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SINGULAR FREE BOUNDARY PROBLEM FROM IMAGE PROCESSING ´ ANNA LISA AMADORI AND JUAN LUIS VAZQUEZ Abstract. We study a degenerate nonlinear parabolic equation with moving boundaries which arises in the study of the technique of contour enhancement in image processing. In order to obtain mass concentration at the contour, singular data must be imposed at the free boundary, leading to a non-standard free boundary problem. Our main results are: (i) the well-posedness for the singular problem, without monotonicity assumptions on the initial datum, and (ii) the convergence of the approximation by combustion-type free-boundary problems.

1. Introduction This paper deals with some degenerate parabolic equation with moving boundaries which has arisen in the study of the technique of contour enhancement in image processing following the model of Malladi and Sethian [11]. According to [2] the contour is described in one dimension by an equation that controls the grey level, u(x, t) ∈ (0, 1) in the region D contained between two moving boundaries, x = ξ 0 (t), where u = 0, and x = ξ 1 (t), where u = 1. The whole problem reads  ut = Φ(ux )x , x ∈ (ξ 0 (t), ξ 1 (t)), t ∈ (0, T ),        i = 0, 1, t ∈ (0, T ),  u(ξ i (t), t) = i, Φ(ux )(ξ i (t), t) = Φ∞ , (1.1)   u(x, 0) = uo (x), x ∈ (`, r),       0 ξ (0) = `, ξ 1 (0) = r. This is a free boundary problem because the unknowns are the function u and the moving boundaries, ξ 0 (t) and ξ 1 (t), which must be determined from the problem thanks to the extra boundary conditions. For this reason the lines x = ξ0 (t) and x = ξ1 (t) are also called free boundaries. The space function u(x, t) for fixed t is called the front profile; it is limited on both sides by the free boundaries. In the image analogy, it is continued by the value u = 1 to the right, x > ξ1 (t), by zero to the left, x < ξ0 (t). We are interested in seeing whether the initial front concentrates with time, i.e., its profile sharpens as t grows. It can even happen that the two moving boundaries meet after a finite time, in which case a vertical front is formed, representing maximal contour enhancement. Date: March 24, 2004. 1991 Mathematics Subject Classification. Primary: ; Secondary: 1

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` A.L. AMADORI AND J.L. VAZQUEZ

In our study the function Φ is of class C 2 (R) and strictly increasing, so that the equation is in principle parabolic. But we also assume that Φ has finite limit at infinity: lim Φ(v) = Φ∞ < ∞.

v→+∞

This is a basic assumption of the model, and the prototype of function Φ is given by Φ0 (v) =

1 , (1 + v 2 )1+α

1 α>− , 2

The main cases are α = 0 and α = 1, cf. [2, 3]. Note that we can shift Φ by a constant without changing the problem. We use this property to normalize Φ(0) = 0. Then, Φ∞ > 0. We keep this normalization in the paper. Problem (1.1) is singular at the free boundary with respect to u since ux → ∞, and degenerate as a parabolic equation since Φ0 (ux ) → 0 (in some integral sense). This is rather non-standard in free boundary theory, and the reason for the mathematical difficulties and for our interest in the problem. With respect to the initial datum, the mathematical theory in the case of monotone uo was treated in [3]. However, the functional transformation used in that paper to solve the free boundary problem does not allow to study non-monotonic fronts, which is the first purpose of the present paper. Non-monotonic fronts are already proposed in Barenblatt’s paper [2], which contains the results of numerical computations by Chertock. We assume in our study that uo is a Lipschitz-continuous function defined on [`, r], with uo (`) = 0, uo (r) = 1, and 0 < uo < 1 on (`, r), not necessarily monotone. We shall assume for simplicity that it has a finite number of local extremal points and nonzero derivative at the end points, ` and r. Approximate problems. We propose to attack the difficulty created by the singular boundary conditions by looking at the modified problems:  ut = Φ(ux )x , x ∈ (ξε0 (t), ξε1 (t)), t ∈ (0, T ),        i = 0, 1, t ∈ (0, T ),  u(ξεi (t), t) = i, Φ(ux )(ξεi (t), t) = Φ(1/ε), (1.1.ε)   u(x, 0) = uoε (x), x ∈ (l, r),       0 ξ (0) = `, ξ 1 (0) = r, where ε > 0 and uoε is a suitable approximation of uo . The main difference of (1.1.ε) with respect to the original problem lies in the finite boundary condition imposed at the free boundaries, which is taken from combustion theory, which have been object of intensive study in recent years. It is usually referred to as the “combustion boundary condition”. We refer to the survey [16] and to references therein for more details about the topic. Relating problems (1.1) and (1.1.ε) is the second objective we have in mind.

SINGULAR FREE BOUNDARY PROBLEMS

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Summing up, our plan is to establish well-posedness for the singular problem (1.1), and to show that the combustion-type problems (1.1.ε) converge to the original singular problem, when ε → 0 so that ux → ∞ at the boundaries. To this aim, we state the problem in a slightly more general form, and then use a problem reformulation. Indeed, we consider the nonlinear diffusion problems of the type:  ut = Φ(ux )x , x ∈ (ξ 0 (t), ξ 1 (t)), t ∈ (0, T ),        i = 0, 1, t ∈ (0, T ),  u(ξ i (t), t) = i, Φ(ux )(ξ i (t), t) = f i (t), (1.2)   u(x, 0) = uo (x), x ∈ (`, r),       0 ξ (0) = `, ξ 1 (0) = r, where the boundary data f 0 and f 1 take values inside the range of Φ. Now, this freeboundary problem can be regarded as an elliptic-parabolic problem   b(u)t = Φ(ux )x , x ∈ (a0 , a1 ), t ∈ (0, T ),     Φ(ux (ai , t)) = f i (t), i = 0, 1, t ∈ (0, T ), (1.3)      b(u(x, 0)) = v (x), x ∈ (a0 , a1 ), o where the new nonlinearity b is defined as   0 b(u) := u  1

if u ≤ 0, if u ∈ (0, 1), if u ≥ 1,

We set vo := b(uo ), and a0 , a1 are taken by convenience, respectively, suitably small and large. The free boundaries of the former problem can be recovered a-posteriori by means of ξ 0 (t) := sup {x ∈ (a0 , a1 ) : u(y, t) = 0 as y ∈ (a0 , x)} , ξ 1 (t) := inf {x ∈ (a0 , a1 ) : u(y, t) = 1 as y ∈ (x, a1 )} . Elliptic-parabolic problems have been studied by a number of authors in the literature. We mention [1, 12] for the general well-posedness theory. The equivalence with the free boundary problem and the qualitative behavior and regularity of solutions and free boundaries were studied by researchers of the Leiden group [5, 8, 9, 15] in the case of linear diffusion, Φ(ux ) = ux , and we use many of their results and ideas. The passage to the limit is rather ticklish and needs a separate analysis of interior and boundary convergence, which forms the most delicate part of our analysis. 2. Statement of main results We study Problem (1.3), we derive from it conclusions about Problem (1.1.ε), and finally send ε to zero to obtain the solution of (1.1). Sending ε to zero presents some

` A.L. AMADORI AND J.L. VAZQUEZ

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difficulties: energy based techniques merely provide a bound of the L1 -norms of uε,x , while the structure of problem (1.1) needs ux to be a function. On the other hand, such bound can even be optimal; [3] shows that the free boundaries can collapse after a finite time, in the monotone case: this means that the limit of uε becomes discontinuous; more precisely, the solution develops a vertical front. Such a difficulty was overcome in [4], when dealing with a slightly different problem, by introducing an ad hoc notion of solution, which only requests u to have bounded variation; further regularity is recovered in a second stage. Here we are able to obtain more regularity for u, by taking advantage of the particular “parabolic regularization” of (1.1) implied by (1.1.ε). We shall produce a classical solution in the following sense: Definition 2.1. Let ξ 0 < ξ 1 two continuous real curves defined on [0, T ) with ξ 0 (0) = `, ξ 1 (0) = r, and u a function defined on © ª D := (x, t) : x ∈ (ξ 0 (t), ξ 1 (t)), t ∈ (0, T ) . We say that the triplet (u, ξ 0 , ξ 1 ) is a classical solution to problem (1.1) if (i) u ∈ C 2,1 (D) satisfies ut = Φ(ux )x in classical sense, (ii) u is continuous up to the parabolic boundary with u(ξ i (t), t) = i, u(x, 0) = uo , (iii) Φ(ux ) is continuous up to the free boundaries with Φ(ux )(ξ i (t), t) = Φ∞ . For later use we introduce the notations © ª ˜ := D ∪ (ξ i (t), t) : t ∈ (0, T ), i = 0, 1 , D Do := D ∪ (`, r) × {0}. We list the precise assumptions that shall be in force in all what follows: H.1 The function Φ is of class C 2 on R, strictly increasing with Φ0 (s) > 0 for all s ∈ R, and a finite limit lim Φ(v) = Φ∞ < ∞. v→+∞

H.2 uo is a Lipschitz-continuous function on [`, r] with uo (`) = 0, uo (r) = 1, and 0 < uo < 1 on (`, r). Moreover it has a finite number of local extremal points and there exist ` < `0 < r0 < r and β > 0 so that uo (x2 ) − uo (x1 ) ≥ β(x2 − x1 ) if ` ≤ x1 ≤ x2 ≤ `0 or r0 ≤ x1 ≤ x2 ≤ r. Hypothesis H.2 allows to approximate uo by means of a sequence uε satisfying: H.2.ε uoε ∈ C ∞ (`, r) ∩ W 3,∞ (`, r) ∩ C 1 ([`, r]) converges to uo in C([`, r]) with uoε (`) = 0, uoε (r) = 1, and m := sup{the largest local maximum value of uoε in (`, r)} < 1, ε>0

m := inf {the smallest local minimum value of uoε in (`, r)} > 0. ε>0

SINGULAR FREE BOUNDARY PROBLEMS

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Moreover, there exist 0 < β ≤ b < 1/ε and ` < `0 < r0 < r (not depending by ε) such that |u0oε (x)| ≤ b u0oε (x) ≥ β

if x ∈ [`, r], if x ∈ [`, `0 ] ∪ [r0 , r],

˜ ε , Doε In what follows, we shall denote by (uε , ξε0 , ξε1 ) the solution to (1.1.ε), and by Dε , D the respective sets. The convergence result can be stated as follows. Theorem 2.1 (Convergence). Let assumptions H.1, H.2, and H.2.ε hold. Then there exists a time T such that the solution (uε , ξε0 , ξε1 ) of the nonsingular free boundary problem (1.1.ε) converges as ε → 0 to a classical solution (u, ξ 0 , ξ 1 ) of (1.1) in the following sense, 1+α,α (i) the functions uε converge to u in Cloc (D) for α < 1, and in Cloc (Do ), i i (ii) the free boundaries ξε tend to ξ in Cloc ([0, T )), ˜ (iii) the flux functions Φ(uε,x ) tend to Φ(ux ) in Cloc (D).

The maximal time T may be finite or infinite, depending on Φ. To this purpose, an important role is played by the quantity A(Φ) := lim s2 Φ0 (s). s→∞

The maximal time T is infinite if A(Φ) < ∞, and it is a finite constant, depending on uo and Φ, if A(Φ) = ∞. The proof of Theorem 2.1 splits into two steps: Theorems 5.5 and 5.7. Combining the convergence result with the comparison principle established by Theorem 4.3 provides the well posedness of Problem (1.1). Theorem 2.2 (Well posedness). Under assumptions H.1, H.2, the singular free boundary problem (1.1) admits a unique classical solution defined in a maximal time interval T > 0. The Comparison Principle holds. 3. Nonsingular problems: elliptic-parabolic formulation In this section we collect some results about the elliptic-parabolic problem (1.3). Most of the arguments extend the classical theory from the already mentioned references to the present situation of nonlinear diffusion given by a function Φ. Our main interest stands in obtaining, via the elliptic parabolic formulation, well posedness for the non-singular free boundary problem (1.2). Concerning the boundary data, we assume that H.3 The functions f i ∈ Lip(0, T ), i = 0, 1, take values inside the range of Φ. Moreover f i > Φ(0). Notice that the boundary condition of problem (1.1.ε), f i = Φ(1/ε), satisfies the request H.3. As we have already said, equation b(u)t = Φ(ux )x is unchanged after shifting Φ by

` A.L. AMADORI AND J.L. VAZQUEZ

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a constant. On the other hand, the boundary condition Φ(ux ) = f i makes sense only for a particular choice of that shift. In all what follows, we shall normalize Φ(0) = 0. We shall write I for (a0 , a1 ) and QT for I × (0, T ). We begin by recalling the notion of weak solution to the elliptic-parabolic problem (1.3) ¡ ¢ Definition 3.1. A function u ∈ L2 0, T ; H 1 (I) is a weak solution to the ellipticparabolic problem (1.3) if (i) b(u) ∈ C(QT ), (ii) for all ϕ ∈ C 1 (QT ) which vanish at t = T , Z [−b(u)ϕt + Φ(ux )ϕx ] dxdt = Z Z T £ 1 ¤ vo (x)ϕ(x, 0)dx + f (t)ϕ(a1 , t) − f 0 (t)ϕ(a0 , t) dt. QT

(3.1)

I

0

Remark 3.1. It is easily seen that (3.1) is equivalent to Z Z Z (3.2) {−b(u)ϕt + [Φ(ux ) − f ] ϕx } dxdt = vo (x)ϕ(x, 0)dx + QT

I

fx ϕdxdt QT

where f (x, t) = f 0 (t) + (f 1 (t)−f 0 (t))(x − a0 )/(a1 − a0 ). By a weak solution to the free boundary problem (1.2) we mean a solution to the elliptic parabolic problem (1.3) whose interfaces ξ i do not touch the artificial boundaries ai . More precisely, we give the following definition, in the spirit of [6]. Definition 3.2. Let ξ 0 < ξ 1 two continuous real curves defined on [0, T ) with ξ 0 (0) = `, ξ 1 (0) = r, and u a function defined on © ª D := (x, t) : x ∈ (ξ 0 (t), ξ 1 (t)), t ∈ (0, T ) . We say that the triplet (u, ξ 0 , ξ 1 ) is a weak solution to the free boundary problem (1.2) if (i) u is continuous on D ∪ {(ξ i (t), t) : t ∈ [0, T ], i = 0, 1} with u(ξ i (t), t) = i, (ii) ux ∈ L∞ (D) and the weak form stated in Definition 3.1(ii) is satisfied for all a0 ≤ inf ξ 0 , a1 ≥ sup ξ 1 . Here it is understood that the function u is extended linearly outside the free boundaries by u(x, t) = 1 + Φ−1 (f 1 (t))(x − ξ 1 (t)) if x > ξ 1 (t), and u(x, t) = Φ−1 (f 0 (t))(x − ξ 0 (t)) if x < ξ 0 (t). Existence of solution to (1.3) can be obtained by a standard approximation technique. We set ¢ ¡ f = max kΦ−1 (f 0 )k∞ ), kΦ−1 (f 1 )k∞ , ku0o k∞ , and approximate (uniformly on compact sets) b and Φ by means of two sequences bn and Φn of class C ∞ with bn ≡ b on (0, 1), Φn ≡ Φ on [−2f − n, 2f + n], b0n , Φ0n > 1/n,

SINGULAR FREE BOUNDARY PROBLEMS

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and b0n , Φ0n ≤ b. We also approximate the data uo and f i in Lip(0, T ) by means of two sequences uon , fni of class C ∞ . Next we look at   bn (u)t = Φn (ux )x , x ∈ I, t ∈ (0, T ),     Φn (ux (ai , t)) = fni , i = 0, 1, t ∈ (0, T ), (3.3)      u(x, 0) = u (x), x ∈ I, on By standard parabolic theory, for all n (3.3) has an unique classical solution un . In order to list some uniform estimates for (3.3), we introduce the so called saturation time, i.e. ½ ¾ Z a1 Z T 1 0 Ts := min T > 0 : vo dx + (f − f )dt ∈ {0, a1 − a0 } . a0

0

By means of well-known barrier and energy techniques one can prove that: Lemma 3.2 (Estimates). For all integers n and T < Ts we have (3.4)

min{essinf u0on , Φ−1 (min fni )} ≤ un,x ≤ max{esssup u0on , Φ−1 (max fni )}.

Moreover there exist c1 , · · · , c4 , not depending on n, such that (3.5)

kun kL∞ (QT ) ≤ c1 ,

(3.6)

© ª sup kbn (un (x))kC 1/2 [0,T ] : x ∈ I ≤ c2 ,

where un (x) denotes un (x, t) as a function of t, for fixed x, (3.7)

kΦ(un,x )x kL2 (QT ) ≤ c3 ,

(3.8)

kun,xx kL2 (QT ) ≤ c4 .

Proof. (3.4) follows by applying maximum principle to the equation satisfied by w = un,x , namely  Φ0 (w) Φ00 (w) 2 Φ0n (w)b00n (u)   wxx + 0n w − wwx ,  wt = 0n bn (u) bn (u) x b0n (u)2 w(i, t) = Φ−1 (fni ),    w(x, 0) = u0 (x). on The mass balance says that for all t < T Z Z Z T bn (un (t))dx = bn (uon )dx + (fn1 − fn0 )dt, I

I

0

and the right-hand side in the last formula lies in the interval (0, a1 −a0 ). Hence, as bn (un ) is continuous, there exists y = y(t) such that bn (un (y, t)) ∈ (0, 1) and so un (y, t) ∈ (0, 1);

` A.L. AMADORI AND J.L. VAZQUEZ

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then (3.4) yields (3.5). Concerning the regularity w.r.t. t, we fix x1 , x2 ∈ I, t1 , t2 ∈ [0, T ] and compute ¯Z x2 ¯ ¯Z t2 ¯ ¯ ¯ ¯ ¯ ¯ [bn (un )(t1 ) − bn (un )(t2 )] dx¯¯ = ¯¯ [Φ(un,x )(x2 ) − Φ(un,x )(x1 )] dt¯¯ ¯ x1

t1

≤ 2Φ∞ |t1 − t2 |;

eventually (3.6) follows by invoking [15, Proposition 1]. Regarding (3.7), we have Z TZ Z TZ 2 (Φ(un,x )x ) dxdt ≤ b Φ(un,x )x un,t dxdt 0 I 0 I Z Z o n T£ T Z ¤ 0 1 Φ(un,x )un,xt dxdt =b fn un,t (a1 ) − fn un,t (a0 ) dt − Z 0 I X0 i ≤b kfn kW 1,1 (QT ) kun kL∞ (QT ) + b [F (u0on ) − F (un,x )(T )] dx, with F (v) := Z

Rv

QT

I

i=0,1

Φ(s)ds ∈ [0, Φ∞ |v|]. Therefore X kfni kW 1,1 (QT ) kun kL∞ (QT ) + bΦ∞ ku0on kL1 (I) , Φ(un,x )2x dxdt ≤ b

0

i=0,1

which is bounded, uniformly w.r.t. n, by (3.5). Lastly, take a = inf{Φ0 (v) : |v| ≤ f} and note that a > 0 by H.1; so Z Z TZ 1 2 Φ(un,x )2x dxdt. un,xx dxdt ≤ 2 a 0 I QT Eventually (3.7) yields (3.8).

¤

Remark 3.3. In particular, for all n, un solves problem (3.3) with Φ instead of Φn . These estimates allow to obtain the following convergence result, by standard compactness arguments (see [14]). ¡ 0+1, 12 ∞ Proposition 3.4. Let T < T . There exist three functions v ∈ C ), u ∈ L (Q 0, T ; s T ¢ ¡ ¢ ¡ ¢ 1,∞ 2 2 2 1 W (I) ∩ L 0, T ; H (I) , and p ∈ L 0, T ; H (I) and three subsequences of bn (un ), un , and Φ(un,x ), that we still denote by bn (un ), un , and Φ(un,x ), such that (3.9) (3.10)

bn (un ) → v un → u

(3.11)

Φ(un,x ) → p

strongly in C α,α/2 (QT ) as α < 1, ¡ ¢ weakly in L2 0, T ; H 2 (I) , ¡ ¢ weakly in L2 0, T ; H 1 (I) .

In order to pass to the limit inside the equation, it is needed to relate v, u, and p. First of all we notice that 0 ≤ v ≤ 1 by construction. Next, following [1], we introduce the functions Z un Z u Bn (un ) := [bn (un ) − b(s)] ds, B(u) := [b(u) − b(s)] ds. 0

0

SINGULAR FREE BOUNDARY PROBLEMS

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Lemma 3.5. The function v coincides with b(u) almost everywhere in QT . Moreover bn (un )t → b(u)t weakly in L2 (QT ), up to an extracted sequence. Next B(u) ∈ L∞ (0, T ; L1 (I)) with Z Z Z tZ (3.12) B(u)(x, t)dx = B(uo )(x)dx + b(u)t udxds, I

I

0

I

for almost all t, and lim inf [Bn (un ) − B(u)] ≥ 0, n→∞

for almost all x, t. Proof. The first part of the statement follows as in [15, Lemma 5]. Next, bn (un )t = Φ(un,x )x is equibounded in L2 (QT ) by the estimate (3.7). So, keeping in mind the convergence stated in (3.9), there is an extracted sequence which tends to b(u)t weakly in L2 (QT ). Afterwards [1, Lemma 1.5] gives (3.12). Lastly, by the monotonicity of bn we have Z u

Bn − B ≥ [bn (un ) − b(u)] u +

[bn (s) − b(s)] ds 0

which vanishes as n → 0 by (3.9)

¤ ¡ ¢ Lemma 3.6. The functions un and Φ(un,x ) tend to u and Φ(ux ) strongly in L2 0, T ; H 1 and in L2 (QT ), respectively. Proof. By taking advantage of hypothesis H.1, it suffices to check that un,x → ux strongly in L2 (QT ). We set a = inf{Φ0 (v) : |v| ≤ f} and use un − u − fn /a as a test function in (3.1). Writing for simplicity of notations hn := un − u we have Z Z 1 2 [Bn − B](t)dx + akhn,x kL2 (QT ) − fn hn,x dxdt − kfn k2L2 (QT ) a I Z QT Z Z T [bn (un )t −b(u)t ]udt − Φ(ux )hn,x dxdt + fn hn dxdt ≤ 0

QT

QT

Lemma 3.5 and the convergence result (3.10) imply that theR right-hand side vanishes as n → 0. The thesis follows because, after estimating the term fn hn,x by Cauchy Schwartz inequality we obtain n o a 2 lim kBn − BkL∞ (0,T ;L1 ) + khn,x kL2 (QT ) ≤ 0. n→∞ 2 ¤ Summing up, we have proved: Theorem 3.7 (Existence). The function u constructed 3.4 is a weak so¡ in Proposition ¢ ¡ ¢ lution to (3.1). Moreover u ∈ L∞ 0, T ; W 1∞ (I) ∩ L2 0, T ; H 2 (I) . Concerning uniqueness of solution, the same arguments of [12] produce the following result:

` A.L. AMADORI AND J.L. VAZQUEZ

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Theorem 3.8 (Uniqueness). Let u1 and u2 be two solutions to (1.3) with equal boundary data f and respective initial data v1o , v2o . Then for all t > 0 Z Z [b(u1 ) − b(u2 )]+ (t)dx ≤ [v1o − v2o ]+ dx. I

I

In particular, (1.3) has an unique solution, in the sense that the function b(u) is uniquely determined. 3.1. Some qualitative properties. We now investigate the regularity and qualitative behavior of solutions, in the spirit of similar results in [5, 8, 9, 15]. As a byproduct, we obtain the well posedness of the free boundary problem (1.2) via elliptic parabolic formulation. To this aim we set ξ 0 (t) := sup {x ∈ I : b(u) ≡ 0 on (a0 , x] × {t}} , ξ 1 (t) := inf {x ∈ I : b(u) ≡ 1 on [x, a1 ) × {t}} , ˆ := {(x, t) ∈ QT : v(x, t) ∈ (0, 1)} , D P := {(x, t) ∈ QT : v(x, t) = i, i = 0, 1} . Proposition 3.9. We have (i) (ii) (iii)

ˆ = D = {(x, t) : x ∈ (ξ 0 (t), ξ 1 (t)), t ∈ [0, T )}, D uxx = 0 a.e. in the interior of P. u is a classical solution in D.

ˆ then there Proof. With respect to (i), it is sufficient to check that, if (x1 , to ), (x2 , to ) ∈ D, ˆ By continuity, there exists δ > 0 such that exists δ > 0 such that [x1 , x1 ] × (to−δ, to ] ⊂ D. 0 < u(xi , t) < 1 as i = 1, 2 and t ∈ [to −δ, to ]. We thus look at the parabolic problem  x ∈ (x1 , x2 ), t ∈ (to −δ, to ),  wt = Φ(wx )x , w = u, x = x1 , x2 , t ∈ (0, T ),  w = b(u), x ∈ (x1 , x2 ), t = to −δ. By standard parabolic theory (see Remark 3.3), there exists an unique classical solution w. Moreover, the strong maximum principle implies 0 < w < 1 on [x1 , x1 ] × (to −δ, to ]. Therefore, actually w solves  x ∈ (x1 , x2 ), t ∈ (to −δ, to ),  b(w)t = Φ(wx )x , w = u, x = x1 , x2 , t ∈ (0, T ),  b(w) = b(u), x ∈ (x1 , x2 ), t = to −δ, as well as u. Eventually, [1, Theorem 2.2] establishes that u = w ∈ (0, 1) on [x1 , x1 ] × (to −δ, to ]. Concerning (ii) and (iii), they can be proved as in [9, Theorem 4] (see also [15, Theorem 5]). ¤

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Remark 3.10. In force of the regularity established by Theorem 3.7 and Proposition 3.9(iii), we may assume after redefining u on a set of zero measure that u(x, t) = Φ−1 (f 0 (t)) (x − ξ 0 (t)) u(x, t) = 1 + Φ−1 (f 1 (t)) (x − ξ 1 (t))

if x ≤ ξ 0 (t), if x ≥ ξ 1 (t).

In order to study the convergence of problems (1.3.ε), it is useful to obtain some comparison result about solutions to (1.3) with different data. Proposition 3.11 (Comparison). Let uk , as k = 1, 2, be weak solutions to (1.3) with data fk0 , fk1 , and vok , respectively. Under assumptions H.1–3, for all t ∈ (0, T ) we have Z Z [b(u1 ) − b(u2 )]+ (x, t)dx ≤ [vo1 − vo2 ]+ dx I I Z tn (3.13) £ 1 ¤ £ 0 ¤ o 1 + f1 − f2 + + f1 − f20 − ds. 0

Proof. If assumptions H.1–3 holds, it suffices to check that (3.13) holds after replacing b(uk ) by vkn := bn (ukn ). For all t we have Z © ª (v1n − v2n )t sgn + (v1n − v2n ) (x, t)dx = Z I ª © (Φ(u1n,x ) − Φ(u2n,x ))x sgn + (v1n − v2n ) (x, t)dx. I

+

AsR sgn is Lipschitz continuous and vin are smooth, the left-hand side is equal to d [v1n − v2n ]+ (x, t)dx. Concerning the right-hand side, we notice that the set (a0 , a1 )r dt I {x : v1n − v2n ≤ 0} splits into open intervals where v1n > v2n , because vin (t) are of class C 1 with respect to x. Hence u1n > u2n , as bn is strictly increasing. Take (a, b) one of such intervals: Z b © ª (Φ(u1n,x ) − Φ(u2n,x ))x sgn + (v1n − v2n ) (x, t)dx = a

(Φ(u1n,x ) − Φ(u2n,x )) (b, t) − (Φ(u1n,x ) − Φ(u2n,x )) (a, t). If b = a1 , the first term on the right-hand side is equal to f11 (t) − f21 (t) and henceforth less or equal to its positive part. If b ∈ I, it is a minimum point for the smooth function u1n − u2n on [a, b] and therefore u1n,x ≤ u2n,x . Because Φ is nonincreasing, it yields that the first term on the right-hand side is less or equal to zero. Similar arguments applied to a give that Z £ ¤ £ ¤ d [v1n − v2n ]+ (x, t)dx ≤ f11 − f21 + (t) + f10 − f20 − (t) dt I for all t ∈ (0, T ). The thesis follows after integrating on (0, t). ¤ Remark 3.12. It is easily seen that, under the same assumptions of Proposition 3.11, the following contraction property holds X° ° °f1i − f0i ° 1 . (3.14) kb(u1 )(t) − b(u2 )(t)k 1 ≤ kvo1 − vo2 k 1 + L (I)

L (I)

L (0,T )

i=0,1

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A first byproduct of Proposition 3.11 is that, up to choose a0 sufficiently small and a1 sufficiently large, there is no contact between the free boundaries ξ i and the artificial boundaries ai . It follows immediately by next result. Proposition 3.13. For any T > 0, ξ 1 (resp., ξ 0 ) is bounded from above (resp., from below) for t ∈ [0, T ]. The bound does not depend by a0 , a1 . Proof. We only deal with the right free boundary ξ 1 . The same arguments applied to (1.3) with initial datum 1 − uo (−x) give the respective bound for ξ 0 . We look for a subsolution to (1.3) of travelling wave type: u(x, t) = g(x − ct), where c is a parameter to be determined. The function g ∈ C 1 (R) is chosen in such a way that g(s) = Φ−1 (c + k)(s − r),

s ≤ r,

g 0 (s) = Φ−1 (k + c(1 − g(s))),

r < s < r,

g(s) = 1 + Φ−1 (k)(s − r),

s ≥ r,

with k = min f 1 , c = max f 0 − k, and Z

1

r=r+ 0

du 1 =r+ −1 Φ (k + c(1 − u)) c

Φ−1 Z(c+k)

Φ0 (y) dy ∈ (r, ∞). y

Φ−1 (k)

Eventually Proposition 3.11 implies that b(u) ≥ b(u): hence ξ 1 (t) ≤ r +ct, as wanted. ¤ Remark 3.14. When dealing with the approximating problems (1.3.ε), hypothesis H.3 is trivially satisfied. Besides, it comes out by the same proof that the conclusions of Proposition 3.13 still hold without assumptions about the sign of f 0 and f 1 , if the diffusion Φ degenerate also at zero. We now establish the continuity of the free boundaries ξ i , by means of barrier argument inspired by [8]. Proposition 3.15. The curves t 7→ ξ i (t) are uniformly continuous on [0, Ts ). Proof. We only consider ξ 1 , being the case of ξ 0 completely analogous. The proof is very next to the one given in [8]: changes are only needed to produce a continuous function ω with ω(0) = 0 so that (3.15)

ξ 1 (t) ≤ ξ 1 (to ) + ω(t − to )

if 0 ≤ to ≤ t < Ts .

In view of Proposition 3.13, we may assume that ξ 1 (to ) < a1 ; for any 0 < h < (a1 − ξ 1 (to ))/2, we construct a subsolution u of the form u(x, t; h) = v(x; h) + a(h)(t − to ),

SINGULAR FREE BOUNDARY PROBLEMS

13

where a(h) is a positive constant depending on h and v ∈ C 1 [a0 , a1 ] is defined by v 0 (x; h) = l0 , v(x; h) = g(x−ξ 1 (to ); h), v 0 (x; h) = l1 ,

in [a0 , ξ 1 (to )], in [ξ 1 (to ), ξ 1 (to )+h], in [ξ 1 (to )+h, a1 ].

Here, l0 = max (max Φ−1 (f 0 ), max Φ−1 (f 1 ), ku0o k∞ ) ≥ 0 and l1 = min Φ−1 (f 1 ) ∈ [0, l0 ]. The function g ∈ C 2 [0, h] will be chosen so that g(h; h) = 1, g 0 (0, h) = l0 and g 0 (h; h) = l1 . Now u solves ut = Φ(ux )x on (ξ 1 (to ), ξ 1 (to )+h) × (to , Ts ) if Z h g(y; h) = 1 − Φ−1 (Φ(l0 ) − a(h)η)dη. y

Thus the item g 0 (h; h) = l1 is assured by choosing A(h) = [Φ(l0 ) − Φ(l1 )] /h ≥ 0. Note that v(ξ 1 (to ), to ) ≤ 1 and vx (x, to ) ≥ esssup ux (to ) if a0 < x < ξ 1 (to ) in view of (3.4). Therefore Proposition 3.11 implies that b(u) ≥ b(u) at t ≥ to . In particular ξ 1 (t) ≤ ξ 1 (t), where ξ 1 stands for the right free boundary of u. For each h, ξ 1 (t) = ξ 1 (to ) + 2h for t = to + l1 h2 / [Φ(l0 ) − Φ(l1 )]. Inverting the dependence between t and h in this last relation provides the function w which fulfills (3.15). ¤ Note that the proof of Proposition 3.15 shows that the interfaces are at least Holdercontinuous with Holder exponent equal to 1/2. Eventually we have obtained the well posedness of the free boundary problem (1.2). Theorem 3.16. Under assumptions H.1–3 and for all T < Ts , the nonsingular free boundary problem (1.2) admits a unique solution in the sense of Definition 3.2. Moreover the solution u has the following regularity properties: (i) u ∈ C 2,1 (D) solves ut = Φ(ux )x in classical sense, 7 ux (x, t) is continuous till x = ξ i (t) with ux (ξ i (t), t) = (ii) for almost any t, x → Φ−1 (f i (t)). Proof. Let u be the solution to the elliptic-parabolic problem (1.3), where ai are chosen in such a way that the respective set D does not touch {ai }×[0, T ), according to Proposition 3.13. The comparison result obtained in Proposition 3.11 implies that set D and the function u restricted to D does not depend by the choice of a0 , a1 . Afterwards Theorem 3.7 and Propositions 3.9, 3.15 yields that the triplet (u, ξ 0 , ξ 1 ) solves (1.2) in the sense of Definition 3.2, and has the stated regularity. Concerning uniqueness, it follows from Theorem 3.8. ¤ We explicitly remark that Ts = ∞ for the approximating problems (1.1.ε).

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4. Singular problem, I: Uniqueness and comparison Before showing the convergence of the combustion-type Problem (1.1.ε) to (1.1), we settle the question of uniqueness and comparison. In order to do that we need some preliminaries. Thus, it will be useful to give another proof of the comparison and contraction properties for elliptic-parabolic problem, stated by (3.13) and (3.14), which does not need the use of uniformly parabolic approximation. Definition 4.1. We say that a (weak) solution to (1.3) is strong if it satisfies, beneath Definition 3.1, (iii) b(u)t ∈ L2 (QT ), (iv) for a.a. t, the set D(t) := {x : 0 < b(u)(x, t) < 1} is connected, and the function x 7→ Φ(ux )(x, t) is continuous, equal to f 0 (t) at the left of D(t) and to f 1 (t) at the right of D(t). Note that a strong solution satisfies the equation almost everywhere inside QT . Now we have that Proposition 4.1. Let uk , as k = 1, 2, be strong solutions to the elliptic-parabolic problem (1.3) with data fk0 , fk1 , and vok , respectively. Then the comparison property (3.13) and the contraction property (3.14) hold. Proof. Since b(uk )t ∈ L2 , we can multiply the equation for the difference b(u1 ) − b(u2 ) by sgn + (b(u1 ) − b(u2 )) and integrate over I. For almost any t we get Z Z © ª d + sgn (b(u1 )−b(u2 )) (x, t)dx = (Φ(u1,x )−Φ(u2,x ))x sgn + (b(u1 )−b(u2 )) (x, t)dx, dt I I and we may assume that Definition 4.1 (iv) is satisfied at that t. Concerning the righthand side, we notice that {x : b(u1 ) > b(u2 )} splits into open intervals. Take (c, d) one of such intervals: Z d © ª (Φ(u1,x )−Φ(u2,x ))x sgn + (b(u1 )−b(u2 )) (x, t)dx = c

(Φ(u1,x ) − Φ(u2,x )) (d, t) − (Φ(u1,x ) − Φ(u2,x )) (c, t). We claim that (Φ(u1,x ) − Φ(u2,x )) ≤ 0 at all that points d, except at most one where it is equal to f11 (t) − f21 (t). Similar arguments applied to the other extreme c and integration over time give (3.13), while (3.14) follows after switching u1 and u2 . In order to check our claim, we set Dk (t) = {x : 0 < b(uk )(x, t) < 1} = (ξk0 , ξk1 ). By construction b(uk ) ≡ 0 at the left of ξk0 and b(uk ) ≡ 1 at the right of ξk1 , so that it is needed that min{ξ10 , ξ20 } < d ≤ max{ξ11 , ξ21 }. Indeed, also the item min{ξ10 , ξ20 } < d ≤ max{ξ10 , ξ20 } has to be rejected: otherwise the continuity of b(uk ) would imply b(u1 ) = 0 = b(u2 ) at d and therefore ξ10 = ξ20 = d. If d ∈ D1 (t) ∩ D2 (t), the same arguments of Proposition 3.11 yields Φ(u1,x ) ≤ Φ(u2,x ). If min{ξ11 , ξ21 } ≤ d ≤ max{ξ11 , ξ21 }, then actually ξ11 ≤ d ≤ ξ21 , because by construction 1 ≥ b(u1 ) > b(u2 ) at the left of d. In particular b(u1 ) = 1

SINGULAR FREE BOUNDARY PROBLEMS

15

and therefore by continuity b(u2 ) = 1 at d. Summing up, the only possibility is that d = ξ21 ≥ ξ11 : thus Definition 4.1 yields (Φ(u1,x ) − Φ(u2,x )) (d, t) = f11 (t) − f21 (t). ¤ We will also need a geometrical property of all classical solutions. Lemma 4.2. Let u be a classical solution to (1.1) defined for t < T , and let 0 < τ < T . Then there exists δ > 0 such that, for any t ∈ [0, τ ], the interval (ξ 0 (t), ξ 1 (t)) splits into three subintervals (ξ 0 (t), ξ 0δ (t)), (ξ 0δ (t), ξ 1δ (t)), and (ξ 1δ (t), ξ 1 (t)) with 0 < u(x, t) < δ δ < u(x, t) < 1 − δ 1 − δ < u(x, t) < 1

if ξ 0 (t) < x < ξ 0δ (t), if ξ 0δ (t) < x < ξ 1δ (t), if ξ 1δ (t) < x < ξ 1 (t).

Moreover, the curves ξ iδ are of class C 1 and converge uniformly to the free boundaries as δ → 0. Proof. We define ξ iδ as the curves respectively given by ξ 0δ (t) := min{x ≥ ξ 0 (t) : uk (x, t) = δ}, ξ 1δ (t) := max{x ≤ ξ 1 (t) : u(x, t) = 1 − δ}. for sufficiently small δ > 0. We only examine the upper region 1 − δ < u < 1, since the lower one is completely analogous. The concept of classical solution implies that there is a small neighborhood of the line x = ξ 1 (t) where ux ≥ c À 0. Therefore, no local maxima may exist in that neighborhood, let us say for x ∈ Iε1 (t) = (ξ 1 (t) − ε, ξ 1 (t)), 0 < t ≤ τ . Note that ε depends only on τ . If m(t) is the supremum of local maximum points of the front x 7→ u(x, t) on (ξ 0 (t), ξ 1 (t)) at time t, then it is elementary to check that, since u and ux are continuous, the value m(t) is attained at some point xm (t) ∈ [ξ 0 (t), ξ 1 (t)] with ux (xm (t), t) = 0, hence for x ≤ ξ 1 (t) − ε. Let us call m := sup{m(t) : t ≤ τ }. This value is less than 1, say 1 − 2δ. With this δ we prove that the curves x = ξ 0δ (t), ξ 1δ (t) are C 1 by the Implicit Function Theorem. They converge uniformly to the free boundaries as δ → 0. ¤ We are now ready to prove the main result of this section. Theorem 4.3 (Uniqueness and comparison for classical solutions). Assume H.1, and take two functions uo1 and uo2 , both satisfying hypothesis H.2. Let (u1 , ξ10 , ξ11 ) and (u2 , ξ20 , ξ21 ) be two classical solutions of (1.1) with initial data uo1 and uo2 , respectively. If uo1 ≤ uo2 in the common domain of definition, then ξ20 ≤ ξ10 and ξ21 ≤ ξ11 for all t, and u1 ≤ u2 in the intersection of the respective domains. In particular the free boundary Problem (1.1) has at most one classical solution. Proof of Theorem 4.3. We extend the functions u1 and u2 outside the free boundaries by setting them equal to 0 to the left, and equal to 1 to the right; next we show that

` A.L. AMADORI AND J.L. VAZQUEZ

16

comparison holds between these extended functions, i.e. Z Z + sgn (u1 − u2 ) (x, t)dx ≤ sgn + (uo1 − uo2 ) dx, I

I

for all t < T . Here I is any interval containing the projections on the x–axis of the domains of u1 and u2 . We fix τ < T and set for any δ > 0  if u ≤ δ,  δ u if δ < u < 1 − δ, bδ (u) :=  1−δ if u ≥ 1 − δ, For properly small δ, Lemma 4.2 yields   δ δ uk (x, t) b (uk )(x, t) =  1−δ

if x ≤ ξk0δ (t), t ≤ τ, if ξk0δ (t) < x < ξk1δ (t), t ≤ τ, if x ≥ ξk1δ (t), t ≤ τ,

as k = 1, 2. Next, we extend uk outside of the region {ξk0δ (t) < x < ξk1δ (t)} in a linear way:  if x ≤ ξk0δ (t), t ≤ τ,  δ − uk,x (ξk0δ (t), t)(ξk0δ (t) − x) uk (x, t) if ξk0δ (t) < x < ξk1δ (t), t ≤ τ, uδk (x, t) :=  1δ 1δ 1 − δ + uk,x (ξk (t), t)(x − ξk (t)) if x ≥ ξk1δ (t), t ≤ τ. In this way we ensure that uδk is C 1 in space. Note also that bδ (uδk ) = bδ (uk ). Besides, by taking a0 , a1 so that a0 < ξk0 (t) and a1 > ξk1 (t) for all t ∈ [0, T ] and k = 1, 2, we obtain that uδk solve in the strong sense the following elliptic-parabolic problem:  δ δ b (uk )t = Φ(uδk,x )x , x ∈ I, t ∈ (0, τ ],      Φ(uδk,x )(ai , t) = Φ(uk,x )(ξkiδ (t), t) i = 0, 1, t ∈ (0, τ ],      δ δ b (uk )(x, 0) = bδ (uo )(x), x ∈ I. Proposition 4.1 yields Z Z ¡ δ δ ¢ ¡ ¢ + δ δ sgn b (u1 ) − b (u2 ) (x, t)dx ≤ sgn + bδ (uo1 ) − bδ (uo2 ) dx I

I

+

XZ i=0,1

τ 0

|Φ(u1,x )(ξ1iδ (t), t) − Φ(u2,x )(ξ2iδ (t), t)|dt

The last term on the right hand side vanishes as δ → 0: it follows by Lebesgue Theorem because ξkiδ → ξki pointwise and Φ(uk,x ) are continuous up to t = 0. Moreover bδ (uk ) and bδ (uok ) tend uniformly to the extended uk and uok , as k = 1, 2. Therefore passing to the limit as δ → 0 implies the comparison result. ¤

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17

5. Singular problem, II: Passage to the limit and existence of solutions We now want to pass to the limit in problems (1.1.ε), in order to get a solution to the singular free boundary problem (1.1). We contend that the approximation by ellipticparabolic problems (1.3.ε) provides a candidate. Proposition 5.1.¢ There exists a function v ∈ L∞ (QT ) such that b(uε ) → v strongly ¡ in L∞ 0, T ; L1 (I) . Moreover, v ∈ BVx (QT ) and b(uε )x → vx weakly in the sense of measures, up to an extracted sequence. Proof. Because Φ(1/ε) % Φ∞ as ε¡ → 0, the ¢ contraction property (3.14) guarantees that b(uε ) is a Cauchy sequence in L∞ 0, T ; L1 . In order to check that b(uε )x is bounded in L1 (QT ), we recall that the conservation of energy gives Z Z 1 Φ(un,x )b(uε )x dxdt ≤ vo2 dx + T Φ∞ . 2 I QT This is obtained as usual by multiplication of the equation satisfied by uε by b(uε ) and integration by parts. So for any α > 0 we have Z Z 1 |b(uε )x |dxdt ≤ α|QT | + Φ(uε,x )b(uε )x dxdt. Φ(α) QT QT The sequence b(uε )x is uniformly bounded in L1 (QT ), and the result follows.

¤

With the aim of recovering the limit problem, we introduce the notations (5.1)

ξ 0 (t) := lim sup ξε0 (s), (ε,s)→(0,t)

ξ 1 (t) := lim inf ξε1 (s), (ε,s)→(0,t)

so that ξ 0 (t) is l.s.c. and ξ 1 (t) is u.s.c. We denote by © ª D := (x, t) : ξ 0 (t) < x < ξ 1 (t), 0 < t < T , © ª ˜ := D ∪ (ξ i (t), t) : 0 < t < T, i = 0, 1 , D © ª Do := D ∪ (x, 0) : ξ 0 (0) < x < ξ 1 (0) , and by u the restriction to the domain D of the limit function produced in Proposition 5.1. The motivation for choosing that particular limit for the free boundaries is made clear by the following topological lemma: Lemma 5.2. Let KTbe a compact subset of D (resp., Do ); then there exists εo > 0 so that K is contained in Dε (resp., Doε ). ε 0 and δ > 0 so that d0 ≥ ξε0 (t) + δ for all ε < εo and t ∈ [t0 , t1 ]: the right side can be dealt with exactly in the same way. Set 2δ = min{d0 − ξ 0 (t) : t ∈ [t0 , t1 ]}: by definition of ξ 0 , for any t ∈ [t0 , t1 ] there exist ε(t) and ρ(t) such that d0 ≥ sup{ξε0 (s) : ε < ε(t), |s − t| < ρ(t)} + δ. By compactness, there exists a finite set {t1 , · · · , tn } so that [t0 , t1 ] is covered by (ti − ρ(ti ), ti + ρ(ti )); setting

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18

εo = min{ε(t1 ), · · · , ε(tn )} we obtain d0 ≥ maxi=1,··· ,n sup{ξε0 (s) : ε < εo , |s − ti | < ρ(ti )} + δ ≥ ξε0 (s) + δ for all s ∈ [t0 , t1 ] and ε < εo . ¤ By taking advantage of Lemma 5.2, one can prove as in [4, Corollary 4.2] that Corollary 5.3. Up to an extracted sequence, uε tends to u in BVloc (D) and in Lploc (Do ) for all 1 ≤ p < ∞. The convergence of the combustion type problems (1.1.ε) to (1.1) is achieved by adopting two different techniques according either dwelling in the interior of the definition domain or near at the free boundary. In both cases an intermediate step consists of looking at uε,x as a function of uε , but the line of argument is different. Before tackling this task, we recall an interesting property of the free boundaries of the approximating problems (1.1.ε). A first consequence of imposing large derivatives at the boundary is the monotonicity of the free boundary themselves, so that the domain shrinks with time. Proposition 5.4. For small ε, the curves [0, T ) 3 t 7→ ξεi (t) are monotone: actually ξε0 is nondecreasing and ξε1 is nonincreasing. Proof. We only perform the proof of the monotonicity for ξε0 , being the one for ξε1 completely analogous. For simplicity of notations, we shall omit the dependence by ε. For ε sufficiently small, the derivative bound (3.4) yields that u(x, to ) ≤ (x − ξ 0 (to ))/ε as x ≥ ξ 1 (to ), for any to ≥ 0. Besides, it is trivial to check that u(x, t) = (x − ξ 0 (to ))/ε solves (1.3) for t > to with fixed free boundary x = ξ 0 (to ). We thus apply the comparison result, Proposition 3.11, with u1 = u and u2 = u and obtain that for all t > to Z Z £ ¤ [b(u) − b(u)]+ (t)dx ≤ vo − b((x − ξ 0 (to ))/ε) + dx ≤ 0. I

I

Therefore b(u) ≤ b(u) for all t ≥ to : in particular b(u) ≡ 0 on (a0 , ξ 0 (to )] × [to , T ), so that ξ 0 (t) ≥ ξ 0 (to ). ¤ 5.1. Convergence in the interior. The main result of this subsection is the convergence of the solutions of (1.1.ε) to a classical solution of ut = Φ(ux )x in the domain D. 1+α,α Theorem 5.5. There exists T > 0 so that the sequence uε tends to u in Cloc (D), for α < 1, and in Cloc (Do ). Moreover, u ∈ C 2,1 (D) satisfies ut = Φ(ux )x in classical sense in D and u(x, 0) = uo (x) if ξ 0 (0) < x < ξ 1 (0).

The proof of this result needs switching to a new independent variable. To this purpose we explicitly mention that uε,x (x, t) ∈ [−b, 1/ε] on Dε , because of estimate (3.4). In particular uε + 2bx is strictly increasing. This allows us to introduce the new variable (5.2)

y := uε (x, t) + 2bx,

and the function pε (y, t) := Φ(uε,x (x, t)).

SINGULAR FREE BOUNDARY PROBLEMS

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To derive an equation for pε , we compute, omitting the subscripts ε, · ¸ ut Φ(ux )x 0 pt = Φ(ux )t − Φ(ux )x = Φ (ux ) uxt − uxx ux +2b ux + 2b £ ¤ 0 0 = Φ (ux ) [Φ(ux )xx − uxx py ] = Φ (ux ) (py (ux +2b))x − uxx py = Φ0 (ux )(ux + 2b)2 pyy , for all t > 0 and y between 2bξε0 (t) and 1 + 2bξε1 (t). It can be written as an ellipticparabolic equation by setting  1  , if p ∈ (Φ(−2b), Φ∞ ), − −1 c(p) := Φ (p) + 2b  0 if p ≥ Φ . ∞

Precisely we consider the problem   c(p)t = pyy ,     p(ηεi (t), t) = Φ(1/ε), (5.3.ε)      c(p(y, 0)) = c(p ), oε where

ηεi (t) := i + 2bξεi (t) poε (y) := Φ(u0oε )(x)

y ∈ (ηε0 (t), ηε1 (t)), t ∈ (0, T ), t ∈ (0, T ), y ∈ (ηε0 (0), ηε1 (0)), as i = 0, 1, if y = uoε (x) + 2bx

To avoid ambiguity, we explicitly remark that now ηεi (t) are given curves, and not free boundaries to be determined. Thus (5.3.ε) is an elliptic-parabolic problem posed in a non-cylindrical domain. Note that, if uo is strictly increasing by itself, we may chose b = 0 and therefore the domain is actually cylindrical. It is worth mentioning that pε ∈ [Φ(−b), Φ(1/ε)] for any ε > 0, by (3.4); therefore (5.3.ε) is uniformly parabolic. A similar problem is studied in [4], where a parabolic regularization is performed by modifying the function c. In both cases the limit problem is  c(p)t = pyy , y ∈ (η0 (t), η1 (t)), t ∈ (0, T ),      p(ηi (t), t) = Φ∞ , t ∈ (0, T ), (5.3)      c(p)(y, 0) = c(p )(y), y ∈ (η 0 (0), η 1 (0)), o where po (y) := Φ(u0o (x))

if y = uo (x) + 2bx

and η i are some limit of ηεi . We have to make sure that the limit function p does not overcome the upper value Φ∞ , thus falling into the ellipticity region. We now prove that this can not happen in the

20

` A.L. AMADORI AND J.L. VAZQUEZ

interior of the domain, by establishing a uniform estimate for φ(uε,x ). In this connection a role is played by the behavior of Φ at infinity, namely by the quantity A(Φ) := lim s2 Φ0 (s). s→∞

If A(Φ) < ∞, we shall obtain an uniform bound for uε,x for all times, while in the opposite case this estimate blows up in finite time. This is consistent with [3], which shows that the solution to (1.1) becomes vertical in finite time, in the monotone case. Lemma 5.6. Let ε be small enough. For any δ ∈ (0, 1) there exists ρ(δ) > 0, not depending on ε, such that φ(uε,x )(x, t) ≤ Φ∞ − ρ , whenever δ ≤ uε (x, t) ≤ 1 − δ and t ≤ T − δ. Here T can be any positive time if A(Φ) < ∞, or is a suitable constant if A(Φ) = ∞. Proof. The change of variables (5.2) maps the set © ª (x, t) : δ ≤ uε (x, t) ≤ 1 − δ, t ∈ [0, T − δ] into a subset of

© ª Q := (y, t) : ηε0 (t) + δ ≤ y ≤ ηε1 (t) − δ, t ∈ [0, T − δ] .

Because ηε0 and ηε1 are respectively nondecreasing and nonincreasing (see Proposition 5.4), Q can be covered by a finite number of rectangles of type R(θ) := [ηε0 (θ) + δ/2, ηε1 (θ) − δ/2] × [0, θ], with different θ ≤ T −δ/2. Therefore it suffices to check that there exists ρ, only depending by δ, such that pε ≤ Φ∞ − ρ in R(θ). So we set d0 = ηε0 (θ) + δ/4, d1 = ηε1 (θ) − δ/4, and consider the problem  c(p)t = pyy , y ∈ (d0 , d1 ), t ∈ (0, θ),      p(d0 , t) = p(d1 , t) = Φ(1/ε), t ∈ (0, θ),      c(p(y, 0)) = c(p ), y ∈ (d0 , d1 ). oε By (3.4) pε is a subsolution to this problem, hence the thesis is attained by producing a suitable supersolution laying in the range [Φ(−b), Φ(1/ε)], where the equation is uniformly parabolic. First we take γ, εo > 0 so that ku0oε k∞ ≤ 1/ε − γ for all ε < εo , and we set d := π/(d1 − d0 ). Proposition 5.4 guarantees that d is bounded, uniformly with respect to ε, namely π/(1−δ/2+2b(r−`)) ≤ d ≤ π/(1−δ/2). Next we argue separately from case to case. The case A(Φ) < ∞. An easy computation shows that 2

pε (y, t) := Φ(1/ε) − γe−ad t sin(d(y − d0 )), is a supersolution if a = sup {(s + 2b)2 Φ0 (s) : s > −2b}. The thesis follows after noticing 2 that, in the rectangle R(θ), we have pε ≤ Φ(1/ε) − γe−ad (T −δ/2) sin(δd/4), where d is bounded uniformly with respect to ε.

SINGULAR FREE BOUNDARY PROBLEMS

21

The case A(Φ) = ∞. We now show that there exists T > 0 such that ³ ´ 2 − Td−t pε (y, t) := Φ 1/ε − e sin(d(y − d0 )) is a supersolution on (0, T ). To simplify notations, we set τ = T − t and h(y, τ ) := d2

e− τ sin(d(y − d0 )). Since Φ is increasing and has limit at infinity, there is a suitable n > 0 such that Φ0 (s) ≤ 1 and Φ00 (s) ≤ 0 as s ≥ n. Besides h(y, τ ) → 0 as τ → 0, uniformly w.r.t. y; therefore we may assume that 1/ε − h ≥ max{n, b}, up to choosing T and ε sufficiently small. In particular pε (y, 0) ≥ Φ(b) ≥ poε . Afterwards we compute pε,yy = d2 hΦ0 (1/ε−h) + (dh cos(d(y − d0 )))2 Φ00 (1/ε−h) ≤ d2 hΦ0 (1/ε−h). Hence

·

¸ 1 0 c(pε )t − pε,yy ≥ 2 − Φ (1/ε−h) d2 h. τ (1/ε−h+2b)2

Thus one can show that pε is a supersolution by producing T > 0 such that f (y, τ ) := 1/τ 2 (1/ε−h+2b)2 − Φ0 (1/ε−h) ≥ 0 for all τ ∈ (0, T ). This holds because f is continuous and lim f (y, τ ) = +∞, uniformly w.r.t. y. τ →0

Eventually the thesis follows because pε ≤ Φ(1/ε) − e−2d R(θ), and d is bounded uniformly with respect to ε.

2 /δ

sin(δd/4) in the rectangle ¤

The uniform bounds obtained in Lemma 5.6 provide the convergence stated in Theorem 5.5 arguing as follows. Proof of Theorem 5.5. Let T K be a compact subsetS of D: by Lemma 5.2 there exists εo > 0 so that K is contained in Dε . Because K = Kδ , where ε 0. Now the uniform estimates obtained in Proposition 5.6 allows to use the standard machinery of uniformly parabolic theory (cf. [10]) to get the result. ¤ 5.2. Boundary convergence. We now make the second step of our program, by showing that the combustion type problem (1.1.ε) approximate (1.1) also at the free boundaries. Proposition 5.7. As ε tends to zero, ξεi → ξ i in Cloc ([0, T )), and Φ(uε,x ) → Φ(ux ) in ˜ Cloc (D). In order to prove Proposition 5.7, we first observe that uε are strictly increasing with respect to x near at the free boundaries. This enable us to make use of the hodograph change of variable and to study some conjugate problem, on the stream of similar arguments in [3]. The strict monotonicity of uε (·, t) can be seen by means of the so called

22

` A.L. AMADORI AND J.L. VAZQUEZ

lap-number theory, cf. [13]. To this aim, we set δo = min{m, 1 − m}, where m and m appeared in H.2.ε. For any δ < δo , we define ξεiδ as the curves respectively given by ξε0δ (t) := min{x ≥ ξε0 (t) : uε (x, t) = δ}, ξε1δ (t) := max{x ≤ ξε1 (t) : uε (x, t) = 1 − δ}. By construction uε (x, t) < δ < m at the left of ξε0δ (t) and uε (x, t) > 1 − δ > m at the right of ξε1δ (t). Next the standard lap-number theory implies that δ < uε (x, t) < 1 − δ inside (ξε0δ (t), ξε1δ (t)). Beside we have Lemma 5.8. Let δ < δo . Then uε,x > 0 in (ξε0 (t), ξε0δ (t)] and in [ξε1δ (t), ξε1 (t)), for all t. Moreover the curves t 7→ ξεiδ (t) are of class C 1 . Proof. Let t > 0 be fixed; we first show that x 7→ uε (x, t) is non decreasing. Suppose by contradiction that uε (·, t) has a strict local minimum in the set J(δ) = (ξε0 (t), ξε0δ (t)): by construction the value of such minimum should be strictly smaller than m. This contradicts the standard lap-number theory. As a consequence, uε,x ≥ 0. Next, let 0 x ≤ ξε0δ (t). By Sard Lemma, there exists δ 0 > δ so that the curve t 7→ ξ 0δ (t) is of class C ∞ ; in particular, there exists a parabolic neighborhood of (x, t) where uε,x ≥ 0. Therefore strong maximum principle implies that uε,x (x, t) > 0. Lastly, the Implicit Function Theorem yields that ξεiδ (t) is of class C 1 . ¤ The monotonicity established by Lemma 5.8 allows us to introduce the hodograph variable (5.4)

(ξε0 (t), ξε0δ (t)) 3 x 7→ y := uε (x, t) ∈ (0, δ), (ξε1δ (t), ξε1 (t)) 3 x 7→ y := uε (x, t) ∈ (1 − δ, 1),

and to consider the conjugate formulation for vε (y, t) = x and wε (y, t) = vε,y (y, t):  vt = Ψ(vy )y y ∈ (0, δ), t > 0,      (5.5) vy (0, t) = ε, vy (δ, t) = 1/uε,x (ξε0δ (t), t), t > 0,      v(y, 0) = v (y), y ∈ (0, δ), oε

(5.6)

 wt = Ψ(w)yy      w(0, t) = ε, w(δ, t) = 1/uε,x (ξε0δ (t), t)      w(y, 0) = w (y), oε

y ∈ (0, δ), t > 0, t > 0, y ∈ (0, δ),

where Ψ(s) := −Φ(1/s), voε (y) := u−1 oε (y),

0 woε (y) := voε (y) = 1/u0oε (u−1 oε (y)).

SINGULAR FREE BOUNDARY PROBLEMS

23

Here and in what follows we only deal with the set 0 < y < δ (correspondingly, ξ 0 (t) < x < ξ 0δ (t))). The case 1−δ < y < 1 (correspondingly, ξ 1δ (t) < x < ξ 1 (t))) can be handled by similar arguments. Assumption H.2 allows to suppose w.l.g. that kwoε kL∞ (0,δ) ≤ 1/β for all ε > 0. Next Lemma shows that we may assume that wε are equibounded in L∞ ((0, δ) × (0, T )). Lemma 5.9. For all δ < δo , there exists c > 0 so that wε ≤ 1/c in [0, δ] × [0, T ). Proof. The conclusion follows by comparison with the supersolution w(y, t) = Ψ−1 (−Φ(β)(δ − y)/δ) . ¤ Eventually, as in [3], for all ε > 0 we set Z zε (y, t) = [wε dy + Ψ(wε )y dt] − ξε0δ (0), Γ

where Γ is any smooth curve joining (δ, 0) to (y, t). Because zε,y = wε > 0, we can invert the dependence between zε and y to get a function uˆε defined by uˆε (x, t) = y We also set

⇔

zε (y, t) = x.

ξˆε0 (t) = lim zε (y, t). y→0

In this way, we have got the solution uε of the free boundary problems (1.1.ε) previously constructed via elliptic-parabolic formulation. Lemma 5.10. For all ε > 0, uˆε equals uε , on the set where they are both defined. In particular, ξˆε0 = ξε0 Proof. By construction 0 < uˆε < δ if ξˆε0 (t) < x < ξε0δ (t). We extend it at the left of ξˆε0 (t) by setting uˆε (x, t) = (x − ξˆε0 (t))/ε, and at the right of ξε0δ (t) by setting uˆε (x, t) = ux (ξε0δ (t), t))(x − ξε0δ (t)). A mere computation shows that uˆε satisfies in strong sense  ˜b(ˆ u)t = Φ(ˆ ux )x , x ∈ (a0 , a1 ), t ∈ (0, T ),      uˆx (a0 , t) = 1/ε, uˆx (a1 , t) = uε,x (ξε0δ (t), t) t ∈ (0, T ),     ˜ b(ˆ u)(x, 0) = ˜b(uoε )(x), x ∈ (a0 , a1 ), where

  0 ˜b(u) = u  δ

if u ≤ 0, if u ∈ (0, δ), if u ≥ δ,

and a0 , a1 can be arbitrarily large. Proposition 4.1 implies that ˜b(ˆ uε ) = ˜b(uε ), hence 0δ 0 ¤ uˆε = uε as ξˆε (t) < x < ξε (t) , t < T .

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` A.L. AMADORI AND J.L. VAZQUEZ

In the sequel we shall omit the hat over uε and ξε , because actually they equal the functions introduced in Section 3. We next turn to the conjugate limit problem  wt = Ψ(w)yy y ∈ (0, δ), t > 0,      (5.7) w(0, t) = 0, w(δ, t) = 1/ux (ξ 0δ (t), t) t > 0,      w(y, 0) = w (y), y ∈ (0, δ), o were u is the limit function constructed in Proposition 5.1 and ξ 0δ (t) the curve where it takes the value δ (recall that the interior regularity of such u has been proved in Theorem 5.5). As we have done for the approximating problem, we set Z z(y, t) := [wdy + Ψ(w)y dt] − ξ 0δ (0), Γ

uˆ(x, t) := y

⇔

z(y, t) = x,

ξˆ0 (t) := lim z(y, t). y→0

A simple computation shows that the function uˆ satisfies  uˆt = Φ(ˆ ux )x , x ∈ (ξˆ0 (t), ξ 0δ (t)], t ∈ (0, T ),      uˆ = 0, Φ(ˆ ux ) = Φ ∞ , x = ξˆ0 (t), t ∈ (0, T ),      uˆ(x, 0) = uo (x), x ∈ (ξ 0 (0), ξ 0δ (0)]. Afterwards we recall a well-known convergence result. Lemma 5.11. Up to an extracted sequence, wε → w in Cloc ((0, δ] × [0, T )) and Ψ(wε ) → Ψ(w) in Cloc ([0, δ] × (0, T )). Here w is the classical solution to (5.7). Proof. Using Lemma 5.6, one shows that wε is bounded away from zero in any [ρ, δ] × [0, T −ρ], uniformly w.r.t. ε. Therefore standard parabolic theory yields that wε converges in Cloc ((0, δ] × [0, T )) to some classical solution of wt = Ψ(w)yy , which in addition satisfies w(·, 0) = wo and w(δ, ·) = 1/ux (ξ 0δ (·), ·). With respect to the convergence near at y = 0, we take K a compact subset of [0, δ) × (0, T ) and set qε = Φ∞ + Ψ(wε ). Note that qε = 0 for wε = 0. It is clear that the thesis is equivalent to the equicontinuity of qε on K. Let β(q) = Ψ−1 (q − Φ∞ ), then we have   β(qε )t = qε,yy y ∈ (0, δ), t > 0,     qε (0, t) = Φ∞ − Φ(1/ε), t > 0,      q (y, 0) = Φ − Φ(v (y)), y ∈ (0, δ). ε ∞ oε

SINGULAR FREE BOUNDARY PROBLEMS

25

Lemma 5.9 yields that qε are equibounded in L∞ (K). Moreover after computation we have β 0 (q) = 1/Φ0 (Φ−1 (Φ∞ − q))(Φ−1 (Φ∞ − q))2 , so that β 0 (q) > 0 if q > 0 and lim β 0 (q) = lim 1/Φ0 (s)s2 . In any case qε is equicontinuous q→0

s→∞

on K: it follows by standard parabolic theory if lim Φ0 (s)s2 > 0, otherwise by [7]. s→∞

¤

At the present moment we have two candidates for being the free boundary of (1.1): the curve ξ 0 produced as a limit in (5.1), and the curve ξˆ0 produced by the conjugate problem. The result stated by Proposition 5.7 is recovered by checking that the two coincide. Proof of Proposition 5.7. We first check that zε → z locally uniformly in [0, δ] × (0, T ). Concerning the convergence on (0, δ] × (0, T ), it suffices to check that wε and Ψ(wε )y are uniformly locally bounded and then recall Lemma 5.11. The bound for wε has been established in Lemma 5.9, while the one for Ψ(wε )y can be obtained as in [4, Lemma 5.3]. As for the convergence near at y = 0, for any ε1 , ε2 > 0, y1 < y2 ∈ (0, δ), and t ∈ [0, T ) we have |zε1 (y1 , t) − zε2 (y1 , t)| ≤ |zε1 (y1 , t) − zε1 (y2 , t)| + |zε2 (y1 , t) − zε2 (y2 , t)| + |zε1 (y2 , t) − zε2 (y2 , t)| ≤ 2 sup kwε kL∞ + kzε1 − zε2 kC(K) , ε≥0

where K is, for instance, [y2 /2, δ] × [0, (T + t)/2]. Sending y1 to zero gives the thesis. Therefore, uˆ coincides with the limit function u produced in Proposition 5.1, and ξε0 = lim zε (y, ·) converges to ξˆ0 in C([0, T )). In particular ξˆ0 = ξ 0 , ξ 0 (0) = `, and u(ξ 0 (t), t) = 0. y→0 © ª ˜ δ = (x, t) : ξ 0 (t) ≤ x ≤ ξ 0δ (t), t ∈ (0, T ) . Keeping in mind that, Afterwards, we set D under assumption H.2.ε, Φ(uε,x ) are continuous till ξεi , the same arguments of [4, Lemma ˜ δ ) to some function Φ. Eventually Theorem 5.5 4.3] yields that Φ(uε,x ) tends in Cloc (D yields that Φ = Φ(ux ) in D, while by construction Φ(ξ 0 (t), t) = lim Φ(uε,x )(ξε0 (t), t) = ε→0 Φ∞ . ¤

Acknowledgments. This work has been done while the first author was visiting the Department of Mathematics of the Universidad Aut´onoma de Madrid, supported by the Research Training Network HYKE, HPRN-CT-2002-00282, and by Proyecto de Investigaci´on MCYT BMF2002-04572-C02-02. She further wishes to thank all the staff of Universidad Aut´onoma de Madrid for their kind hospitality. J. L. V. also supported by MCYT Project BFM2002-04572-C02-02 (Spain) Keywords and Phrases. Nonlinear diffusion, image enhancement, singular solutions, free boundaries. AMS Subject Classification. 35K55, 35K65. Secondary: 68U10.

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References [1] H. W. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Math. Z., 183(3):311–341, 1983. [2] G. I. Barenblatt. Self-similar intermediate asymptotics for nonlinear degenerate parabolic freeboundary problems that occur in image processing. Proc. Natl. Acad. Sci. USA, 98(23):12878–12881 (electronic), 2001. [3] G. I. Barenblatt and J. L. V´azquez. Nonlinear diffusion and image contour enhancement. Interfaces and Free Boundaries, to appear. Preprint, Center for Pure and Applied Mathematics, Univ. of Berkeley, 2003. [4] M. Bertsch and R. Dal Passo. Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rational Mech. Anal., 117(4):349–387, 1992. [5] M. Bertsch and J. Hulshof. Regularity results for an elliptic-parabolic free boundary problem. Trans. Amer. Math. Soc., 297(1):337–350, 1986. [6] L. A. Caffarelli and J. L. V´azquez. A free-boundary problem for the heat equation arising in flame propagation. Trans. Am. Math. Soc., 347(2):411–441, 1995. [7] E. DiBenedetto. A boundary modulus of continuity for a class of singular parabolic equations. J. Differential Equations, 63(3):418–447, 1986. [8] J. Hulshof. An elliptic-parabolic free boundary problem: continuity of the interface. Proc. Roy. Soc. Edinburgh Sect. A, 106(3-4):327–339, 1987. [9] J. Hulshof and L. A. Peletier. An elliptic-parabolic free boundary problem. Nonlinear Anal., 10(12):1327–1346, 1986. [10] O. A. Ladyˇzenskaja, V. A. Solonnikov, and N. N. Ural0 ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1967. [11] R. Malladi and J. Sethian. Image processing: Flows under min/max curvature and mean curvature. Graphical Models and Image Processing, 1996. [12] F. Otto. L1 -contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differential Equations, 131(1):20–38, 1996. [13] D. H. Sattinger. On the total variation of solutions of parabolic equations. Math. Ann., 183:78–92, 1969. [14] J. Simon. Compact sets in the space Lp (0, T ; B). Ann. Mat. Pura Appl. (4), 146:65–96, 1987. [15] C. J. van Duijn and L. A. Peletier. Nonstationary filtration in partially saturated porous media. Arch. Rational Mech. Anal., 78(2):173–198, 1982. [16] J. L. V´azquez. The free boundary problem for the heat equation with fixed gradient condition. In Free boundary problems, theory and applications (Zakopane, 1995), volume 363 of Pitman Res. Notes Math. Ser., pages 277–302. Longman, Harlow, 1996. (Anna Lisa Amadori) IAC - CNR: Viale del Policlinico, 137 00161 Roma, Italy and DIIMA - Univ. of Salerno: via Ponte don Melillo 84084 Fisciano (SA), Italy E-mail address: [email protected] URL: http://www.iac.cnr.it/~amadori/ (Juan Luis V´azquez) ´ ticas - Univ. Auto ´ noma de Madrid Dpto. de Matema Campus de Cantoblanco, 28049 Madrid, Spain E-mail address: [email protected] URL: http://www.uam.es/juanluis.vazquez