Svadlenka, Karel

EPrint Series of Department of Mathematics, Hokkaido University

Analysis of the motion of a membrane touching a solid plane (abstract)
Karel Svadlenka
The 35th Sapporo Symposium
on Partial Dierential Equations
Hokkaido University, August 23, 2010
We are interested in the motion of a membrane that is in contact with a rigid plane.
γ
γSL
γSV
θ
In many cases, the membrane is described by some partial dierential equation (such as heat equation) and
on the free boundary (points where the membrane touches the plane) a contact angle condition is prescribed
which originates in the physical properties of the materials in contact (i.e., surface tensions ; SV ; SL).
A pioneering beautiful paper on the mathematical aspect of the problem by Alt and Caarelli (1981) deals
with the stationary case
u = 0 in 
 \ fu > 0g; jruj = Q; u = 0 on 
 \ @fu > 0g:
They study the functional Z


 jruj2 +Q2u>0 dx
and show that it possesses minima which are Lipschitz continuous and have linear growth away from the free
boundary. For such harmonic functions they nd a representation formula and show that the minima are weak
solutions, while the free boundary is a smooth surface except of a set of zero (n   1)-dimensional Hausdor
measure.
On the other hand, Caarelli and Vazquez (1995) studied the evolutionary problem
ut  u = 0 in fu > 0g; jruj = 1; u = 0 on @fu > 0g
by a dierent technique. They regularize the problem by adding an absorption term in the following way
u"t  u" =   120"(u"); u"  0:
Here, " is a smoothing of the characteristic function in the interval (0; "). The authors show uniform estimates
for the solution of the regularized equation (Lipschitz in space and Holder in time) and use them to construct
a weak solution of the original problem. They also study the regularity of free boundary in case of shrinking
support.
We are interested in the study of the evolutionary problem with volume constraintZ


u(t; x) dx = V 8t;
which appears, for example, in the free boundary problem modelling the motion of bubbles or droplets on a
surface. The problem becomes
ut  u =  in fu > 0g; jruj2 = 2 on @fu > 0g
1
and its regularized version is
ut = u  0"(u) + u>0" in (0; T ) 
; where " =
Z



utu+ jruj2 + 0"(u)u

dx:
Here  (or ") are nonlocal terms coming from the volume constraint.
In the regularized problem the volume constraint gives rise to an obstacle-type problem with a nonlocal
obstacle function. Accordingly, the sharp contact angle limit "! 0 is expected to have two factors inuencing
the behaviour on the free boundary: the stronger linear growth due to contact angle condition and the weaker
quadratic growth (curvature) originating in the volume constraint.
With the view of numerical approximation and because of the presence of the global constraint we analyse
the regularized obstacle problem by a minimization method introduced by K. Rektorys and developed by N.
Kikuchi. In this method time variable is discretized and the functional
Jn(u) =
Z


 ju  un 1j2
2h
+
1
2
jruj2 + "(u)

dx
is minimized. Here we dene a special constrained space
K = u 2 H10 (
) ; Z


(u)u dx = V
	
as the admissible space for minimization.
The (regularization of) characteristic function in the admissible space is essential in order to satisfy the
obstacle condition. Indeed, the minimizers are shown to exist and be nonnegative. The weak solution is then
constructed by deriving uniform estimates in h and  and taking h;  ! 0. (See, Svadlenka & Omata, 2009 for
details.)
The analysis for the sharp limit "! 0 is yet to be done. Yamaura constructed L2 - generalized minimizing
movement corresponding to the considered energy without taking into account the volume constraint. It is
expected that a similar technique will basically work for the constrained problem.
Our future plan is to consider the application of phase-eld approximation to the contact angle problem.
The phase-eld method is superior to the above scalar approach in the sense that it addresses surfaces and
can therefore express contact angles larger than right angle. It is assumed that one can derive a boundary
parabolic monotonicity formula (see [5]) and thus rigorously construct a hypersurface evolving according to its
mean curvature with a prescribed contact angle on the rigid boundary.
Another challenging task is the contact problem arising, e.g., in the modelling of collision of elastic curves
with an obstacle. For this phenomenon, there are numerical results but a suitable mathematical approach to
this hyperbolic free boundary problem is still unknown.
References
[1] H. W. Alt, L. A. Caarelli: Existence and regularity for a minimum problem with free boundary, J. Reine
Angew. Math. 325 (1981), 105-144.
[2] L. A. Caarelli, J. L. Vazquez: A free-boundary problem for the heat equation arising in ame propagation,
Trans. Amer. Math. Soc. 347 (1995), 411-441.
[3] K. Svadlenka, S. Omata: Mathematical analysis of a constrained parabolic free boundary problem describing
droplet motion on a surface, Indiana Univ. Math. J. 58 (2009), 2073-2102.
[4] Y. Yamaura, preprint.
[5] Y. Tonegawa: Domain dependent monotonicity formula for a singular perturbation problem, Indiana Univ.
Math. J. 52 (2003), 69-83.
2


Analysis of the motion of a membrane touching a solid plane

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EPrint Series of Department of Mathematics, Hokkaido University