 it suffices to prove the theorem only for $v=1$. To this end,
let $E\subseteq H^{+}$ be such that $|E|=1$. We choose
$x_{0}\in\mathrm{int}\,(W_{\varphi})$ and $\lambda^{\prime}=\lambda+x_{0}\cdot
e_{d}$ such that the assertions of Lemma 3.3 hold true. Using Lemma
3.1,(3.5)(ii), and (3.6)(ii) we obtain

$\displaystyle F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))$
$\displaystyle=P_{\varphi_{\lambda}}(W_{\lambda,\varphi}(1))=P_{\varphi_{\lambda}}(W_{\lambda^{\prime},\varphi_{x_{0}}}(1))=P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(W_{\lambda^{\prime},\varphi_{x_{0}}}(1))$

and, for any set $E$ of finite perimeter,

$\displaystyle F_{\lambda,\varphi}(E)$ $\displaystyle\geq
P_{\varphi_{\lambda}}(E)=P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)\,.$

Finally applying Lemma 3.2 to $\varphi_{x_{0}}$ and the positive
$\lambda^{\prime}$ gives
$P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(W_{\lambda^{\prime},\varphi_{x_{0}}}(1))\leq
P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)$ if $|E|=1$, and we have shown

$\displaystyle F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))\leq
F_{\lambda,\varphi}(E)$

for such $E$. In case of equality, we also have
$P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(W_{\lambda^{\prime},\varphi_{x_{0}}}(1))=P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)$
and Lemma 3.2 implies
$E=W_{\lambda^{\prime},\varphi_{x_{0}}}(1)+\tau^{\prime}=W_{\lambda,\varphi}(1)+\tau+\tau^{\prime}$
for $\tau$ as above and some $\tau^{\prime}\in H^{+}\cup H$. As also
$P_{\varphi_{\lambda}}(E)=F_{\lambda,\varphi}(E)$, necessarily
$\tau+\tau^{\prime}\in H$. This concludes the proof. ∎

We are now in position to prove Theorem 2.5 establishing the stability of the
Winterbottom shape.

###### Proof of Theorem 2.5.

Let $x_{0}\in\mathrm{int}(W_{\varphi})$ and
$\lambda^{\prime}=\lambda+x_{0}\cdot e_{d}$ be such that the assertions of
Lemma 3.3 hold true. By properly rescaling we may without loss of generality
assume $|E|=1$. In the following, we set $W=W_{\lambda,\varphi}(1)$ and
$W^{\prime}=W_{\lambda^{\prime},\varphi_{x_{0}}}(1)$. We first want to prove
that there exists $\tau\in\mathbb{R}^{d}$ such that

$\displaystyle|E\triangle(W+\tau)|^{2}\leq
C(F_{\lambda,\varphi}(E)-F_{\lambda,\varphi}(W))\,.$ (3.7)

The quantitative isoperimetric inequality [7, Theorem 1.1] for the
unconstrained functional $P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}$
yields $C>0$ such that for a suitable $\tau^{\prime}\in\mathbb{R}^{d}$ we have

$\displaystyle|E\Delta(W^{\prime}+\tau^{\prime})|^{2}\leq
C\left(P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(E)-P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(W^{\prime})\right)\,.$

Recalling that, by (3.6)(ii), there holds $W=W^{\prime}+\hat{\tau}$ for some
$\hat{\tau}\in H$ and setting $\tau=\tau^{\prime}-\hat{\tau}$, we obtain

$\displaystyle|E\Delta(W+\tau)|^{2}=|E\Delta(W^{\prime}+\tau^{\prime})|^{2}\leq
C\left(P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(E)-P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(W^{\prime})\right)\,.$
(3.8)

Now, by Lemma 3.1, Lemma 3.3, and, as $\lambda^{\prime}>0$, by Lemma 3.2 we
have

$\displaystyle P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(E)\leq
P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(E)=P_{\varphi_{\lambda}}(E)\leq
F_{\lambda,\varphi}(E)$

and

$\displaystyle
P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(W^{\prime})=P_{(\varphi_{x_{0}})_{\lambda^{\prime}}}(W^{\prime})=P_{\varphi_{\lambda}}(W^{\prime})=P_{\varphi_{\lambda}}(W)=F_{\lambda,\varphi}(W)\,,$

where we again used that $W=W^{\prime}+\hat{\tau}$ with $\hat{\tau}\in H$.
Therefore

$\displaystyle
P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(E)-P_{(\varphi_{x_{0}})_{\lambda^{\prime}}^{**}}(W^{\prime})\leq
F_{\lambda,\varphi}(E)-F_{\lambda,\varphi}(W)\,.$

Together with (3.8), this yields (3.7).

The challenge that remains is to show that $\tau$ can be chosen in $H$, i.e.,
such that $\tau_{d}=0$. In order to do that we distinguish cases.

Case 1. Assume (3.7) holds true with $\tau_{d}<0$. By convexity of $W$ we have

$\displaystyle|E\triangle(W+\tau)|\geq|\\{x\in\tau+W\colon
x_{d}<0\\}|=|\\{x\in W\colon x_{d}<-\tau_{d}\\}|\geq c\min\\{-\tau_{d},1\\}$

for a constant $c>0$. Since also

$\displaystyle|(W+\tau)\triangle(W+\tau-\tau_{d}e_{d})|\leq
C\min\\{-\tau_{d},1\\}\,$

(note that trivially $|(W+\tau)\triangle(W+\tau-\tau_{d}e_{d})|\leq 2|W|\leq
C$), we find that

$\displaystyle|E\triangle(W+\tau-\tau_{d}e_{d})|\leq|E\triangle(W+\tau)|+|(W+\tau)\triangle(W+\tau-\tau_{d}e_{d})|=(1+Cc^{-1})|E\triangle(W+\tau)|\,.$

This implies the claim in view of (3.7).

Case 2. Assume (3.7) holds true with $\tau_{d}>0$. In the following we denote
by

$W_{\varphi}(v)=\left(\frac{v}{|W_{\varphi}|}\right)^{\frac{1}{d}}W_{\varphi}$

the rescaled Wulff set of $\varphi$ with volume $v>0$. Since in the partial
drying/wetting regime $W_{\varphi}(1)$ (placed in $H^{+}$) is not optimal for
$F_{\lambda,\varphi}$, we have
$P_{\varphi}(W_{\varphi}(1))>F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))$ and
so we can define a positive constant $c_{0}$ by asking that

$\displaystyle\left(1-c_{0}\right)^{\frac{d-1}{d}}P_{\varphi}(W_{\varphi}(1))-(\varphi(e_{d})+\varphi(-e_{d}))c_{0}=F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))+\delta\,,$

where
$\delta:=\frac{1}{2}(P_{\varphi}(W_{\varphi}(1))-F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))>0$.

Case 2a. Suppose that $|\\{x\in E\colon x_{d}<\tau_{d}\\}|\geq
c_{0}\min\\{\tau_{d},1\\}$. Then

$\displaystyle|E\triangle(W+\tau)|\geq c_{0}\min\\{\tau_{d},1\\}$

and, similarly as above, from

$\displaystyle|(W+\tau)\triangle(W+\tau-\tau_{d}e_{d})|\leq
C\min\\{\tau_{d},1\\},$

we conclude

$\displaystyle|E\triangle(W+\tau-\tau_{d}e_{d})|\leq|E\triangle(W+\tau)|+|(W+\tau)\triangle(W+\tau-\tau_{d}e_{d})|=(1+Cc_{0}^{-1})|E\triangle(W+\tau)|\,.$

Again this finishes the proof with the help of (3.7).

Case 2b. Assume that $|\\{x\in E\colon
x_{d}<\tau_{d}\\}|<c_{0}\min\\{\tau_{d},1\\}$. Then there is an
$\varepsilon\in(0,\tau_{d})$ such that

$\mathcal{H}^{d-1}(E\cap(\varepsilon e_{d}+H))\leq
c_{0}\min\\{1,\tau_{d}^{-1}\\}.$

We cut along this hyperplane and set $\tilde{E}=\\{x\in
E:x_{d}>\varepsilon\\}$. With this set we then get

$F_{\lambda,\varphi}(E)\geq
F_{\lambda,\varphi}(E\setminus\tilde{E})+P_{\varphi}(\tilde{E})-(\varphi(e_{d})+\varphi(-e_{d}))\mathcal{H}^{d-1}(E\cap(\varepsilon
e_{d}+H)).$

By Remark 2.4(2), the first term on the right hand side is non-negative and we
get the bound

$\displaystyle F_{\lambda,\varphi}(E)$ $\displaystyle\geq
P_{\varphi}(\tilde{E})-(\varphi(e_{d})+\varphi(-e_{d}))c_{0}\min\\{1,\tau_{d}^{-1}\\}$
$\displaystyle\geq
P_{\varphi}(W_{\varphi}(|\tilde{E}|))-(\varphi(e_{d})+\varphi(-e_{d}))c_{0}$
$\displaystyle=|\tilde{E}|^{\frac{d-1}{d}}P_{\varphi}(W_{\varphi}(1))-(\varphi(e_{d})+\varphi(-e_{d}))c_{0}\,,$

where in the second step we have used that the Wulff shape
$W_{\varphi}(|\tilde{E}|)$ minimizes the unconstrained functional
$P_{\varphi}$ among sets of volume $|\tilde{E}|$. As by assumption
$|\tilde{E}|\geq 1-c_{0}$, it follows that

$\displaystyle F_{\lambda,\varphi}(E)$
$\displaystyle\geq\left(1-c_{0}\right)^{\frac{d-1}{d}}P_{\varphi}(W_{\varphi}(1))-(\varphi(e_{d})+\varphi(-e_{d}))c_{0}\geq
F_{\lambda,\varphi}(W_{\lambda,\varphi}(1))+\delta$ (3.9)

by our choice of $c_{0}$. Thus, for any $C\geq 4\delta^{-1}$ we have even for
$\tau=0$

$|E\triangle W|^{2}\leq(|E|+|W|)^{2}=4\leq
C(F_{\lambda,\varphi}(E)-F_{\lambda,\varphi}(W)).\qed$

## Acknowledgements

The research of LK was supported by the DFG through the Emmy Noether Programme
(project number 509436910).

## References

  * [1] L. Ambrosio, N. Fusco, D. Pallara.  Functions of bounded variation and free discontinuity problems.  Oxford University Press  (2000).
  * [2] T. Bonnesen.  Sur une amélioration de l’inégalité isopérimetrique du cercle et la démonstration d’une inégalité de Minkowski.  C. R. Acad. Sci. Paris  172 (1921), 1087–1089.
  * [3] M. Cicalese, G. P. Leonardi.  Selection principle for the sharp quantitative isoperimetric inequality.  Arch. Ration. Mech. Anal.  206(2) (2012), 617–643.
  * [4] E. Davoli, P. Piovano.  Analytical validation of the Young-Dupré law for epitaxially-strained thin films.  Math. Models Methods Appl. Sci.  29 (2019), 2183–2223.
  * [5] A. Dinghas.  Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen.  Zeitschrift für Kristallographie  105 (1944), 304–314.
  * [6] L. C Evans, R. F. Gariepy.  Measure theory and fine properties of functions.  CRC Press, Boca Raton $\cdot$ London $\cdot$ New York $\cdot$ Washington, D.C. 1992.
  * [7] A. Figalli, F. Maggi, A. Pratelli.  A mass transportation approach to quantitative isoperimetric inequalities.  Invent. Math.  182(1) (2010), 167–211.
  * [8] I. Fonseca.  The Wulff theorem revisited.  Proc. R. Soc. Lond. 432  432 (1991), 125–145.
  * [9] I. Fonseca, S. Müller.  A uniqueness proof for the Wulff theorem.  Proc. Roy. Soc. Edinburgh Sect.  119 (1991), 125–136.
  * [10] N. Fusco, F. Maggi, A. Pratelli.  The sharp quantitative isoperimetric inequality.  Ann. Math.  (2) 168(3) (2008), 941–980.
  * [11] C. Herring.  Some theorems on the free energies of crystal surfaces.  Phys. Rev.  82 (1951), 87–93.
  * [12] G. De Philippis, F. Maggi.  Regularity of free boundaries in anisotropic capillarity problems and the validity of Young’s law.  Arch. Rat. Mech. Anal.  216(2) (2015) 473–568.
  * [13] R. Kotecky, C. Pfister.  Equilibrium shapes of crystals attached to walls.  J. Stat. Phys.]  76 (1994) 419–446.
  * [14] J. E. Taylor.  Existence and structure of solutions to a class of nonelliptic variational problems  Sympos. Math.  14 (1974), 499–508.
  * [15] J. E. Taylor.  Unique structure of solutions to a class of nonellipic variational problems  Proc. Sympos. Pure Math.  27 (1975), 419–427.
  * [16] J. E. Taylor.  Crystalline variational problems.  Bull. Amer. Math. Soc.  84 (1978), 568–588.
  * [17] W. L. Winterbottom.  Equilibrium shape of a small particle in contact with a foreign substrate.  Acta Metallurgica  15 (1967), 303–310.
  * [18] G. Wulff.  Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen.  Z. Krist.  34 (1901), 449–530.
  * [19] T. Young.  An essay on the cohesion of fluids.  Philos. Trans. Roy. Soc. London  95 (1805), 65–87.