数理解析研究所講究録
第 1842 巻 2013 年 61-66

61

On a volume-preserving free boundary problem
Karel \v{S}vadlenka(Kanazawa University, Graduate School of Natural Science and Technology)

We are interested in the motion of a membrane that is in contact with a rigid plane. In this report, westate some important previous results, explain our new results and present several related open problems.

In many cases, the membrane is described by some partial differential equation (such as heat equation)and on the free boundary (points where the membrane touches the plane) a contact angle condition isprescribed which originates in the physical properties of the materials in contact (i.e., surface tensions. typical example of such free boundary condition is Young’s equation$\gamma,$$\gamma_{SV)}\gamma_{SL})$$A$$\gamma\cos\theta=\gamma_{SV}-\gamma_{SL}.$A pioneering beautiful paper related to this phenomenon by Alt and Caffarelli (1981) [1] deals withthe stationary problem$\triangle u=0$

in $\Omega\cap\{u>0\},$$|\nabla u|=Q,$

$u=0$

on

$\Omega\cap\partial\{u>0\}.$

The authors study the functional$\int_{\Omega}(|\nabla u|^{2}+Q^{2}\chi_{u>0})dx,$

is characteristic function of the set $\{u>0\}=\{x\in\Omega;u(x)>0\}$ , and show that it possesses
minima which are Lipschitz continuous and have linear growth away from the free boundary. For suchharmonic functions they find the representation formulaand show that the minima
are weak solutions, while the free boundary is a smooth surface except of a set of zero $(n-1)$ -dimensional
Hausdorff measure. Of course, the smoothness depends on the smoothness of the datum : for example,
if is H\"older continuous then the function whose graph determines locally the shape of the free boundaryhas H\"older continuous first derivatives.
On the other hand, Caffarelli and V\’azquez (1995) [2] studied the evolutionary problem
where$\chi_{u>0}$$\triangle u=q_{u}\mathcal{H}^{n-1}L_{\partial\{u>0\}}$

$Q$

$Q$

$u_{t}-\triangle u=0$$in$$\{u>0\},$$|\nabla u|=1,$$u=0$$on$ $\partial\{u>0\}$by a different technique. They regularize the problem by adding an absorption term in the following way$u_{t}^{\epsilon}- \triangle u^{\epsilon}=-\frac{1}{2}\chi_{\epsilon}’(u^{\epsilon}) , u^{\epsilon}\geq 0.$

Here,$\chi_{\epsilon}$is an appropriate smoothing of the characteristic function in the interval$(0, \epsilon)$:62

They show uniform estimates for the solution of the regularized equation (Lipschitz in space andH\"older in time) and use them to construct a weak solution of the original problem. The regularity of freeboundary is also studied in case of shrinking support. The above results were extended and generalized
by several researchers later on.
We are interested in the study of evolutionary problem with $vo$ lume constraint$\int_{\Omega}u(t, x)dx=V \forall t,$which appears, for example, in the free boundary problem modelling the motion of bubbles or droplets
on a surface (see [6]).

The full model equation is$\chi_{u>0}\beta u_{tt}+\mu u_{t}=\Delta u-\gamma\chi_{\epsilon}’(u)+\chi_{u>0}\lambda_{\epsilon}(u)$in$\Omega\cross(0, T)$,(1)

where describes the shape of the bubble under the assumption that it can be represented as graph ofscalar function. It is derived from the surface energy functional$u$$\gamma_{g}\int_{\Omega}\sqrt{1+|\nabla u|^{2}}\chi_{u>0}dx+\int_{\Omega}\gamma_{s}\chi_{u>0}dx\approx\frac{\gamma_{g}}{2}\int_{\Omega}|\nabla u|^{2}dx+\int_{\Omega}\gamma_{s}\chi_{\epsilon}(u)dx$

by applying Hamilton’s principle and taking into account the constraint and the presence of obstacle.
The second time derivative term has a degenerating coefficient (see [17]) andis a function of time only
representing a Lagrange multiplier for the volume constraint.
If we minimize the unapproximated surface energy under volume constraint, we discover that the, which is identical to Young’s equation. However, if thestationary contact angle satisfies
shape of the membrane evolves in time, there is no generally accepted physical theory for the dynamic
contact angle. The advantage of our model is the fact that the dynamic contact angle is not prescribedbut is determined implicitly from other data of the model.
Based on the mathematical analysis described below, a numerical scheme for the computation of thisequation has been developed and simulation of various phenomena was attempted (see, e.g., [6] [10] [16][8] .
$\lambda_{\epsilon}$$\theta$$\gamma_{s}=-\gamma_{g}\cos\theta$$)$To start the mathematical analysis of the above model equation, we consider the following parabolic
problem:$u_{t}-\Delta u=\lambda$

in $\{u>0\},$$|\nabla u|^{2}=2\gamma$

on $\partial\{u>0\}$63
The regularized version is$in$$u_{t}=\triangle u-\gamma\chi_{\epsilon}’(u)+\chi_{u>0}\lambda_{\epsilon}$$(0, T)\cross\Omega,$where$\lambda_{\epsilon}=\int_{\Omega}[u_{t}u+|\nabla u|^{2}+\gamma\chi_{\epsilon}’(u)u]dx$

is the nonlocal Lagrange multiplier coming from the volume constraint. Notice that the second derivative
term in the original model has been neglected, which can be interpreted as considering relatively slow
motion.
One can see by maximum principle that solutions to the regular problem are nonnegative because theright-hand side vanishes for negative values of . This is the mathematical reason for multiplying thenonlocal term by characteristic function. The physical reason is that the outer force or source representing$u$the volume constraint should not act on the region where the solution vanishes.
In the regularized problem the volume constraint gives rise to an obstacle-type problem with a nonlocal
is expectedobstacle function depending on the solution. Accordingly, the sharp contact angle limit
to have two factors influencing the behaviour on the free boundary: the stronger linear growth due tocontact angle condition and the quadratic growth (curvature) originating in the volume constraint.
With the view of numerical approximation and because of the presence of the global constraint weanalyse the regularized obstacle problem by a minimization method, where time variable is discretizedand the functional(2)$\epsilonarrow 0$$J_{n}(u)= \int_{\Omega}(\frac{|u-u_{n-1}|^{2}}{2h}+\frac{1}{2}|\nabla u|^{2}+\gamma\chi_{\epsilon}(u))dx$

is minimized. Here$h$is the time step and $u_{n-1}$ refers to the minimizer on the previous time level. The, whereis given as initial datum. This gives
and minimizers
on the following time levels are computed inductively. Finally, minimizers are interpolated in time asin the following picture to obtain functions
and :computation starts from minimizing$J_{1}$$u_{1}$$u_{0}$$u_{n}$$u^{h}$$\overline{u}^{h}$$uA:$$\overline{u}^{h}$:::$rightarrow u2$$-::.:$$ul rightarrow$$rightarrow u_{3}$$u4$$\bullet:^{u_{0}}$$t$$::^{u_{0}}$:$0:$:$t$$s\cdots\cdots\cdots\cdots|\cdots\cdots\cdots\cdot\cdot\{\cdots\cdots\cdots\cdots|\cdots\cdots\cdots\cdot\cdot\{\cdots\cdots\cdots\cdots\prime |\cdots\cdots\cdots\cdots\}\cdots\cdots\cdots\cdot\cdot\{\cdots\cdots\cdots\cdots I\cdots\cdots\cdots\cdot\cdot i\cdots\cdots\cdots\cdots\sim$

:

$0:$

If we minimize
satisfy$J_{n}$in$h$$H_{0}^{1}$

$2h$

$3h$

$4h$

$h$

$2h$

$3h$

$4h$

, it is easy to find from the first variation that the interpolated functions$u_{t}^{h}=\triangle\overline{u}^{h}-\gamma\chi_{\epsilon}’(\overline{u}^{h})$in the weak sense and are, therefore, candidates for approximate solutions. However, in the case ofproblems with volume constraint we have to restrain the space of functions admissible for minimization.
This was done together with regularity analysis in the paper [11] for parabolic problems and in the paper[12] for hyperbolic problems.

Here we have an additional constraint represented by the obstacle and hence we define a special
constrained space(3)$\mathcal{K}^{\delta}=\{u\in H_{0}^{1}(\Omega) ; \int_{\Omega}\chi_{\delta}(u)udx=V\}$

as the admissible space for minimization.

64
The characteristic function in the constraint of the admissible space is essential in order to satisfy
the obstacle condition, while the regularization thereof is necessary to obtain an equality from the firstvariation. Indeed, the minimizers are shown to exist and be nonnegative. The weak solution is then(see [13] for details).
constructed by deriving uniform estimates in and and taking
It is to be noted that this minimization approach avoids direct treatment of the complicated nonlocal
term , naturally discards incorrect solutions mentioned in [2] and provides a theoretical backgroundfor numerical computation of this type of problems.
We show here two steps from the existence proof, namely the existence of minimizers and theirsuch that
nonnegativity. Since the functional (2) is nonnegative, there is a minimizing sequence, there is a subsequence converging weakly. As this sequence is bounded ininand strongly into some function. However, here we have to assume thatdomain is bounded. From the weak lower semicontinuity ofinone can say thatis aminimizer, if it belongs todefined in (3). Therefore, we compute$h$$h,$ $\deltaarrow 0$$\delta$$\lambda_{\epsilon}$$\{u^{k}\}$$H^{1}(\Omega)$$J_{n}(u^{k}) \downarrow\inf_{u\in \mathcal{K}^{\delta}}J_{n}(u)$$H^{1}(\Omega)$$u_{n}\in H^{1}(\Omega)$$L^{2}(\Omega)$$\Omega$$J_{n}$$H^{1}(\Omega)$$u_{n}$$\mathcal{K}^{\delta}$$| \int_{\Omega}\chi_{\delta}(u)udx-V| = |\int_{\Omega}(\chi_{\delta}(u)u-\chi_{\delta}(u^{k})u^{k})dx|$$\leq$$\int_{\Omega}|\frac{d}{du}(\chi_{\delta}(u)u)|_{u=\overline{u}}|u-u^{k}|dxarrow 0$

as$karrow\infty.$

To show that minimizers are nonnegative a.e., let us assume that a minimizer
is negative on a setof positive measure and define a new function
by. Then it is easy to check thatbelongs towhich is in contradiction with minimality under the condition thatHowever,fulfills the constraint because the smoothing of characteristic function causes that only
positive values of given function are taken into account and thus ”cutting off‘ negative part as in thecase of does not change the fact that the constraint is satisfied.
The analysis for the sharp contact angle limit
is yet to be done. Yamaura [15] constructed-generalized minimizing movement corresponding to the considered energy without taking into accountthe volume constraint. It is expected that a similar technique would basically work for the constrained
problem. However, there is a problem closely related to the global constraint. Specifically, if we takein order to use Bemstein’s technique to show uniform Lipschitz continuity which is indispensablefor the existence proof, we are not able to proof the existence of minimizers as the above presented proof
fails, i.e., the volume of the minimizing functions
may “leak out to infinity”. On the other hand, if
is taken bounded, the application of Bernstein’s method becomes difficult.
Since in the present $mo$ del the contact angle cannot be larger than right angle, our future plan is toextend the theory from scalar functions to hypersurfaces. The goal is to rigorously derive the motion ofa hypersurface according to a model equation with a given contact angle on the obstacle. To this end,
we plan to consider the application of phase-field approximation, where the hypersurface is constructed
as the limit of a layer between regions where a function is identically equal to and 1, for example.
Specifically, to obtain the mean curvature flow, the following energy is considered:$u_{n}$$\tilde{u}_{n}$$\tilde{u}_{n}=u_{n}\chi_{u_{n}>0}$$\mathcal{K}^{\delta}.$$J_{n}(\tilde{u}_{n})<J_{n}(u_{n})$$\tilde{u}_{n}$$\tilde{u}_{n}$$\tilde{u}_{n}$$L^{2}$$\epsilonarrow 0$$\Omega=\mathbb{R}^{m}$$u^{k}$$\Omega$$u$$0$$\int_{\Omega}(\epsilon^{2}|\nabla u|^{2}+W(u))dx+\int_{\partial\Omega}\epsilon\sigma(u)d\mathcal{H}^{n-1},$

where is a small parameter corresponding to the width of the layer, $W$ is a double-well potential withminima at and 1 and is a function describing contact energy. In order to prove that the limit of
the layer as
is a smooth hypersurface, the foremost task is to prepare a parabolic monotonicity
formula holding up to the boundary. This might be possible if one consults the ideas regarding interior
parabolicity formula derived in [4] and stationary boundary formula given in [14].
$\epsilon$$0$$\sigma$$\epsilonarrow 0$

65
An obvious future task is to analyze the full hyperbolic model equation (1). Extracting a part of thefeatures of the model equation we obtain slightly simpler problems. One of them is the equation$\chi_{u>0}u_{tt}+\alpha u_{t}=\Delta u$related in [17] to the vibration of a string with obstacle. Another similar problem$u_{tt}-\triangle u=0$$in$$\{u>0\},$$|\nabla u|^{2}-u_{t}^{2}=Q^{2}$$on$ $\partial\{u>0\}$describes the peeling of a tape from a plane. The classical analysis of this problem is given in [9] andinteresting numerical results were reported in [5]. Both problems were solved to some extent only in spacedimension one, the same being true for the analysis of the model equation (1) in [3]. However, there issome doubt whether these results touch the core of the problems since in dimension one it is possible touse special tools such as Sobolev imbedding theorem or D’Alembert’s formula.
Another challenging task is the analysis of contact problems arising, e.g., in the modelling of collisionof elastic curves with an obstacle. One such example is mentioned in [7].

References[1] H. W. Alt, L. A. Caffarelli: Existence and regularety for a minimum problem with free boundary, J.

Reine Angew. Math. 325 (1981), 105-144.
[2] L. A. Caffarelli, J. L. V\’azquez: A free-boundary problem for the heat equation
propagation, Trans. Amer. Math. Soc. 347 (1995), 411-441.

$ar\cdot\iota sing$in flame[3] E. Ginder, K. Svadlenka: A variational approach to a constrained hyperbolic free boundary problem,

Nonlinear Anal., Theory Methods Appl., 71/12 (2009), 1527-1537.
[4] T. Ilmanen: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J.
Differential Geom. 38/2 (1993), 417-461.

[5] H. Imai, K. Kikuchi, K. Nakane, S. Omata, T. Tachikawa: A numencal approach to the asymptotic

behaviour of solutions of a one-dimensional free boundary problem of hyperbolic type, Japan J. Indust.
Appl. Math. 18 (2001), 43-58.
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Sciences and Application 29 (2008), 383-398.
[7] M. Kazama: Doctor thesis, Kanazawa University (2010).
[8] M. Kazama, S. Omata: Modeling and computation of fluid-membrane interaction, Nonlinear Analysis

71 (2009), e1553-e1559.
[9] K, Kikuchi, S. Omata: A free boundary problem for a one dimensional hyperbolic equation, Adv.
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66[10] S. Omata, M. Kazama, H. Nakagawa: Variational approach to evolutionary free boundary problems,Nonlinear Analysis 71 (2009), e1547-e1552.
[11] K. Svadlenka, S. Omata: Construction of solutions to heat-type problems with volume constraint via
the discrete Morse flow, Funkc. Ekvacioj 50/2 (2007), 261-285.
[12] K. Svadlenka, S. Omata: Mathematical modelling of surface vibration with volume constmint and itsanalysis, Nonlinear Anal., Theory Methods Appl. 69/9 (2008), 3202-3212.

[13] K. Svadlenka, S. Omata: Mathematical analysis of a constrained pambolic free boundary problem
descmbing droplet motion on a surface, Indiana Univ. Math. J. 58 (2009), 2073-2102.
[14] Y. Tonegawa: Domain dependent monotonicity fomula for a singular perturbation problem, Indiana
Univ. Math. J. 52 (2003), 69-83.
[15] Y. Yamaura, preprint.
[16] T. Yamazaki, S. Omata, K. Svadlenka, K. Ohara: Construction of approximate solution to a hyperbolic free boundary problem with volume constraint and its numem cal computation, Adv. Math. Sci.
Appl. 16/1 (2006), 57-67.

[17] H. Yoshiuchi, S. Omata, K. Svadlenka, K. Ohara: Numemcal solution of film vibration with obstacle,
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