Svadlenka, Karel

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TitleOn a volume-preserving free boundary problem (GeometricAspect of Partial Differential Equations and ConservationLaws)Author(s)Svadlenka, KarelCitation数理解析研究所講究録 (2013), 1842: 61-66Issue Date2013-07URL http://hdl.handle.net/2433/194993RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn a volume-preserving free boundary problemKarel \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$\gamma,$ $\gamma_{SV)}\gamma_{SL})$ . $A$ typical example of such free boundary condition is Young’s equation$\gamma\cos\theta=\gamma_{SV}-\gamma_{SL}.$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$\int_{\Omega}(|\nabla u|^{2}+Q^{2}\chi_{u>0})dx,$where $\chi_{u>0}$ is characteristic function of the set $\{u>0\}=\{x\in\Omega;u(x)>0\}$ , and show that it possessesminima which are Lipschitz continuous and have linear growth away from the free boundary. For suchharmonic functions they find the representation formula $\triangle u=q_{u}\mathcal{H}^{n-1}L_{\partial\{u>0\}}$ and show that the minimaare weak solutions, while the free boundary is a smooth surface except of a set of zero $(n-1)$-dimensionalHausdorff measure. Of course, the smoothness depends on the smoothness of the datum $Q$ : for example,if $Q$ 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$u_{t}-\triangle u=0$ $in$ $\{u>0\},$ $|\nabla u|=1,$ $u=0$ $on$ $\partial\{u>0\}$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 $(0, \epsilon)$ :1842 2013 61-66 61They 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 generalizedby 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 dropletson 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 $u$ 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$\gamma_{g}\int_{\Omega}\sqrt{1+|\nabla u|^{2}}\chi_{u>0}dx+\int_{\Omega}\gamma_{s}\chi_{u>0}dx\approx\frac{\gamma_{g}}{2}\int_{\Omega}|\nabla u|^{2}dx+\int_{\Omega}\gamma_{s}\chi_{\epsilon}(u)dx$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]) and $\lambda_{\epsilon}$ is a function of time onlyrepresenting a Lagrange multiplier for the volume constraint.If we minimize the unapproximated surface energy under volume constraint, we discover that thestationary contact angle $\theta$ satisfies $\gamma_{s}=-\gamma_{g}\cos\theta$ , which is identical to Young’s equation. However, if theshape of the membrane evolves in time, there is no generally accepted physical theory for the dynamiccontact 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] $)$ .To start the mathematical analysis of the above model equation, we consider the following parabolicproblem:$u_{t}-\Delta u=\lambda$ in $\{u>0\},$ $|\nabla u|^{2}=2\gamma$ on $\partial\{u>0\}$62The regularized version is$u_{t}=\triangle u-\gamma\chi_{\epsilon}’(u)+\chi_{u>0}\lambda_{\epsilon}$ $in$ $(0, T)\cross\Omega,$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 derivativeterm in the original model has been neglected, which can be interpreted as considering relatively slowmotion.One can see by maximum principle that solutions to the regular problem are nonnegative because theright-hand side vanishes for negative values of $u$ . This is the mathematical reason for multiplying thenonlocal term by characteristic function. The physical reason is that the outer force or source representingthe 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 nonlocalobstacle function depending on the solution. Accordingly, the sharp contact angle limit $\epsilonarrow 0$ is expectedto 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$J_{n}(u)= \int_{\Omega}(\frac{|u-u_{n-1}|^{2}}{2h}+\frac{1}{2}|\nabla u|^{2}+\gamma\chi_{\epsilon}(u))dx$ (2)is minimized. Here $h$ is the time step and $u_{n-1}$ refers to the minimizer on the previous time level. Thecomputation starts from minimizing $J_{1}$ , where $u_{0}$ is given as initial datum. This gives $u_{1}$ and minimizers$u_{n}$ on the following time levels are computed inductively. Finally, minimizers are interpolated in time asin the following picture to obtain functions $u^{h}$ and $\overline{u}^{h}$ :$uA:$:$\overline{u}^{h}$ : $rightarrow u2$$-::.:$:$rightarrow u_{3}$$ul rightarrow$$::^{u_{0}}$$t$$\bullet:^{u_{0}}$:$u4$$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:$: $h$ $2h$ $3h$ $4h$ $0:$: $h$ $2h$ $3h$ $4h$If we minimize $J_{n}$ in $H_{0}^{1}$ , it is easy to find from the first variation that the interpolated functionssatisfy$u_{t}^{h}=\triangle\overline{u}^{h}-\gamma\chi_{\epsilon}’(\overline{u}^{h})$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 specialconstrained space$\mathcal{K}^{\delta}=\{u\in H_{0}^{1}(\Omega) ; \int_{\Omega}\chi_{\delta}(u)udx=V\}$ (3)as the admissible space for minimization.63The characteristic function in the constraint of the admissible space is essential in order to satisfythe 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 thenconstructed by deriving uniform estimates in $h$ and $\delta$ and taking $h,$ $\deltaarrow 0$ (see [13] for details).It is to be noted that this minimization approach avoids direct treatment of the complicated nonlocalterm $\lambda_{\epsilon}$ , 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 theirnonnegativity. Since the functional (2) is nonnegative, there is a minimizing sequence $\{u^{k}\}$ such that$J_{n}(u^{k}) \downarrow\inf_{u\in \mathcal{K}^{\delta}}J_{n}(u)$ . As this sequence is bounded in $H^{1}(\Omega)$ , there is a subsequence converging weaklyin $H^{1}(\Omega)$ and strongly in $L^{2}(\Omega)$ to some function $u_{n}\in H^{1}(\Omega)$ . However, here we have to assume thatdomain $\Omega$ is bounded. From the weak lower semicontinuity of $J_{n}$ in $H^{1}(\Omega)$ one can say that $u_{n}$ is aminimizer, if it belongs to $\mathcal{K}^{\delta}$ defined in (3). Therefore, we compute$| \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 $u_{n}$ is negative on a setof positive measure and define a new function $\tilde{u}_{n}$ by $\tilde{u}_{n}=u_{n}\chi_{u_{n}>0}$ . Then it is easy to check that$J_{n}(\tilde{u}_{n})<J_{n}(u_{n})$ which is in contradiction with minimality under the condition that $\tilde{u}_{n}$ belongs to $\mathcal{K}^{\delta}.$However, $\tilde{u}_{n}$ fulfills the constraint because the smoothing of characteristic function causes that onlypositive values of given function are taken into account and thus ”cutting off‘ negative part as in thecase of $\tilde{u}_{n}$ does not change the fact that the constraint is satisfied.The analysis for the sharp contact angle limit $\epsilonarrow 0$ is yet to be done. Yamaura [15] constructed $L^{2}$-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 constrainedproblem. However, there is a problem closely related to the global constraint. Specifically, if we take$\Omega=\mathbb{R}^{m}$ in 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 prooffails, i.e., the volume of the minimizing functions $u^{k}$ may “leak out to infinity”. On the other hand, if $\Omega$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 constructedas the limit of a layer between regions where a function $u$ is identically equal to $0$ and 1, for example.Specifically, to obtain the mean curvature flow, the following energy is considered:$\int_{\Omega}(\epsilon^{2}|\nabla u|^{2}+W(u))dx+\int_{\partial\Omega}\epsilon\sigma(u)d\mathcal{H}^{n-1},$where $\epsilon$ is a small parameter corresponding to the width of the layer, $W$ is a double-well potential withminima at $0$ and 1 and $\sigma$ is a function describing contact energy. In order to prove that the limit ofthe layer as $\epsilonarrow 0$ is a smooth hypersurface, the foremost task is to prepare a parabolic monotonicityformula holding up to the boundary. This might be possible if one consults the ideas regarding interiorparabolicity formula derived in [4] and stationary boundary formula given in [14].64An 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 $ar\cdot\iota sing$ in flamepropagation, Trans. Amer. Math. Soc. 347 (1995), 411-441.[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 asymptoticbehaviour of solutions of a one-dimensional free boundary problem of hyperbolic type, Japan J. Indust.Appl. Math. 18 (2001), 43-58.[6] K. Ito, M. Kazama, H. Nakagawa, K. Svadlenka: Numerical solution of a volume-constrained freeboundary problem by the discrete Morse flow method, Gakuto International Series, MathematicalSciences and Application 29 (2008), 383-398.[7] M. Kazama: Doctor thesis, Kanazawa University (2010).[8] M. Kazama, S. Omata: Modeling and computation offluid-membrane interaction, Nonlinear Analysis71 (2009), e1553-e1559.[9] K, Kikuchi, S. Omata: A free boundary problem for a one dimensional hyperbolic equation, Adv.Math. Sci. Appl. 9/2 (1999), 775-786.65[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 viathe 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 problemdescmbing 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, IndianaUniv. 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 hyper-bolic free boundary problem with volume constraint and its numemcal 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,Adv. Math. Sci. Appl. 16/1 (2006), 33-43.66

On a volume-preserving free boundary problem (Geometric Aspect of Partial Differential Equations and Conservation Laws)

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On a volume-preserving free boundary problem (Geometric Aspect of Partial Differential Equations and Conservation Laws)

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