A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
Abstract. In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments. Draft version: Monday 13th April, 2009.
1. Introduction Level set methods (cf. [24]) have become a popular device for solving shape optimization, inverse obstacle, and free boundary problems due to their topological flexibility and computational efficiency (cf. [9] and the references therein). The main idea behind the level set approach is an implicit representation of evolving shapes in the form Σ(t) = {x ∈ Rd | ϕ(x, t) < 0}.
(1.1)
The motion of the shape with normal velocity Vn on Γ(t) = ∂Σ(t) is translated into an equation for ϕ, after extending the Vn to Rd one solves ∂ϕ + Vn |∇ϕ| = 0, (1.2) ∂t which can be interpreted as an Eulerian way of computing the shape evolution. The standard approach for the construction of level set based optimization methods consists in deriving a geometric gradient flow, i.e., choosing the normal velocity of the evolving shape in appropriate dependence of the shape gradient (cf. [27, 25, 7]). For local problems such as area minimization this strategy yields well-known geometric flows such as motion by mean curvature, which can be analyzed using viscosity solutions techniques (cf. e.g [16]), or higher order flows such as surface diffusion, which can be analyzed in regular situations (cf. e.g [14]). In most practical applications however, one encounters nonlocal problems, i.e., the shape functional depending on the solution of some partial differential equations (referred to as the state equation). Here the unknown shape enters as the discontinuity set of coefficients or as an inner or outer boundary. In such a case one can still construct level set methods by formal calculations, and the existing numerical examples confirm the expected behavior in most case, i.e., convergence of the flow to an optimal shape (cf. e.g [2, 5, 7, 27, 25, 29]). By analogous formal reasoning one can even construct fast Newton-type methods (cf. e.g. [8, 18]) and carry out efficient computations. The mathematical analysis by far lacks behind the computational results. Most rigorous results, e.g., decay of the shape functional, can be given only in regular situations and the needed regularity of the shape during the flow cannot be verified (cf. e.g. [7, 8]). For a simple class of inverse problems, a flow that can be analyzed with respect to convergence and stability has been Key words and phrases. Level set methods, free boundary problems, obstacle problem. 2000 Mathematics Subject Classification 35R35, 49Q10, 65K10, 49Q12. 1
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
constructed in [6]. However, the existence of the flow is carried out in terms of weak set evolution (similar to [30]), for which one cannot show that the evolving shape is independent of the level set representation. The known results for independence of level set representations and non-fattening rely on viscosity solution techniques and again regularity of the normal velocity (cf. [3]). In [7] a general framework has been constructed, which puts the problem of choosing appropriate normal velocities on a functional-analytical basis, but still no rigorous convergence results were provided, again in particular due to missing regularity of shapes. Due to the big gaps in the analysis it seems reasonable to look for a simple model problem, where regularity issues are well-understood and to investigate the construction of level set methods in this case. In this paper we make such an effort for an elliptic obstacle problem of the form ∆u = χΩ in D, u>0 in Ω, (1.3) u = |∇u| = 0 in Σ := D \ Ω. u=f on ∂D, where χΩ denotes the indicator function of the set Ω ⊂ D, f is a nonnegative function on the boundary, and the Poisson equation is satisfied in the sense of distributions. The existence and uniqueness of a solution u with associated shape Ω has been shown (cf. [10]), as well as the regularity of ∂Ω (cf. [10]). It is obvious that level set methods are not the only numerical method for the solution of (1.3). Attractive alternatives are based on finite element discretization of the equation for u only with suitable inequality constraints to be solved by semi-smooth Newton methods (cf. [19]), which might be even more efficient in this case. However, as mentioned above, our primary goal is not just to solve (1.3), but to gain further insight into the analysis of level set methods for shape optimization problems. For this sake we shall reformulate (1.3) in the framework of shape optimization below, compute and analyze its shape sensitivities, and finally construct a convergent level set method based on the framework developed in [7]. As we shall see below, this approach yields a convergent level set method, which can be analyzed rigorously with respect to existence and convergence. Due to our motivation being to gain insight into the analysis of level set methods for shape optimization, in the following we shall not only highlight the main results, but also present the strategy of how to obtain them. This could become a successful algorithmic recipe to construct and analyze level set methods for more general shape optimization problems: (1) Compute the shape derivative and shape Hessian. (2) Analyze the continuity and mapping properties of the shape derivative (as a linear functional acting on normal velocities) and of the shape Hessian (as a linear operator acting on normal velocities) at the optimal shape. (3) Find a suitable Hilbert space (and scalar product) of admissible normal velocities corresponding to these continuity and mapping properties. (4) With the scalar product and shape gradient, construct a geometric gradient flow based on the general framework in [7]. (5) Derive suitable energy estimates from the gradient flow structure, and obtain compactness properties for evolving quantities (e.g. solutions of the state equation). (6) Use the compactness to derive existence of a weak set evolution and to extract convergent subsequences as time increases to infinity. (7) Show that the limit of subsequences is a solution of the shape optimization problem. (8) Verify that the weak set evolution can be realized via a level set method independent of the choice of the initial representation. For many applications, the first five steps seem reasonably understood (though they still require a lot of insight and analytical effort), leaving the three remaining ones as the key issues. First of all, the quantities appearing in energy estimates for which one can derive compactness properties, are not easy to handle. A typical form of a Lyapunov functional is some squared norm of a state variable on the (changing) shape boundary, from which one can neither control the geometric properties of the shape boundary nor a suitable norm of the state variable. A way out of this difficulty might be the use of gradient flows in fractional Sobolev norms of the order k − 12 (k ∈ Z)
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
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on Γ(t), proposed and used in this context in [7]. Due to trace theorems for Sobolev spaces (cf. e.g. [1]), such a fractional Sobolev space on Γ(t) can be related to a Sobolev space of order k on Ω(t) or some domain covering it. In this way it might provide a possibility to control a full norm of the state variable (and in the obstacle problem considered here it indeed does), which can then be used to carry out steps (6) and (7). If steps (1)-(7) are solved, one can at least guarantee that there exists a weak set evolution converging to an optimal state, but it is not yet clear that this weak set evolution can be realized via a level set method (and in particular independent of the choice of the initial level set representation). A key issue in this respect is the regularity of the normal velocity, which is usually related to the regularity of state and adjoint variables. In the standard theory, the normal velocity needs to be a Lipschitz function of the spatial variable in order to verify non-fattening and independence properties of the zero level set. Also in this respect, 1 the use of fractional order Sobolev spaces might help, in particular the Sobolev space H − 2 (Γ(t)), whose norm is realized via Dirichlet-to-Neumann maps. As a consequence, a geometric gradient flow in this space yields a velocity Vn = ∂w ∂n for some auxiliary function w. With this specific form one has an immediate candidate for a full velocity field chosen as V = ∇w. Since a tangential component of the velocity field does not change the evolution of the shape, one can also consider the evolution in such a velocity field, realized in level set formulation via solving ∂ϕ + V · ∇ϕ = 0. ∂t
(1.4)
Equation (1.4) is a Hamilton-Jacobi equation with Hamiltonian H(x, t, p) := V(x, t) · p, which is linear and thus convex in p. For convex Hamiltonians, the regularity requirements for the Hamiltonian (and consequently the velocity field in our case) can be weakened (cf. [11]), and we shall use this fact heavily to carry out step (8) in the case of the obstacle problem (1.3). This paper is organized as follows: In Section 2 we provide a shape optimization formulation of the obstacle problem and discuss its shape sensitivity analysis in Section 3. In Section 4 we introduce our level set approach, and provide the detailed analysis in Section 5. Finally, we discuss the numerical implementation and computational examples in Section 6. 2. Shape Optimization Formulation In order to reformulate (1.3) as a shape optimization problem, we consider the following functional: Z Z Z 1 J˜ (w, Σ) := |∇w|2 dx + w dx − w dx. 2 D D Σ If J˜ is minimized with respect to w ∈ Hf1 (D) := { ϕ ∈ H 1 (D) : ϕ|∂D = f } for fixed Σ, then the minimizer u satisfies ∆u = 1 − χΣ
in D,
subject to u = f on ∂D. On the other hand, suppose that w ∈ Hf1 (D) is fixed, then we clearly have Z Z Z − w dx ≤ − min{w, 0} dx = − w dx. Σ
Σ
{w≤0}
˜ i.e., a minimizer u ˆ Thus, if we look for a saddle point of J, ˆ with respect to u and a maximizer Σ ˆ with respect to Σ, then (ˆ u, Σ) is a solution of (1.3). ˆ satisfying Due to this motivation we look for a pair (ˆ u, Σ) Z ! Z Z 1 2 ˜ ˆ |∇w| dx + w dx − w dx J u ˆ, Σ = sup inf (2.1) w∈Hf1 (D) 2 D Σ D Σ
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
The inner problem with respect to w can be solved for a fixed Σ. By standard arguments one can show that there exists a unique minimizer uΣ ∈ Hf1 (D), which is the weak solution of the boundary value problem ∆uΣ = 1 − χΣ = χΩ with Ω = D \ Σ, i.e., Z
Z ∇uΣ · ∇ϕ dx = −
D
in D,
u=f
Z ϕ dx +
D
ϕ dx
on ∂D,
∀ ϕ ∈ H01 (D).
(2.2)
Σ
By substituting the solution of (2.1) we can rewrite the problem as an optimization over Σ only, i.e., Z Z Z 1 |∇uΣ |2 dx + J˜ (uΣ , Σ) = sup uΣ dx − uΣ dx . 2 D Σ D Σ Finally, we equivalently rewrite the maximization as a minimization over the negative functional, i.e., Z Z Z 1 2 |∇uΣ | dx − uΣ dx + uΣ dx → min . (2.3) J(Σ) = − Σ 2 D D Σ In the following we shall always use the notation Γ := ∂Σ \ ∂D
(2.4)
for the unknown part of the boundary of Σ and we use the orientation such the unit normal vector n to Γ points into Ω = D \ Σ. Note that the remaining part of the boundary of Σ is known from the boundary data as ∂Σ ∩ ∂D = {x ∈ ∂D | f (x) = 0}, and we assume that this set has positive (d − 1)-dimensional Hausdorff measure. Remark 2.1. Let us also note, the since the right hand side of the following equation is bounded ∆u = 1 − χΣ
in D,
from standard elliptic theory we have that the solution u of the equation belongs to W 2,p ∩ C 1,α in D. Moreover we have 1 |∇u(x) − ∇u(y)| ≤ c|x − y| log |x − y| for any x, y ∈ K ⊂⊂ D with |x − y| ≤ 1/2 and C = C(n, K, D) (see for example [23] Theorem 2.5.1, page 47). 3. Shape Sensitivity Analysis In the following we provide a shape sensitivity analysis of the functional J defined via (2.3), using classical techniques for shape derivatives such as in particular the speed method (cf. [28, 12]), which fits better to the subsequently used level set schemes than other equivalent (cf. [12]) approaches. Since the shape sensitivities and their structure are only used as motivations for the subsequent analysis, we only provide a rather formal analysis without rigorous proofs. 3.1. Shape Derivative. Let us start by computing the first shape derivative of the functional J defined in (2.3), i.e., Z Z Z 1 |∇uΣ |2 dx − uΣ dx + uΣ dx. J (Σ) := J˜ (uΣ , Σ) = − 2 D D Σ For a sufficiently smooth set Σ and a sufficiently smooth velocity field V¯ , the shape sensitivity of J (Σ) is given by J (Σs ) − J (Σ) dJ Σ, V¯ := lim , s&0 s if this limit exists. Here we use the notation dξ (τ ) s ¯ Σ := ξ (s) | = V (ξ (τ )) , τ ∈ (0, s) , ξ (0) ∈ Σ dτ
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
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for the shapes changed by the speed or velocity method. Using the chain rule and the transport theorem (cf. [12]) we get Z Z Z Z dJ Σ, V¯ = − ∇u0Σ · ∇uΣ dx − u0Σ dx + u0Σ dx + div uΣ V¯ dx, D
where
u0Σ
D
Σ
Σ
is the variation of the state variable with respect to the shape change: d 0 . uΣs uΣ = ds s=0
By differentiating (2.2) one can show that u0Σ ∈ H01 (D) satisfies Z Z ∇u0Σ ∇ϕdx = div ϕV¯ dx D
(3.1)
Σ
for all ϕ ∈ H01 (D). From (2.2), taking ϕ = u0Σ as a test function, we conclude Z Z Z ∇uΣ ∇u0Σ dx + u0Σ dx − u0Σ dx = 0, D
D
and hence dJ Σ, V¯ =
Z
Σ
div uΣ V¯ dx =
Z
uΣ V¯ · ndσ.
(3.2)
∂Σ
Σ
Since the function uΣ belongs to H 1 (D), a trace theorem for Sobolev functions implies uΣ |∂Σ ∈ H 1/2 (∂Σ) . Denoting by h·, ·i−1/2,1/2 the dual product of H 1/2 (∂Σ) and its dual space H −1/2 (∂Σ), we have
dJ Σ, V¯ = V¯ · n, uΣ −1/2,1/2 . In particular, dJ (Σ, ·) can be extended to a continuous linear functional of the normal velocity V¯ · n in H −1/2 (∂Σ). This linear functional is usually called the shape derivative J 0 (Σ) : H −1/2 (Σ) → R,
J 0 (Σ)Vn = hVn , uΣ i−1/2,1/2 .
(3.3)
3.2. Shape Hessian. Now let us calculate the second shape sensitivity of J, the so-called shape Hessian. The second shape sensitivity can be obtained from the iterated definition: ¯ − dJ Σ, W ¯ dJ Σs , W 2 ¯ ¯ . d J Σ, V , W := lim s&0 s ¯ ) we obtain the second shape variation Using (3.2) (with V¯ replaced by W Z Z ¯ dx ¯ ¯ dx + d2 J Σ, V¯ , W = div u0Σ W div V¯ div uΣ W ZΣ ZΣ ¯ dσ. ¯ dx + V¯ · n div uΣ W = div u0Σ W ∂Σ
Σ
As usual for second variations also d J is a symmetric bilinear form of the velocity fields V¯ and ¯ (cf. [12]), although the symmetry is not visible immediately from the above formula. W It is instructive to analyze the local behavior around a solution u ˆΣ of the obstacle problem, ˆ V¯ ) = 0. The local behavior for a shape Σ ˆ s close which is also a stationary point of J, i.e. dJ(Σ, ˆ to Σ is described by 2 ˆ s ) = J(Σ) ˆ + s d2 J(Σ, ˆ V¯ , V¯ ) + O(s3 ), J(Σ 2 ˆ V¯ , V¯ ). From the fact that uΣ is a solution of (1.3) we conclude and hence we study d2 J(Σ, ¯ =V¯ we obtain uΣ |∂Σ = ∇uΣ |∂Σ = 0. Thus we have div uΣ V¯ = 0 on ∂Σ, and with W Z d2 J Σ, V¯ , V¯ = div u0Σ V¯ dx. 2
Σ
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
Now we insert (2.2) with ϕ = u0Σ to conclude Z Z d2 J Σ, V¯ , V¯ = div u0Σ V¯ dx = |∇u0Σ |2 dx > 0. Σ
(3.4)
D
Hence, the second shape sensitivity is coercive at a solution of the obstacle problem. In order to clarify the relation of u0Σ and the normal velocity in the second sensitivity, we recall the strong formulation of (2.2), which is the boundary value problem −∆u0Σ
=
0
in D \ ∂Σ
u0 0 Σ ∂uΣ
=
0
on ∂D
∂n
= V¯ · n
on ∂Σ \ ∂D,
where [.] denotes the jump over the interface. Hence, the trace theorem for Sobolev functions implies that ku0Σ kH 1 (D) is equivalent to the norm of H −1/2 (∂Σ \ ∂D). Due to the PoincareFriedrichs-inequality the seminorm sZ |∇v|2 dx |v|H 1 (D) := D
is equivalent to the H 1 -norm on the subspace H01 (D). Hence, we obtain the following result: ˆ for Σ ˆ being a solution of the Lemma 3.1. Inequality (3.4) implies that the shape Hessian J 00 (Σ), −1/2 obstacle problem, is coercive in the H (∂Σ) norm, i.e., there exists a constant C > 0 such that
ˆ V¯ · n, V¯ · n) = d2 J Σ, ˆ V¯ , V¯ ≥ C V¯ · n 2 −1/2 ˆ , J 00 (Σ)( H (Γ) ˆ for all normal velocities V¯ · n ∈ H −1/2 (Γ). We can also interpret the H 1 -seminorm of u0Σ as an equivalent realization of the norm in H −1/2 (Γ) of V¯ · n, which is actually more easy to handle than the original definition. The fact that the shape derivative is a continuous linear functional on H −1/2 (Γ) and that the shape Hessian is coercive (and also continuous due to the above norm-equivalence) in this space indicates that H −1/2 (Γ) might be the correct functional space to use for normal velocities. We shall exploit this idea in the following to construct a level set scheme based on a velocity choice via the scalar product of H −1/2 (Γ). 4. The Level Set Approach In the following we discuss the construction of a level set scheme for the solution of the obstacle problem (1.3), or more precisely of the reformulation (2.3) as a shape optimization problem. The latter allows the application of a standard framework developed for such problems (cf. [7, 9]) 4.1. Construction of an Evolution. As common in the application of level set methods to shape optimization problems (cf. [9] and the references therein), we construct an (artificial) time evolution of the form Σ (t) := {ϕ (·, t) < 0} ,
Γ(t) := {ϕ (·, t) = 0} ∩ D,
which ideally converges to a solution of the obstacle problem. The level set function ϕ is advected in a velocity field V¯ and therefore it is determined as a (viscosity) solution of ϕt + V¯ · ∇ϕ = 0. Here V¯ = V¯ (x, t) is a time dependent velocity and Vn = V¯ · ∇ϕ is the velocity component normal |∇ϕ|
to the level sets of ϕ. The desired convergence of the evolution to a solution of the obstacle
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
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problem has to be achieved by an appropriate choice of the velocity field, which is usually done in dependence on shape derivative. One can show that d J (Σ (t)) = dJ Σ (t) , V¯ (t) , dt and therefore it is natural to choose V¯ such that dJ Σ (t) , V¯ (t) = J 0 (Σ(t))(V¯ · n) < 0, ˆ in order to obtain a descent method to a minimizer of the functional J. This unless Σ(t) = Σ, can be realized within a gradient flow approach, following [7] we choose the normal velocity in a Hilbert space H(t) defined on Γ(t), i.e. hV¯ · n, wiH(t) = −J 0 (Σ(t))w
∀ w ∈ H(t).
With this approach one assigns a formal Riemannian structure to a space of shapes, which can be made rigorous in certain cases (cf. [22]). Using this choice we automatically arrive at a descent method due to d J (Σ (t)) = J 0 (Σ(t))(V¯ · n) = −hV¯ · n, V¯ · niH(t) ≤ 0. dt Having found a normal velocity, we can actually choose an arbitrary tangential velocity component V¯T = V¯ − (V¯ · n) · n, since the level set evolution does not depend on the tangential component. In most level set schemes the tangential component is just set to zero, but we shall use a different choice here motivated also by the special scalar product we use for the Hilbert space H(t), as we shall see in the next section. Based on the mapping properties of the shape derivative and the shape Hessian we shall use the Hilbert space H(t) = H −1/2 (Γ(t)) for determining the normal velocity, i.e.
∀w ∈ H −1/2 (∂Σ (t)) . (4.1) V¯ (·, t) · n, w H −1/2 (∂Σ(t)) = −dJ (Σ (t) , w · n) , As noticed above we then obtain the estimate
2
d J (Σ (t)) = − V¯ (·, t) · n H −1/2 (∂Σ(t)) ≤ 0. dt Such a scheme can only stop in a stationary point of the functional J, i.e. d J (Σ (t)) = 0 dt if and only if V¯ (·, t)·n = 0 and J 0 (Σ(t)) ≡ 0. Due to the coercivity of the shape Hessian at solutions of the obstacle problem, one is tempted to think of some local convexity of the functional around stationary points, which would imply that all stationary points are actually local minimizers. Consequently one may hope that the above evolution can stop only if Σ(t) is a solution of the obstacle problem. This statement is mostly true, but one exception, namely the empty set or more precisely Γ(t) = ∅, Σ(t) = ∂Σ(t) ∩ ∂D := Σ∅ . (4.2) To see this, assume that J 0 (Σ(t)) ≡ 0. Then for all w ∈ L2 (Γ(t)) ⊂ H −1/2 (Γ(t)) we have Z uΣ(t) w dσ = 0, ∂Σ(t)
and consequently Γ(t) = ∅ or uΣ(t) ≡ 0 on Γ(t). From the definition of uΣ(t) we conclude ∆uΣ(t) = 0 in Σ (t) , uΣ(t) = 0 on ∂Σ (t) , and the uniqueness for the Dirichlet problem for the Laplace equation implies uΣ(t) = 0 in Σ (t) . If Γ(t) 6= ∅, i.e., Σ(t) has nonempty interior, then we also conclude ∇uΣ(t) ≡ 0 on ∂Σ(t). Let us ¯ (t) , since ∆uΣ(t) = 1 6= 0 in D \ Σ ¯ (t) . We also mention that uΣ(t) 6= 0 almost everywhere in D \ Σ summarize this result in the following:
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
Lemma 4.1. Let Σ(t) be obtained from an evolution via the level set method with a normal velocity chosen via (4.1) and let J 0 (Σ(t)) ≡ 0. Then either Σ(t) is given via (4.2) or Σ(t) is a solution of the obstacle problem. We also mention that the possible degeneracy of the stationary point resulting in an empty set can eventually be avoided by choosing appropriate initial values. If J(Σ(0)) < J(Σ∅ ) then, due to the descent of the objective functional J, the evolution can only stop in a solution of the obstacle problem. In this case J(Σ∅ ) also cannot be a long-time asymptotic limit of the flow. Note also that the condition J(Σ(0)) < J(Σ∅ ) can be checked via the solution of the two boundary value problems for uΣ(0) and uΣ∅ . 4.2. Another Realization of the H −1/2 -Scalar Product. In the following we shall turn to a different realization of the scalar product in H −1/2 , which will also lead to a natural choice for the normal velocity. For a function ψ ∈ H −1/2 (Γ (t)) we define 1 wψ ∈ H0,D (Σ(t)) := {w ∈ H 1 (Σ (t)) | ϕ|∂Σ(t)∩∂D = 0}
via
Z
Z ∇wψ ∇ϕ dx =
Σ(t)
ψϕ dσ,
1 ∀ ϕ ∈ H0,D (Σ(t)).
Γ(t)
Then wψ is a weak solution of the boundary value problem ∆wψ = 0 wψ = 0 ∂wψ ∂n = ψ
in Σ (t) , on ∂Σ(t) ∩ D, on Γ (t) .
Now we define a scalar product on H −1/2 (Γ (t)) as follows: Z hψ1 , ψ2 iH −1/2 (∂Σ(t)) := ∇wψ1 ∇wψ2 dx. Σ(t)
The equivalence of the resulting norm can be checked as for the realization in the previous section by applying a trace theorem and the Poincare-Friedrichs inequality. The gradient flow for that scalar product amounts to choosing the normal velocity, respectively 1 the corresponding function w(t) := wV¯ (t)·n ∈ H0,D (Σ(t)) via Z Z 1 ∇w (t) ∇wψ dx = − uΣ(t) ψdσ, ∀ ψ ∈ H0,D (Σ (t)) . Σ(t)
Γ(t)(t)\∂D
This means that w (t) is the weak solution of the boundary value problem ∆w (t) = 0 w (t) = 0 w (t) = −uΣ(t) V¯ (·, t) · n = ∂w ∂n
in Σ (t) , on ∂Σ (t) ∩ ∂D, on Γ(t), on Γ(t).
Actually we can interpret the first three equations as a system for computing w given uΣ(t) and the last one as the defining relation for the normal velocity. Now note that the function w (t) + uΣ(t) solves the homogeneous problem ∆ w (t) + uΣ(t) = 0 in Σ (t) , w (t) + uΣ(t) = 0 on ∂Σ (t) , and again by uniqueness for the Dirichlet problem for the Laplace operator we conclude w (t) = −uΣ(t) in Σ (t). Hence, the normal velocity is given by ∂uΣ(t) = −∇uΣ(t) · n on ∂Σ (t) . V¯ (·, t) · n = − ∂n Thus, with this choice of scalar product we arrive at a normal velocity that is the negative normal component of the gradient of u. Consequently the same choice for the tangential components seems to suggest itself, i.e., we can use the velocity field V¯ (·, t) = −∇uΣ(t) . (4.3)
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
9
The level set scheme for the obstacle problem is the solution of the coupled system (2.2), (4.3), (1.4) and Σ(t) = {ϕ(·, t) < 0} for the unknowns Σ(t), uΣ(t) , ϕ, and V¯ . In order to perform a rigorous analysis of the scheme, first we shall consider a very special solution of the level set equation , namely 1 1 v(t) := χD\Σ(t) − χΣ(t) = − χΣ (t). (4.4) 2 2 With the short-hand notation u(t) := uΣ(t) the level set scheme can be written as the solution of the evolution problem 1 in D × R+ ∆u(t) − v(t) = 2 ∂v (t) − ∇u(t) · ∇v(t) = 0 in D × R+ ∂t u(t) = f on ∂D × R+ (4.5) sign(f ) v(t) = on ∂D × R+ ∩ {∇u · n ≤ 0} 2 1 v(0) = (χD\Σ0 − χΣ0 ) in D. 2 The well-posedness of this coupled problem as well as its convergence to a solution of the obstacle problem in the large-time limit will be shown in the next section. We also show that actually any level set scheme yields the same evolution of sets Σ(t) and in particular the same long-time limit, thus we obtain the convergence of the level set scheme. Example. In order to gain further insight in the scheme consider a simple example of the obstacle problem, namely D = [0, 1] ⊂ R with the boundary conditions u(0) = 0, u(1) = f , for some f ∈ [0, 21 ]. In this case we can easily compute the solution of the obstacle problem as √ √ ˆ = (0, 1 − 2f ) and u Σ ˆ(x) = 21 ((x − 1 + 2f )+ )2 . Now let γ(0) ∈ (0, 1) and Σ(0) = (0, γ(0)). Then (4.5) is solved by 1 1 1 v = (χ(γ(t),1) − χ(0,γ(t)) ), u = ((x − γ(t))+ )2 + (f − (1 − γ(t))2 )x, 2 2 2 where γ(t) is determined by 1 ∂u γ 0 (t) = − (γ(t), t) = (1 − γ(t))2 − f. ∂x 2 The solution of this ordinary differential equation is of the form p p γ(t) = 1 − 2f tanh( f /2t + c) p for some constant c depending on γ(0). As t → ∞ we obtain tanh( f /2t + c) → 1 and hence, √ γ(t) → 1 − 2f , i.e. the evolution indeed converges to the solution of the obstacle problem. 5. Analysis of the Level Set Scheme In the following we provide a detailed analysis of the level set scheme for the obstacle problems. We start with some properties of the level set evolution and then proceed to the coupled problem. 5.1. Properties of the Level Set Evolution. In the following we analyze some properties of the level set equation (1.4). For this sake it is fundamental to understand the regularities of the velocities during the evolution. We start with a regularity result for the velocity, which follows directly from Remark 2.1 Remark 5.1. Let V¯ = −∇w, where w solves the Poisson problem ∆w = g
in D
with boundary values w = f on ∂D, where 0 ≤ g ≤ 1 a.e. in D. Then there exists a constant C > 0 depending on D and f only, such that 1 kV¯ (x) − V¯ (y)k ≤ C|x − y| log . (5.1) |x − y| Moreover w ∈ W 2,p (D) ∩ C 1,α (D) for any p ∈ [0, ∞).
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
This regularity result is fundamental for the study of trajectories with a velocity as above: Lemma 5.2. Let V¯ satisfy the assumptions of Lemma 5.1. Then, for any T > 0, the ordinary differential equation dξ (t) = V¯ (ξ(t), t) (5.2) dt has a unique solution ξ ∈ C 1 ([0, T ]) for any given initial value ξ(0) as well as for any given terminal value ξ(T ). Proof. Since V¯ is uniformly continuous, existence of a solution follows from Peano’s Theorem. Assume that ξi , i = 1, 2 are two solutions of (5.2) with initial values ξ1 (0) = ξ2 (0). Let e(t) := |ξ1 (t) − ξ2 (t)|2 , then de (t) dt
= = ≤ =
dξ1 dξ2 2 (t) − (t) · (ξ1 (t) − ξ2 (t)) dt dt 2(V¯ (ξ1 (t), t) − V¯ (ξ2 (t), t)) · (ξ1 (t) − ξ2 (t)) 1 2C|ξ1 (t) − ξ2 (t)|2 log |ξ1 (t) − ξ2 (t)| 1 . Ce(t) log e(t)
Ra 1 Since 0 x log 1 dx = ∞ for all a > 0, Osgood’s Theorem implies e ≡ 0, and hence uniqueness of x (5.2). The proof for given terminal value is completely analogous. As a result of the uniqueness of the trajectories we can assign a history to each point. Let (x, s) ∈ D × R+ , then there exists a unique solution of (5.2) with ξ(s) = x. Moreover, either ξ(s) ∈ D for all s > 0 or there exists t0 < t such that ξ(t0 ) ∈ ∂D. We therefore define ξ(0) if ξ(s) ∈ D, ∀ s ∈ (0, t) Tt (x) = (5.3) ξ(t0 ) else, t0 = sup{s ∈ (0, t) | ξ(s) ∈ ∂D}. For ϕ0 ∈ C 1 (D) it is straight-forward to verify by the chain rule that ϕ(x, t) := ϕ0 (Tt (x))
(5.4)
is a classical solution of (1.4). In this case ϕ is also the unique (cf. [30]) weak solution, i.e., Z TZ ∂ψ ¯ ϕ(x, t) + ∇ · (V ψ) dx = 0, ∀ ψ ∈ C0∞ (D × (0, T )), (5.5) ∂t 0 D which follows immediately from Gauss’ Theorem. Here we are also interested in the case of 1 χD\Σ(0) − χΣ(0) , ϕ(·, 0) = v(·, 0) = 2 or more general ϕ(·, 0) ∈ L∞ (D), where we also define the solution via (5.4). Then it is clear that ϕ(·, t) ∈ {− 12 , 12 } almost everywhere in D and hence, there exists Σ(t) ⊂ D such that 1 χD\Σ(t) − χΣ(t) . 2 The definition via (5.4) is also coherent with the definition of a weak solution. To see this, let ϕε0 ∈ C 1 (D) be a sequence converging to ϕ(·, 0) in L2 (D) and hence also pointwise almost everywhere. Then, the solution ϕε (·, t) of (1.4) converges pointwise ϕ(·, t) almost everywhere in D. On the other hand it is easy to see from the uniform boundedness of ϕε that a subsequence ϕε converges weakly in L2 (D × (0, T )) to some function ϕˆ being the weak solution of (1.4). By equality of the limits we conclude ϕˆ = ϕ. Thus, ϕ is a weak solution of (1.4). As a result of these arguments, we can state the following theorem: ϕ(·, t) = v(·, 0) =
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Theorem 5.3. Let V¯ satisfy the assumptions of Lemma 5.1. Then, for any initial value ϕ(·, 0), (1.4) has a unique (weak) solution ϕ given by (5.4). Moreover, if ϕ1 and ϕ2 are two solutions such that {ϕ1 (·, 0) < 0} = {ϕ2 (·, 0) < 0}, then {ϕ1 (·, t) < 0} = {ϕ2 (·, t) < 0}
(5.6)
for all t > 0. Proof. From the analysis above we can conclude the existence and uniqueness of the solution. Since {ϕ(., t) < 0} = Tt−1 ({ϕ(·, 0) < 0}) we can also conclude (5.6).
Theorem (5.3) yields the desired independence of the level set representation. Consequently it is indeed sufficient to consider the level set scheme for the particular initial value in (4.5), for which we shall prove existence and convergence below. The independence of the level set representation then of course implies the convergence for any representation with the same initial zero level set. 5.2. Existence of the Flow. In the following we sake we construct an approximation of the form 1 ∆uε (t) − wε (t) = 2 −ε∆wε (t) + wε (t) − v ε (t) = 0 ∂v ε (t) − ∇uε (t) · ∇v ε (t) = 0 ∂t uε (t) = f
verify the existence of the flow (4.5). For this in D × R+ in D × R+ in D × R+ on ∂D × R+
(5.7) w (t) = 0 on ∂D × R+ sign(f ) ε ε on ∂D × R+ ∩ {∇u · n ≤ 0} v (t) = 2 1 ε v (0) = (χD\Σ0 − χΣ0 ) in D. 2 We shall first prove existence of a solution to (5.7) and subsequently convergence of (uε , v ε ) to a solution (u, v) of level set scheme. Throughout this section we assume W 2,p regularity of the Poisson equation with Dirichlet data f and 0, respectively, i.e. the solutions of ε
∆ϕ = ψ in D satisfies ϕ ∈ W 2,p (D) if ψ ∈ Lp (D) for p ∈ [2, ∞) and D ⊂ Rd , d ≤ 3. Proposition 5.4. For any ε > 0 there exists a solution (uε , v ε , wε ) of (5.7) with 1 a.e. in D, for a.e. t ∈ [0, T ]. 4 Proof. We construct the solution by a fixed-point argument. First of all, let (v ε (t))2 =
F1 : C(0, T ; C 1,1 (D)) ∩ C(0, T ; W 2,4 ) → L∞ (D × (0, T )) × H 1 (0, T ; H 1 (D)), uε 7→ (v ε , wε ) where v ε is the solution of (1.4) with V¯ = −∇uε and the same initial value as in (5.7) and wε as in (5.7). F1 is a continuous operator, the well-definedness being clear from its triangular structure (v ε can be computed first and wε subsequently) and the continuity following from standard continuousε dependence results for linear transport equations and linear elliptic equations. Note that z ε := ∂w ∂t satisfies ∂v ε −ε∆z ε + z ε = = ∇uε ∇v ε ∂t with homogeneous Dirichlet values on ∂D. Thus we have Z Z Z ε 2 ε 2 ε |∇z | dx + (z ) dx = − v ε (∇uε · ∇z ε + ∆uε z ε ) dx D
D
D
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
from which we can indeed conclude the well-definedness and continuity in the image space H 1 (0, T ; H 1 (D)). Now let F2 : L∞ (D × (0, T )) × H 1 (0, T ; H 1 (D)) → C(0, T ; L4 (D))
(v ε , wε ) 7→ wε .
Due to compact embedding of H 1 -spaces F2 is continuous and even compact. Finally define F3 : C(0, T ; L4 (D)) → C(0, T ; C 1,1 (D)) ∩ C(0, T ; W 2,4 ),
wε 7→ u ˆε ,
which is obviously well-defined and continuous due to our assumption on the W 2,p -regularity of the Poisson equation and the continuous embedding of W 2,4 (D) into C 1,1 (D). Now define F = F1 ◦F3 ◦F2 . Then F is a continuous and compact operator on L∞ (D ×(0, T ))× 1 H (0, T ; H 1 (D)). Moreover, F maps a bounded set into itself, which one can easily infer from the a-priori bound on v ε 1 kv ε k∞ ≤ . 2 Hence, Schauder’s fixed point theorem implies the existence of a fixed point uε = F (uε ). With uε = F3 (F2 (v ε , wε )) we obtain a solution of (5.7). Lemma 5.5. Let (uε , v ε , wε ) solve (5.7). Then there exist constants C1 , C2 independent of ε such that 1 kwε kL∞ (Ω×(0,t)) ≤ kv ε kL∞ (Ω×(0,t)) ≤ 2 kuε kL∞ (0,T ;W 2,p (D)) ≤ C1 ∂uε kL∞ (0,T ;W 1,2 (D)) ≤ C2 . k ∂t Proof. As a solution of (1.4) we obtain v ε (x, t) = v0 (Ttε (x)),
v0 =
1 χD\Σ − χΣ , 2
where Ttε is the flow constructed from the velocity V¯ = −∇uε . In particular v ε satisfies (v ε (t)2 = 41 almost everywhere in D for almost every t. Then v ε is bounded uniformly by 12 in L∞ (D). By the maximum principle for the Poisson operator one can show that kwε kL∞ (D×(0,T )) ≤ kv ε kL∞ (D×(0,T )) ≤
1 . 2
As a consequence, uε is uniformly bounded in L∞ (0, T ; W 2,p (D)). From ∆uε = v ε we conclude ∂uε ∂wε ∆ = ∂t ∂t ∂wε ∂wε ∂v ε −ε∆ + = = ∇uε · ∇v ε ∂t ∂t ∂t ε
ε
ε
∂w ∂u and both ∂u ∂t and ∂t vanish on ∂D × (0, T ). Now we multiply the second equation with − ∂t and integrate over D. Then, with some applications of Gauss’s Theorem and insertion of the first equation we conclude Z Z Z ∂wε 2 ∂uε 2 ∂wε ∂wε ∂uε ε ( ) dx + |∇ | dx = (ε∆ − ) dx ∂t ∂t ∂t ∂t D ∂t D D Z ∂uε = (∇uε v ε ∇ · (∇uε ) dx ∂t Z ZD ∂uε ∂uε ε ε = v ∇u · ∇ dx + v ε wε dx ∂t ∂t D D Z Z 1 C2 1 ∂uε 2 ≤ |∇uε |2 dx + 0 |D| + |∇ | dx, 4 D 16 2 D ∂t
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where we have used the uniform bounds for uε and wε and the Friedrichs-Poincare inequality for ∂uε ∂t with constant C0 . Hence, Z Z 1 C2 ∂uε 2 | dx ≤ |∇uε |2 dx + 0 |D|, |∇ ∂t 2 D 8 D and due to the uniform boundedness of uε in L∞ (0, T ; W 2,2 (D)) we obtain the desired uniform ε bound for ∂u ∂t . Theorem 5.6. For any T ∈ R+ , there exists a solution (u, v) of (4.5) satisfying the following properties u(t) ∈ W 2,p (D) ∩ C 1,α (D) 1 v(t)2 = 4
for a.e. t ∈ [0, T ], ∀ p ∈ [1, ∞), ∀ α ∈ (0, 1) a.e. in D, for a.e. t ∈ [0, T ].
Proof. Let (uε , v ε , wε ) solve (5.7). Due to the uniform bounds in Lemma (5.5) there exists a subsequence εn such that uεn converges in the weak-* topology of L∞ (0, T ; W 2,p (D)) and v εn converges in the weak-* topology of L∞ (D × (0, T )). We denote the limit by u and v, respectively. Then it is straight-forward to see that ∆u(t) = v(t) + 21 in D and u(t) = f on ∂D, for almost every t ∈ (0, T ). It remains to verify that v solves (1.4) with V¯ = −∇u. For this sake we use that v ε is the unique weak solution of (1.4) with V¯ = −∇uε and that (v ε )2 = 14 almost everywhere. Then, for any test function ψ we have Z TZ ∂ψ ε ε − ∇ · (∇u ψ) dx dt 0 = v ∂t 0 D Z TZ ∂ψ = vε − ∇uε · ∇ψ − ∆uε ψ dx dt ∂t 0 D Z TZ 1 ∂ψ = − ∇uε · ∇ψ − v ε ψ − ψ dx dt vε ∂t 2 0 D Z TZ Z Z 1 1 T ∂ψ ε ε = − ∇u · ∇ψ − ψ dx dt − ψ dx dt. v ∂t 2 4 0 D 0 D In the last formulation we can pass to the weak-* limit for the subsequence v εn , since by compact embedding ∇uεn converges strongly to ∇u in L1 (D × (0, T )). Hence we conclude Z TZ Z Z ∂ψ 1 1 T 0 = v − ∇u · ∇ψ − ψ dx dt − ψ dx dt ∂t 2 4 0 D 0 D Z TZ Z TZ 1 ∂ψ − ∇ · (ψ∇u) dx dt + v2 − = v ψ dx dt. ∂t 4 0 D 0 D Moreover, v εn (0) converges in the weak-* topology to 12 (χD\Σ(0) − χΣ(0) ). Thus, v is a weak solution of the equation 1 ∂v − ∇u · ∇v = v 2 − . ∂t 4 with initial value 21 (χD\Σ(0) − χΣ(0) ). This equation has the same (unique) characteristics as (1.4) and hence, on each trajectory we have dξ (t) = −∇u(ξ(t), t), dt
d 1 v(ξ(t), t) = v(ξ(t), t) − . dt 4
For v(ξ(0), 0) = ± 21 one observes that the unique solution of the second ODE is given by v(ξ(t), t) = ± 21 . Thus, v 2 = 41 almost everywhere, and hence v is also a solution of (1.4) with Vˆ = −∇u. Together (u, v) is a solution of (4.5) with v = χΣ(t) − 21 for some set Σ(t) ⊂ D.
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
5.3. Energy Estimates. In the following we derive some energy estimates fundamental for the long-time asymptotic of (4.5). Therefore we consider a version of the original functional J, rewritten in terms of u(t) and v(t), namely Z Z 1 1 j(t) = − |∇u(t)|2 dx − u(t)(v(t) + ) dx. (5.8) 2 D 2 D Lemma 5.7. Let (u, v) solve (4.5). Then, for all t > 0, the identity Z dj 1 (t) = (v − )|∇u|2 dx ≤ 0 dt 2 D
(5.9)
holds. Proof. We have dj (t) dt
Z
= =
Z Z ∂u ∂u 1 ∂v (t) dx − (t)(v(t) + ) dx − u(t) (t) dx ∂t 2 ∂t D D ∂t D Z 1 ∂u ∂v (∆u(t) − v(t) − ) (t) dx − u(t) (t) dx 2 ∂t ∂t D D
− Z
∇u(t) · ∇
where we have used Gauss’ Theorem and the fact that the Dirichlet values of ∂u ∂t (t) vanish to manipulate the leading terms. Inserting (4.5) yields Z Z 1 dj (t) = − u(t)∇u(t) · ∇v(t) dx = ∇ · (u(t)∇u(t))(v(t) − ) dx, dt 2 D D where we have used again Gauss’ Theorem and the fact that u(t)(v(t) − 12 ) = 0 on ∂D. Inserting the first equation in (4.5) we further deduce Z Z 1 1 dj (t) = (v(t) − )|∇u(t)|2 dx + (v(t)2 − )u(t) dx dt 2 4 D D and the fact that v(t)2 =
1 4
almost everywhere implies (5.9).
We mention that in the original notation, (5.9) can be rewritten as the identity Z d ∇uΣ(t) 2 dx ≤ 0, J (Σ (t)) = − dt Σ(t) whose direct derivation is more subtle, since the transport theorem has to be applied for differentiating integrals over Σ(t) with respect to time. As a result of (5.9) we conclude that t 7→ j(t) is a non-increasing function and that Z t j(t) + k(s) ds ≤ j(0), 0
with the energy dissipation functional Z k(t) := We also observe that the integral bounded from below: Z tZ
Rt 0
1 ( − v(t))|∇u(t)|2 dx. 2 D
(5.10)
k(s) ds is uniformly bounded in t since j (t) = J (Σ (t)) is 2
|∇u| dxds ≤ C0 := j (0) − inf J (Σ) . 0
Σ
Σ(s)
Thus also the integral over R+ converges and we find Z ∞ k(s) ds ≤ C0 .
(5.11)
0
Since k is nonnegative it follows that there exists a sequence sm → ∞ such that k(sm ) → 0. In order to show that k(s) converges to zero, we estimate its time variation:
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Lemma 5.8. Let (u, v) solve (4.5). Then, there exists a constant C > 0 such that for all t > 0, the estimate p dk (t) ≤ C k(t) (5.12) dt holds. Proof. We have dk (t) dt
Z
= =
Z ∂u ∂v 1 2 − (t)|∇u(t)| dx + 2 ( − v(t))∇u(t) · ∇ (t) dx ∂t ∂t D 2 Z D Z 1 ∂u 1 2 ∇u(t) · ∇( − v(t))|∇u(t)| dx − 2 ∇ · (( − v(t))∇u(t)) (t) dx 2 2 ∂t D D
where we have inserted (4.5) in the first and Gauss’ Theorem (with the boundary values ∂u ∂t = 0 on ∂D) in the second integral. Proceeding further with Gauss’ Theorem of (4.5) we obtain Z Z dk 1 ∂u (t) = (v(t) − )∇ · (∇u(t)|∇u(t)|2 ) dx + 2 ∇v(t) · ∇u(t) (t) dx − dt 2 ∂t D D Z 1 ∂u 2 (( − v(t))∆u(t)) (t) dx 2 ∂t Z D Z 1 1 = (v(t) − )∆u(t)|∇u(t)|2 dx + 2 (v(t) − )∇u(t)T D2 u(t)∇u(t) dx + 2 2 D D Z Z 1 1 ∂u ∂v ∂u (t) (t) dx − 2 (( − v(t))(( + v(t)) (t) dx 2 ∂t ∂t 2 ∂t D 2 Z ZD 1 ∂u ∂u ∆ (t) (t) dx, = 2 (v(t) − )∇u(t)T D2 u(t)∇u(t) dx + 2 2 ∂t ∂t D D where we have inserted ∆ ∂u ∂t (t) =
∂v ∂t (t)
and
1 1 1 (v(t) − )∆u(t) = (v(t) − )(v(t) + ) = 0. 2 2 2 A final application of Gauss’ Theorem to the second integral and subsequent use of the CauchySchwarz inequality yields Z Z ∂u 2 dk 1 T 2 (t) = 2 (v(t) − )∇u(t) D u(t)∇u(t) dx − 2 ∇ ∂t (t) dx dt 2 D D Z 1 ≤ 2 ( − v) ∇uT D2 u∇u dx 2 sDZ sZ 1 1 2 2 2 ≤ 2 ( − v) |D u∇u| dx ( − v) |∇u| dx D 2 D 2 sZ p 2 ≤ 2 |D2 u∇u| dx k (t) D
Note that since according to Remark 2.1 we have u ∈ W 2,p (D), for instance for p = 4, we get the following is uniformly bounded sZ 2 2 |D2 u∇u| dx ≤ C. D
for some constant C > 0, and hence the assertion follows. From (5.12) we conclude
and thus
p dk k (t) (t) ≤ Ck(t) dt Z ∞ Z t 3 3 2 2 2 k (τ ) dτ ≤ k (τ ) dτ → 0, k (t) − k (s) ≤ 3 s s
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MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
for s → ∞, s ≤ t. Now let, as above, sm → ∞ be a sequence such that k (sm ) → 0. For any other sequence tm → ∞ we can find ` (m) → ∞ such that tm ≥ s`(m) and hence 0 ≤ lim sup k (tm ) ≤ lim sup k s`(m) = 0 m→∞
m→∞
thus k (tm ) → 0. Hence, since every subsequence converges to zero, we obtain k(t) → 0 as t → ∞. The decrease of k is of fundamental importance for the convergence to a stationary point as we shall see in the following. 5.4. Convergence of the Level Set Scheme. We are now ready to establish the main result on the level set scheme, namely the convergence to a solution of the obstacle problem in the long-time asymptotic limit: Theorem 5.9. Let (u(t), v(t)) be the evolution obtained from (4.5). Then u(t) → u ˆ weakly in 1 p 2,p ˆ u, Σ) is the solution of the W (D), for arbitrary p, and v(t) → 2 − χΣˆ strongly in L (Ω), where (ˆ obstacle problem. Proof. Since u is uniformly bounded in W 2,p (D) for p arbitrarily large, every sequence tm → ∞ has a subsequence such that u (tm ) converges weakly in W 2,p (D) : u (tm ) * u ˆ in W 2,p (D) . By compact embedding we have u (tm ) → u ˆ in W 1,p (D) . Moreover, χΣ(t) :=
1 2
− v(t) is uniformly bounded in Lp (D) for p < ∞, and 0 ≤ χΣ(t) ≤ 1.
Let vm := v(tm ). Then there exists a subsequence (without loosing the generality we can allow it to be denoted by pm again) such that vm * vˆ in Lp (D) . We also have − 21 ≤ vˆ ≤ 21 . The convergence of vm and u(tm ) imply Z Z 1 1 2 2 k (tm ) = ( − vm ) |∇u (tm )| dx → ( − vˆ) |∇ˆ u| dx. 2 2 D D From the decay k(tm ) → 0 we conclude Z D
1 2 ( − vˆ) |∇ˆ u| dx = 0. 2
ˆ ˆ ⊂ D such that ∇ˆ ˆ and vˆ = 1 a.e. in D \ Σ. Hence, there exists a subset Σ u = 0 a.e. in Σ, 2 Consequently we also have 1 ∆ˆ u = vˆ + 2 ˆ This implies vˆ = 1 − χ ˆ . Hence χΣ(t ) * χ ˆ in Lp (D) , p < ∞ and consequently we a.e. in Σ. m Σ Σ 2 arrive at χΣ(tm ) → χΣˆ in Lp (D) , p < ∞. see [12, p.96, Thm.2.4]. Since Z Z Z Z Z Z 1 1 0= ∇u (tm ) · ∇ϕdx + ϕdx − pm ϕdx → ∇ˆ u · ∇ϕdx + ϕdx − pˆϕdx, 2 D 2 D D D D D for any ϕ ∈ H01 (D), the boundary value problem ∆ˆ u = 1 − χΣˆ |∇ˆ u| = 0
in D, ˆ in Σ
(5.13)
is satisfied by the limit. For the set {f = 0} ⊂ ∂Σ (0) one can show that each connected component ˆ has a nonempty intersection with ∂D and hence of Σ (t) (and finally of Σ) ˆ u ˆ ≡ 0 in Σ. ˆ is a solution of the obstacle problem. Thus, u ˆ, Σ
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17
6. Numerical Solution In the following we briefly comment on the numerical solution of the obstacle problem based on the level set approach. Note that numerical schemes like the one presented in the following have been used for various shape optimization and reconstruction problems (cf. [9] and the references therein). The numerical implementation is rather straight-forward, it consists in the following algorithm (1) Start at t = 0 with Σ(0) = {ϕ(·, 0) < 0} Loop: (2) Solve ∆u(t) = 1 − χΣ(t) , compute V(·, t) = −∇u(t) by an appropriate finite difference or finite element method. (3) Perform a time step of size δt of (1.4) with an upwind or ENO/WENO scheme (cf. [24]). (4) Set t = t + δt and go to (2). In order to obtain flexibility with respect to the shape of D, we choose a finite element approach. All numerical experiments are implemented using the finite element open-source software package Netgen/NgSolve. The Poisson equation as well as the level set equation are solved using a stabilized hybrid discontinuous Galerkin method introduced by Egger and Sch¨oberl in [13]. We rewrite the level set equation (1.4) in conservative formulation as ∂ϕ + div(Vϕ) − div(V)ϕ = 0. ∂t to fit into the framework of [13]. Since V = −∇u we obtain ∂ϕ − div(∇uϕ) + (1 − χΣ )ϕ = 0. (6.1) ∂t We use an implicit Euler method with δt = 10−2 for the time discretization of (1.4). In the first example we consider the solution of the obstacle problem on the domain D = {(x1 , x2 ) | (x1 − 0.5)2 + x22 ≤ 0.5, x2 > 0.5} ∪ [0, 1] × [0.4, 0.5] with boundary data ( 0 f (x1 , x2 ) = 2 e−200(x1 −0.5)
if x2 ≤ 0.5 else.
The Dirichlet boundary conditions and the initial data of the level set function satisfy ϕ(x, y) = y − 0.5.
(6.2)
We discretize the domain using a conforming mesh with 1574 triangles. The evolution of the level set function ϕ as well as the decrease of the energy functional is illustrated in Figure 1. Note that the moving boundary touches the fixed one tangentially, which is a good check with theoretical prediction (see [21]). One also observes the expected decrease and final (approximate) stationarity of the energy functional. In the second example we would like to illustrate the behavior of the free boundary when meeting an obstacle. We consider the domain D = {(x1 , x2 ) | (x1 − 0.5)2 + x22 ≤ 0.5, x2 > 0.5} ∪ ([0, 1] × [0.2, 0.5]\ ([0.3, 0.4] × [0.2, 0.45] ∩ [0.6, 0.75] × [0.2, 0.35])) , discretized into 3502 triangles. The boundary condition for u is given by ( 0 if x2 ≤ 0.5 f (x1 , x2 ) = 2 c e−200(x1 −0.5) else, the initial data by (6.2). Depending on the constant c the free boundary either merges with the obstacle or not. This behavior is illustrated in Figure 2. In the last example the computational domain is a segment of a circle with radius 1 and opening
18
MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
(a) t = 0
(b) t = 0.15
(c) t = 0.3
(d) t = 0.45
(e) t = 0.6
(f) t = 1
(g) Function u
(h) Evolution of the energy functional
Figure 1. Evolution of the level set function in time angle α = 60◦ , discretized into 4312 triangles. The system is supplemented with the following boundary conditions ( 2 c e−400(x1 −0.825) if x21 + x22 = 1 f (x1 , x2 ) = 0 else.
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
(a) t = 0.25
(b) t = 0.25
(c) t = 0.5
(d) t = 0.5
(e) t = 1
(f) t = 1
(g) t = 3
(h) t = 3
19
Figure 2. Left colum: c = 1, right colum c = 0.2 The evolution of the level set function is illustrated in Figure 3. Acknowledgements Part of this work was carried out when the authors were with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, and the Johannes Kepler University
20
MARTIN BURGER, NORAYR MATEVOSYAN, AND MARIE-THERESE WOLFRAM
(a) t = 0
(b) t = 0.5
(c) t = 1
(d) t = 5
(e) u
(f) Evolution of the energy functional
Figure 3. Evolution of the level set function and the energy functional Linz, respectively. The authors thank Heinz Engl (RICAM and University of Vienna) and Peter Markowich (Cambridge University and RICAM) for stimulating this joint research. M. Burger acknowledges financial support by the Austrian Science Foundation FWF through project SFB F 013 / 08, and the German Research Foundation DFG through the project Regularization with Singular Energies. Work of N. Matevosyan was partially supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) project nr.CI06 003, while he were at the University of Vienna. Also the work of N. Matevosyan and M.T. Wolfram is supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). References [1] R.A.Adams, Sobolev Spaces (Academic Press, 1975). [2] G.Allaire, F.Jouve, A.M.Toader, A level set method for shape optimization, C.R. Acad. Sci. Paris, Ser. I, 334, 1125-1130. [3] G.Barles, H.M.Soner, P.E.Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), 439-469. [4] E.N.Barron, R.Jensen, Semicontinuous viscosity solutions of Hamilton-Jacobi equations with convex Hamiltonians, Commun. PDE 15 (1990), 1713-1742.
A LEVEL SET METHOD FOR OBSTACLE PROBLEMS
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