A free-boundary problem for the evolution p -Laplacian equation with a combustion boundary condition
aa r X i v : . [ m a t h . A P ] N ov A FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN EQUATION WITH A COMBUSTION BOUNDARYCONDITION TUNG TO
Abstract.
We study the existence, uniqueness and regularity of solutions of the equation f t = ∆ p f = div ( | Df | p − Df ) under over-determined boundary conditions f = 0 and | Df | = 1. We show that if the initial data is concave and Lipschitz with a bounded andconvex support, then the problem admits a unique solution which exists until it vanishesidentically. Furthermore, the free-boundary of the support of f is smooth for all positivetime. Introduction
Fix a number p >
2. Given a non-negative function f on R n with positive set Ω , wewant to find a non-negative function f ( x, t ) on R n × (0 , T ) with positive set Ω which solvesthe following problem:(P) f t = ∆ p f in Ω = { f > } f = 0 and | Df | = 1 on ∂ Ω ∩ { < t < T } lim t → f ( x , t ) = f ( x ) ∀ x ∈ R n . The operator ∆ p f = div ( | Df | p − Df )is known as the p -Laplacian. In non-divergent form, it can be written as(1.1) ∆ p = | Df | p − ∆ f + ( p − | Df | p − f ij f i f j Note that the Einstein summation notation was used in the last term. It can also be writtenas(1.2) ∆ p = | Df | p − (∆ f + ( p − f νν )where f νν denotes the second derivative of f in the direction of ν = Df / | Df | .In the case p >
2, this operator is nonlinear and degenerate at vanishing points of Df .When p = 2, it is just the regular Laplacian. Mathematics Subject Classification.
Primary: 35R35; Secondary: 35K55, 35K65.
Key words and phrases. p-Laplacian, free-boundary problem, degenerate equation, combustion, regulariza-tion, convex domain.
Due to the over-determined boundary conditions f = 0 and | Df | = 1, the time-sectionof Ω Ω t = { x ∈ R n | f ( x , t ) > } will in general change with time. In other words, the boundary ∂ Ω t moves. It is oftenknown as the moving-boundary or free-boundary.Our work is motivated by the work of Caffarelli and V´azquez [2] in which authors studiedthis problem in the case p = 2. Their result stated essentially that if ∂ Ω ∈ C , f ∈ C (Ω ) and ∆ f ≤
0, then there exists a solution to the problem. Moreover, if Ω iscompact, solutions vanish in finite time. Still in this case, long-time existence, uniquenessand regularity of the free-boundary have been studied by Daskalopoulos and Ki-Ahm Lee [5]or Petrosyan [11, 12] when the initial value is concave or star-shaped with bounded support.Other kinds of solution have also been studied (see also [9]). In the case p >
2, an ellipticversion of the problem has been studied before by Danielli, Petrosyan and Shahgholian [3]or Henrot and Shahgholian [7, 8]. As far as the parabolic problem when p > t is convex and non-decreasingin time. The questions of existence or regularity of the free-boundary were not addressedin that paper.The main result of our work is stated below. Theorem 1.1.
Assume that Ω is a bounded and convex domain. The function f is positiveand concave in Ω . Furthermore, on the boundary ∂ Ω , f satisfies f ( x ) = 0 for all x | Df ( x ) | = 1 for a.e. x . Then the problem (P) has a unique solution up to a finite time T where it vanishes identicallyin the sense that lim t → T f ( x , t ) = 0 ∀ x ∈ R n . Moreover, the free-boundary ∂ Ω t is smooth for all t ∈ (0 , T ) . It is well-known that solutions of the evolution p -Laplacian are only C ,α at points ofvanishing gradient (see for example [6]). Hence, solutions to the problem (P) must bedefined in some weak sense. We will state precisely the meaning of our solution in section2. Our approach to the problem is totally different from [2]. To deal with the degeneracy,we will approximate the p -Laplacian with the following regularized operator(P( ǫ )) ∆ ǫp f = div(( | Df | + ǫ ) q − Df ) . Here and throughout this work, we define q = p/
2. We will establish some properties forsolutions of these regularized problems and then let ǫ go to 0 to obtain a solution to thedegenerate problem.In order to solve this regularized free-boundary problem, we employ a change of coordi-nates that transforms it into a quasilinear equation with Neumann boundary condition on FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 3 a fixed-domain problem. Applying results from standard theory of quasi-linear parabolicequations with oblique boundary condition, we show that this new problem admits a solu-tion for some positive time. Revert back to the original coordinates, we obtain a short-timeexistence result for the regularized problem. This argument is carried out in section 3.In section 4, we prove a simple estimate for the gradient | Df | of solutions of the problem(P( ǫ )). In section 5, we prove a crucial result that the time-section Ω t remains convex andthe function f ( ., t ) remains concave on Ω t for all time t . Convexity of Ω t guarantees thatthe free-boundary ∂ Ω t does not touch itself and also enables us to prove the non-degeneracyof | Df | near the free-boundary.In section 6, we obtain an estimate for higher derivatives of f in a neighborhood the free-boundary ∂ Ω t , uniformly in time t and especially, in ǫ , using the non-degeneracy of | Df | .This fact and the convexity guarantee that singular cannot develop on the free-boundary.The uniqueness for this regularized problem is obtained in section 7. In section 8, we thenobtain a long-time existence result for solution of the regularized problem. Passing ǫ to 0,we then obtain a solution to the degenerate problem in section 9. The uniqueness for thedegenerate problem is then shown in section 10. In the last section, we show that solutionto our degenerate problem vanishes in finite time. Acknowledgement.
I express my gratitude to my thesis advisor, P. Daskalopoulos, forsuggesting this problem, and for her invaluable advices and support during the completionof this work. 2.
Definition of Solution
In this section, we will define precisely what we mean by solution of the problem (P). Westart by introducing some notations. For any 0 < t < t < T , defineΩ ( t ,t ) = Ω ∩ { t < t < t } . First, we require that the free-boundary ∂ Ω t is in C and the function f is in C (0 , T ; C (Ω t )) . The equation f t = ∆ p f in Ωis then defined in the sense that for any test function θ in C ∞ (Ω) and for any 0 < t < t 0) = f are understood in thepointwise sense f ( x , t ) → x → x ∈ ∂ Ω t ,f ( x , t ) → f ( x ) as t → . TUNG TO Finally, the Neumann’s boundary condition | Du | = 1 is defined in the following classicalsense f ν ( x , t ) = lim h → + f ( x + hν ) h = 1 . where x is a point on the free-boundary ∂ Ω t and ν is the spatial inward unit normal vectorat x with regards to ∂ Ω t .3. Short-time Existence for Regularized Problem In this section, we will prove that the regularized free-boundary problem admits a solutionfor some positive time. We do it by a change of coordinates technique that transforms theproblem into a fixed-domain problem. This technique has been used by other authors fordifferent problems before (see for example [5], [4]). Note that concavity is not needed inthis result. Lemma 3.1. Assume that Ω is C ∞ . The function f is in C ∞ (Ω ) and positive in Ω .Furthermore, on the boundary ∂ Ω , f satisfies f = 0 and | Df | = 1 . Then there exists a smooth solution to the regularized problem (P( ǫ )) for some T > .Proof. The argument in this proof works for any dimension, but due to the complexity ofsome computation involved, we will present the proof for the case n = 2 only.A word on notation used in this proof : we use bold-face letters x , y , ... to denote pointsin Euclidean spaces while normal letters x, y, z, ... for real numbers, scalars or componentsof points in Euclidean spaces.Denote by S the smooth surface z = f ( x, y ), ( x, y ) ∈ Ω . Let T = ( T , T , T ) be asmooth vector field on Ω such that T ( x, y ) is not a tangential vector to the surface S atthe point f ( x, y ). Since | Df | = 1 on the boundary ∂ Ω , we can also choose T to be parallelto the plane z = 0 in a small neighborhood of ∂ Ω .It is known that for some positive, small enough η , we can define a change of spatialcoordinates Φ : Ω × [ − η, η ] → R by the formula xyx = Φ uvw = f (cid:18) uv (cid:19) + w T (cid:18) uv (cid:19) . The map Φ defines x, y and z as smooth functions of u, v and w with smooth inverses.The graph of ( x, y, f ( x, y, t )), ( x, y ) ∈ Ω t is then transformed to ( u, v, g ( u, v, t )), ( u, v ) ∈ Ω via this coordinates change for some uniquely-defined g if the surface z = f ( x, y, t ) issufficiently close to S (( x, y, f ( x, y, t )) ∈ Φ(Ω × [ − η, η ]) for all ( x, y ) ∈ Ω t ). When f evolvesas a function of ( x, y ), g evolves as a function of ( u, v ). Importantly, the domain of g isfixed as Ω due to our requirement that T is parallel to the plane z = 0 on ∂ Ω . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 5 We will compute the evolution equation and the boundary condition of g . Denote by x u , x v , x w , y u , y v , y w , z u , z v and z w the partial derivatives of the functions x ( u, v, w ), y ( u, v, w )and z ( u, v, w ). Similarly we denote partial second derivatives of x, y and z by x uu , x uv , ... .We begin with first derivatives. Since x, y and z are functions of u, v and w , while w = g ( u, v, t ) is a function of u, v and t , we have ∂x∂u ∂y∂u∂x∂v ∂y∂v = x u + x w ∂w∂u y u + y w ∂w∂u x v + x w ∂w∂v y v + y w ∂w∂v = x u + x w g u y u + y w g u x v + x w g v y v + y w g v . We can compute the partial derivatives of u ( x, y, t ) and v ( x, y, t ) by ∂u∂x ∂u∂y∂v∂x ∂v∂y = ∂x∂u ∂x∂v∂y∂u ∂y∂v − = 1 D ∂y∂v − ∂x∂v − ∂y∂u ∂x∂u (3.1) = 1 D y v + y w g v − x v − x w g v − y u − y w g u x u + x w g u (3.2)where D = ∂x∂u ∂y∂v − ∂x∂v ∂y∂u = ( x u y v − x v y u ) + ( x w y v − y w x v ) g u + ( y w x u − x w y u ) g v . We then have(3.3) f x f y = ∂z∂x∂z∂y = ∂u∂x ∂v∂x∂u∂y ∂v∂y ∂z∂u∂z∂v = 1 D ∂y∂v − ∂y∂u − ∂x∂v ∂x∂u z u + z w g u z v + z w g v . Next we compute the second-order derivatives. First, we have partial second order deriva-tives of x with regards to u and v . ∂ x∂ u = x uu + 2 x uw ∂w∂u + x ww (cid:18) ∂w∂u (cid:19) + x w ∂ w∂ u = x uu + 2 x wu g u + x w g uu ∂ x∂ v = x vv + 2 x wv g v + x w g vv ∂ x∂u∂v = x uv + x wu g v + x wv g u + x w g uv and similar formulae for y and z .Differentiate (3.3) we have(3.4) f xx = ∂ z∂ x = ∂z∂u ∂ u∂ x + ∂z∂v ∂ v∂ x + ∂ z∂ u (cid:18) ∂u∂x (cid:19) + 2 ∂ z∂u∂v ∂u∂x ∂v∂x + ∂ z∂ v (cid:18) ∂v∂x (cid:19) . TUNG TO We need to compute second order derivatives of u and v with regards to x and y . Theformula (3.4) is true if we substitute any function of u and v in place of z . Because secondorder derivatives of x and y with regards to x are zero0 = ∂x∂u ∂ u∂ x + ∂x∂v ∂ v∂ x + ∂ x∂ u (cid:18) ∂u∂x (cid:19) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:18) ∂v∂x (cid:19) ∂y∂u ∂ u∂ x + ∂y∂v ∂ v∂ x + ∂ y∂ u (cid:18) ∂u∂x (cid:19) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:18) ∂v∂x (cid:19) . In other words ∂x∂u ∂x∂v∂y∂u ∂y∂v ∂ u∂ x∂ v∂ x + ∂ x∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:0) ∂v∂x (cid:1) ∂ y∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:0) ∂v∂x (cid:1) = 0or ∂ u∂ x∂ v∂ x = − ∂x∂u ∂x∂v∂y∂u ∂y∂v − ∂ x∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:0) ∂v∂x (cid:1) ∂ y∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:0) ∂v∂x (cid:1) = − ∂u∂x ∂u∂y∂v∂x ∂v∂y ∂ x∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:0) ∂v∂x (cid:1) ∂ y∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:0) ∂v∂x (cid:1) . We then have ∂z∂u ∂ u∂ x + ∂z∂v ∂ v∂ x = (cid:0) ∂z∂u ∂z∂v (cid:1) ∂ u∂ x∂ v∂ x = − (cid:0) ∂z∂u ∂z∂v (cid:1) ∂u∂x ∂u∂y∂v∂x ∂v∂y ∂ x∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:0) ∂v∂x (cid:1) ∂ y∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:0) ∂v∂x (cid:1) = − (cid:0) f x f y (cid:1) ∂ x∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ x∂u∂v ∂u∂x ∂v∂x + ∂ x∂ v (cid:0) ∂v∂x (cid:1) ∂ y∂ u (cid:0) ∂u∂x (cid:1) + 2 ∂ y∂u∂v ∂u∂x ∂v∂x + ∂ y∂ v (cid:0) ∂v∂x (cid:1) . Substitute into (3.4) f xx = (cid:18) ∂ z∂ u − f x ∂ x∂ u − f y ∂ y∂ u (cid:19) (cid:18) ∂u∂x (cid:19) + (cid:18) ∂ z∂ v − f x ∂ x∂ v − f y ∂ y∂ v (cid:19) (cid:18) ∂v∂x (cid:19) + 2 (cid:18) ∂ z∂u∂v − f x ∂ x∂u∂v − f y ∂ y∂u∂v (cid:19) ∂u∂x ∂v∂x . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 7 Let E = z w − f x x w − f y y w A = ∂ z∂ u − f x ∂ x∂ u − f y ∂ y∂ u = Eg uu + 2( z wu − f x x wu − f y y wu ) g u + ( z uu − f x x uu f y y uu ) B = ∂ z∂ v − f x ∂ x∂ v − f y ∂ y∂ v = Eg vv + 2( z wv − f x x wv − f y y wv ) g v + ( z vv − f x x vv f y y vv ) C = ∂ z∂u∂v − f x ∂ x∂u∂v − f y ∂ y∂u∂v = Eg uv + ( z wv − f x x wv − f y y wv ) g u + ( z wu − f x x wu − f y y wu ) g v + ( z uv − f x x uv − f y y uv ) , then f xx = A (cid:18) ∂u∂x (cid:19) + B (cid:18) ∂v∂x (cid:19) + 2 C ∂u∂x ∂v∂x = E (cid:18) ∂u∂x (cid:19) g uu + (cid:18) ∂v∂x (cid:19) g vv + 2 ∂u∂x ∂v∂x g uv ! + FD where F is a smooth function of u, v, g, g u , g w . We have similar formulae for f xy and f yy f yy = E (cid:18) ∂u∂y (cid:19) g uu + (cid:18) ∂v∂y (cid:19) g vv + 2 ∂u∂y ∂v∂y g uv ! + FD f xy = E (cid:18) ∂u∂x ∂u∂y g uu + ∂v∂x ∂v∂y g vv + (cid:18) ∂u∂x ∂v∂y + ∂v∂x ∂u∂y (cid:19) g uv (cid:19) + FD where F denotes different smooth functions of ( u, v, g, g u , g v ).To compute f t , we differentiate z = f ( x, y, t ) z w ∂w∂t = f t + ( f x x w + f y y w ) w t f t = ( z w − f x x w − f y y w ) g t = Eg t . Substituting into the equation for ff t = ( | Df | + ǫ ) q − ∆ f + ( p − | Df | + ǫ ) q − ( f xx f x + f yy f y + 2 f xy f x f y )and simplifying E from both sides we then obtain an evolution equation for g in the form g t = A ij ( u, v, g, Dg ) g ij + B ( u, v, g, Dg ) . On the other hand, the boundary condition | Df | = 1 becomes C ( u, v, g, Dg ) = 0for some function C . TUNG TO We claim the following is true when g ≡ t = 0) : • A ij , B and C are smooth functions of u, v, g and Dg . • ( A ij ) is positive definite. • C is oblique.Because the surface S and the vector field T are both smooth, it is clear that A ij , B and C are smooth functions of u, v, g and Dg whenever D = ∂x∂u ∂y∂v − ∂x∂v ∂y∂u = 0 E = z w − f x x w − f y y w = 0 . The condition that E = 0 follows from our choice that T is transverse to S . The condition D = 0 is a consequence of the fact that the function Φ is invertible in a neighborhood of S .Next, to show that ( A ij ) is positive definite, we write A ij = ( | Df | + ǫ ) q − A ij + ( p − | Df | + ǫ ) q − A ij where A ij is the coefficient of g ij ( i, j ∈ { u, v } ) obtained from the transformation of ∆ f and A ij from f x f xx + f y f yy + 2 f x f y f xy . We can compute explicitly A = (cid:0) ∂u∂x (cid:1) + (cid:16) ∂u∂y (cid:17) ∂u∂x ∂v∂x + ∂u∂x ∂v∂y∂u∂x ∂v∂x + ∂u∂x ∂v∂y (cid:0) ∂v∂x (cid:1) + (cid:16) ∂v∂y (cid:17) A = (cid:16) f x ∂u∂x + f y ∂u∂y (cid:17) (cid:16) f x ∂u∂x + f y ∂u∂y (cid:17) (cid:16) f x ∂v∂x + f y ∂v∂y (cid:17)(cid:16) f x ∂u∂x + f y ∂u∂y (cid:17) (cid:16) f x ∂v∂x + f y ∂v∂y (cid:17) (cid:16) f x ∂v∂x + f y ∂v∂y (cid:17) . It is obvious that A and A are non-negative definite. Furthermore, if D = 0, A is actuallypositive definite (det( A ) = D ). It then follows readily that ( A ij ) is positive definite.For the proof that C is oblique, we refer to the Appendix of [5].From the continuity, there must exist a positive number δ such that those three claimsare true for all g that satisfies | g | C (Ω ) < δ . It is then a consequence of standard theoryof quasilinear parabolic equation with oblique boundary condition (see for examples [10],Chapter 14) that there exists a solution g up to a positive time T to the problem. g t = A ij g ij + B in Ω × (0 , T ) C ( u, v, g, Dg ) = 0 on ∂ Ω × (0 , T ) g ( ., 0) = 0 . This solution is actually smooth up to the boundary for all t ∈ [0 , T ) since Ω is smoothand C ( u, v, g, Dg ) is a smooth function of ( u, v, g, Dg ). Choose a number T ′ in (0 , T ] suchthat | g | < η on Ω × (0 , T ′ ). Reverting back to the original coordinates system we then FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 9 obtain a solution to the regularized problem (P( ǫ )) up to T ′ . It is clear that the domain Ω t is smooth and the solution f is smooth up to the free-boundary for all time 0 < t < T ′ . (cid:3) Gradient Estimate Lemma 4.1. Assume the same hypotheses as in the Lemma 3.1. Furthermore, f satisfies | Df | ≤ on Ω . If f is a solution of the problem (P( ǫ )) , then | Df ( x , t ) | < for all ( x , t ) ∈ Ω .Proof. We will show an equivalent fact that f λ ( x , t ) < λ .Let a ij ( Df ) = ( | Df | + ǫ ) q − δ ij + 2( q − | Df | + ǫ ) q − f i f j where δ ij is the Kronecker delta function. Recall that we define q = p/ f can be written in non-divergent form as f t = ( | Df | + ǫ ) q − ∆ f + 2( q − | Df | + ǫ ) q − f ij f i f j = a ij f ij . We compute the evolution equation for f λ f λt = a ij f λij + ( f ij Da ij ) Df λ . Since this equation satisfies the Strong Maximum Principle, f λ must attain its maximumvalue on the parabolic boundary of Ω. Because f λ ≤ f λ < λ . (cid:3) Lemma 4.2. Assume the same as in the last lemma, then at any point x on the free-boundary ∂ Ω t f νν ( x , t ) < where ν is the inward normal vector at x with regards to ∂ Ω t .Proof. Apply Hopf’s Lemma to the evolution equation for f ν from the last lemma, observingthat f ν attains the maximum value of 1 at ( x , t ). (cid:3) Convexity In this section we will show that the time-section Ω t remains convex and the function f ( ., t ) remains concave on Ω t . Normally, for this kind of question, the main difficulty liesin showing that Ω t remains convex. The arguments for the case p = 2 as in [11] or [5] donot translate directly to the case p > 2. On the other hand, our argument here can besimplified to give a new and simple proof for the case p = 2. The argument relies heavilyon the Neumann boundary condition | Df | = 1. Lemma 5.1. Assume the same hypotheses as in the Lemma 3.1. Furthermore, assumethat Ω is strictly convex and f is strictly concave on Ω . If f is a solution to the problem (P( ǫ )) up to some positive time T , then Ω t is strictly convex and f ( ., t ) is strictly concavefor all t ∈ [0 , T ) .Proof. We will show that f λλ ( x , t ) < x , t ) ∈ Ω, any unit vector λ and any t ∈ [0 , T ′ ] where T ′ is any numberstrictly less than T . Clearly this implies that Ω t is strictly convex and f is strictly concavefor all t ∈ [0 , T ).First, we compute the evolution equation of f λλ , f t = a ij f ij f λt = a ij f λij + ( f ij Da ij ) · Df λ f λλt = a ij f λλij + 2( f λij Da ij ) · Df λ + ( f ij Da ij ) · Df λλ + ( f ij ( Da ij ) λ ) · Df λ = a ij f λλij + ( f ij Da ij ) · Df λλ + (2 f λij Da ij + f ij D ( a ij ) λ ) · Df λ Since f is smooth for all t ∈ (0 , T ), there exists a finite number C ( T ′ ) such that for anyunit vector λ and any point ( x , t ) ∈ Ω ∩ { < t ≤ T ′ }| f λij Da ij + f ij D ( a ij ) λ | < C. Choose a smooth function v on R n such that ( < v < − ( f ) λλ in Ω v = 0 on ∂ Ω . Such v exists because f is strictly concave on Ω . Let v be the solution of the Cauchy-Dirichlet problem v t = a ij v ij + ( f ij Da ij ) · Dv − Cv in Ω v = 0 on ∂ Ω ∩ { < t < T } v ( ., 0) = v in Ω . Applying Strong Maximum Principle and Hopf’s Lemma to v we easily deduce that ( v > | Dv | > ∂ Ω ∩ { < t < T } . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 11 We are going to show that(5.1) v + f λλ < t ∈ [0 , T ′ ] and all unit vector λ . Assuming that it is not the case, i.e there existssome point ( x ′ , t ′ ) and some unit vector λ ′ such that( v + f λ ′ λ ′ )( x ′ , t ′ ) = 0and v + f λλ < t < t ′ and all unit vector λ . In other words, t ′ is the first time (5.1) fails. We considertwo cases, ( x ′ , t ′ ) is an interior point or a boundary point. But first, note that we have theevolution equation for V = v + f λ ′ λ ′ (5.2) V t = a ij V ij + ( f ij Da ij ) · DV + (2 f λ ′ ij Da ij + f ij D ( a ij ) λ ′ ) · Df λ ′ − Cv. If ( x ′ , t ′ ) is an interior point, then because it is a maximum point of V in Ω t ′ , we have a ij V ij ≤ DV = 0 . Substitute into (5.2) we have V t ≤ (2 f λ ′ ij Da ij + f ij D ( a ij ) ′ λ ) · Df ′ λ − Cv. Because at the point ( x ′ , t ′ ) v + f λ ′ λ ′ ≥ v + f λλ or f λ ′ λ ′ ≥ f λλ for any other unit vector λ , we have f λλ ′ = 0for any λ ⊥ λ ′ . Hence, V t ≤ (cid:0) (2 f λij Da ij + f ij D ( a ij ) λ ) · λ ′ (cid:1) f λ ′ λ ′ − Cv< − Cf λ ′ λ ′ − Cv (remember f λ ′ λ ′ = − v < x ′ , t ′ ) is the first time V = 0. So ( x ′ , t ′ ) cannot bean interior point.If x ′ is on ∂ Ω t ′ . Again, denote by ν the inward normal unit vector to ∂ Ω t ′ at x ′ . Thenat this point we have from definition of ( x ′ , t ′ ) and λ ′ ,( v + f λ ′ λ ′ ) ν ≤ f νλ ′ λ ′ ≤ − v ν < . We will show that on the other hand(5.3) f νλ ′ λ ′ = 0 . We have from the Lemma 4.2 that f νν < . We also have as a consequence of the fact that | Df | = 1 on the free-boundary and | Df | < f νλ = 0 for any tangential unit vector λ . Hence as a consequence ofthe fact f λ ′ λ ′ = 0, λ ′ must be a tangential vector of ∂ Ω t ′ . Otherwise, there would be atangential vector λ that lies on the same plane with ν and λ ′ such that f λλ > x ′ , t ′ ) and λ ′ .Without loss of generality, we can assume that ν = e and λ ′ = e . Because e is theunit normal vector of ∂ Ω t ′ at x ′ , in a small neighborhood of x ′ , we can write ∂ Ω t ′ as thegraph of a smooth function x = γ ( x , x ′ )where x ′ = ( x , ...x n ). From here to the end of the proof, we will use γ ′ and γ ′′ to denotethe first and second derivatives of γ with regards to x . Differentiate f = 0 with regards to e we have f γ ′ + f = 0or γ ′ = 0 since f = 1, f = 0. Differentiate one more time and disregard all termscontaining γ ′ we have f γ ′′ + f = 0and so γ ′′ = 0 since f = 0 due to our assumption. Differentiate Df · Df = 1 twice withregards to e and disregard all terms containing γ ′ or γ ′′ we obtain Df · Df + Df · Df = 0or f + | Df | = 0 . As above, because f ii ≤ f ∀ i , we have f i = 0for all i = 2. But f = 0 as well, so Df = 0. Hence f = 0which is exactly what we want to show in (5.3). We then have a contradiction. In otherwords f λλ < x, t ) ∈ Ω ∩ { < t < T } and all unit vector λ or equivalently, Ω t is strictly convexand f ( ., t ) is strictly concave in Ω t for all t ∈ [0 , T ). (cid:3) FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 13 Regularity near the Free-Boundary In this section, we show that the degeneracy | Df | = 0 is kept away from the free-boundary. Consequently, the free-boundary is smooth, uniformly in ǫ . It enables us to showthat the limiting function obtained by letting ǫ go to 0 satisfies the boundary condition ofthe original problem. The proof depends crucially on the concavity of f .We introduce some notations. We denote by B r ( x ) the disk of radius r around xB r ( x ) = { y ∈ R n | | y − x | < r } when x ∈ R n and r ∈ R . We write B r for B r (0). We also define A r = { ( x , t ) | dist( x , ∂ Ω t ) < r } ∩ Ω . For any point x = ( x , x , ...x n ), we define ψ ( x ) = x ψ ′ ( x ) = ( x , ..., x n ) . Lemma 6.1. Assume all hypotheses as in the Lemma 5.1. Assume also that there existpositive numbers r, R and m and a point x such that B r ( x ) ⊂ Ω t ⊂ B R ( x ) for all t ∈ [0 , T )(6.1) f ( x , t ) > m for all ( x , t ) ∈ B r ( x ) × [0 , T ) . (6.2) Then for any < T < T and k ∈ Z + , there exist positive numbers d ( r, R, m, T ) and C ( d, k ) such that | f ( ., t ) | C k ( A d ∩{ T ≤ t To simplify the notation, we assume that the conditions (6.1) holds for x = 0. Inother words B r ⊂ Ω t ⊂ B R for all t ∈ [0 , T ) f ( x , t ) > m for all ( x , t ) ∈ B r × [0 , T ) . Let ( P, t ) be a point on ∂ Ω for some t ∈ [ T , T ). Fix this value of t from here until theend of this proof. Without loss of generality, we can assume that ψ ( P ) < ψ ′ ( P ) = 0First, we will show that f ( x , t ) is bounded away from 0 in a neighborhood of P in Ω t .Consider any point Q in Ω t that satisfies the following conditions ψ ( Q ) < | ψ ′ ( Q ) | < rf ( Q, t ) < m/ . Let R = (0 , ψ ′ ( Q )). Because R ∈ B r , we have f ( R, t ) > m . We also have f ( Q, t ) < m/ f ( x , t ) decreases in x as a consequence of concavity (here f denotes the first derivativeof f with regards to x ). Thus, m/ < f ( R, t ) − f ( Q, t )= Z ψ ( Q ) f (( x , ψ ′ ( Q )) , t ) dx ≤ | ψ ( Q ) | f ( Q, t ) ≤ Rf ( Q, t ) f ( Q, t ) ≥ m/ R We just showed that if x satisfies ψ ( x ) < | ψ ′ ( x ) | < rf ( x , t ) < m/ f ( x , t ) ≥ m/ R. On the set containing all such x , the Implicit Function Theorem says that there exists afunction g defined on the set B = { ( y, x ′ ) ∈ R × R n − | ≤ y < m/ , | x ′ | < r } × [0 , T )such that ψ ( x ) = g ( f ( x , t ) , ψ ′ ( x )) . We will compute explicitly the evolution equation and boundary condition of gf = 1 g f i = − g i g f t = − g t g f = − g g f i = − g g i − g i g g f ij = − g g ij − g g j g i − g g i g j + g i g j g g . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 15 The boundary condition | Df | = 1 on ∂ Ω t is equivalent to g = n X i =2 g i ! / on { ( y, x ′ ) ∈ R × R n − | y = 0 , | x ′ | < r } × [0 , T ) . Next we compute the evolution for g on B . In all P appearing in the following computations,unless explicitly marked otherwise, indices i and j run from 2 to n . Let M = 1 + X g i . Then | Df | = Mg ∆ f = − g (cid:16) g + g X g ii − g X g i g i + g X g i (cid:17) = − g (cid:16) M g + g X g ii − g X g i g i (cid:17) f i f j f ij = − g (cid:16) g − X g i ( g g i − g i g )+ X g i g j ( g g ij − g g i g j − g g j g i + g i g j g ) (cid:17) = − g (cid:16) g M − g M X g i g i + g X g i g j g ij (cid:17) . Substitute into the equation for f t , g t = ( M + ǫ ) q − g q (cid:16) M g + g X g ii − g X g i g i (cid:17) + 2( q − 1) ( M + ǫ ) q − g q (cid:16) g M − g M X g i g i + g X g i g j g ij (cid:17) = b ij g ij . We want to show that there exist positive numbers λ , Λ, independent of ǫ and t such that λ ≤ b ij ξ i ξ j ≤ Λfor all unit vector ξ ∈ R n . First, because1 ≥ | Df | ≥ f ≥ m/ R, we have 1 ≤ M / ≤ g ≤ R/m. The upper bound Λ then is obvious. For the lower bound, because M ξ − g M X g i ξ ξ i + g X g i g j ξ i ξ j = ( M ξ − X g i ξ i ) , it is enough to show that M ξ − g X g i ξ ξ i + g X ξ i ≥ λ for some positive λ . We have (cid:18) g − n (cid:19) (cid:18) g i + 12 n (cid:19) − ( g g i ) = 12 n (cid:18) g − g i − n (cid:19) ≥ g ≥ M = 1 + P g i and so (cid:18) n + g i (cid:19) ξ − g g i ξ ξ i + ( g − n ) ξ i ≥ ≤ i ≤ n. Summing up we obtain (cid:18) n − n + X g i (cid:19) ξ − g X g i ξ ξ i + (cid:18) g − n (cid:19) X ξ i ≥ M ξ − g X g i ξ ξ i + g X ξ i ≥ n . From the theory of quasi-linear parabolic equation with oblique boundary condition wecan choose d < min( r, m/ 2) such that g ( ., t ) is in C ∞ on the set { ( y, x ′ ) ∈ R × R n − | ≤ y ≤ d, | x ′ | ≤ d } × [ T , T )and for any k , the norm | g ( ., t ) | C k depends only on k, d, r, R, m and T , not on ǫ , t or g .Revert back to f , we conclude that f is smooth on the set { ( x , t ) ∈ Ω | | ψ ′ ( x ) | ≤ d, ψ ( x ) < , f ( x , t ) ≤ d } ∩ { T ≤ t < T } and again, for any k , the norm | f ( ., t ) | C k on this set depends only on k, d, r, R, m and T .Note that the above set includes the set B ( P, d ) ∩ Ω t . The conclusion is of course true for any point on ∂ Ω t in place of P where t ∈ [ T , T ). Thelemma then follows. (cid:3) Comparison Principle In the first lemma here, we show that if f ′ is strictly greater than f , then a solution tothe problem (P( ǫ )) with initial value f ′ remains strictly greater than a solution with initialvalue f . Lemma 7.1. Suppose that f and f ′ are solutions up to some finite time T to the problemP( ǫ ) and P( ǫ ′ ) respectively for some ǫ ≥ ǫ ′ > . Suppose also that at the time t = 0 , Ω ⊂ Ω ′ f ( x ) < f ′ ( x ) in Ω . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 17 Then for any t ∈ (0 , T ) , Ω t ⊂ Ω ′ t f ( x , t ) < f ′ ( x , t ) in Ω t . Proof. Since f ( x ) < f ′ ( x ) in Ω we can choose a positive number m such that f ( x ) + m < f ′ ( x ) in Ω . Choose a positive number δ such that δT < m . We will show thatΩ t ⊂ Ω ′ t f ′ ( x , t ) − f ( x , t ) − m + δt > t for all t ∈ [0 , T ). Assuming it is not the case, there must be a first time t such that at leastone of the two above conditions is violated. Assume that the first condition is violated at t . In other words, ∂ Ω t and ∂ Ω ′ t touches at some point x , then at that point f ′ ( x , t ) − f ( x , t ) − m + δt = − m + δt < t , contradicting ourchoice of t . Hence Ω t ⊂ Ω ′ t for all t ∈ [0 , t ]. The second condition is violated implies that there is a point x ∈ Ω t such that f ′ ( x , t ) − f ( x , t ) − m + δt = 0 . We consider the case x ∈ ∂ Ω t ⊂ Ω ′ t first. Let ν be the inward unit normal to ∂ Ω t at x . From the definition of ( x , t ) we must have( f ′ ( x , t ) − f ( x , t ) − m + δt ) ν ≥ f ′ ν ( x , t ) − f ν ( x , t ) ≥ f ′ ν ( x , t ) ≥ x ∈ Ω t ⊂ Ω ′ t , then because it is an minimum point for f ′ − f on Ω t , we have Df ′ ( x , t ) = Df ( x , t )0 ≥ ∆ f ′ ( x , t ) ≥ ∆ f ( x , t )0 ≥ f ′ νν ( x , t ) ≥ f νν ( x , t ) . Plug into the equation for f ′ t and f t , recalling that ǫ ≥ ǫ ′ we obtain f ′ t = ( | Df ′ | + ǫ ′ ) q − ∆ f ′ + 2( q − | Df ′ | + ǫ ′ ) q − | Df ′ | f νν ≥ ( | Df | + ǫ ) q − ∆ f + 2( q − | Df ′ | + ǫ ) q − | Df | fνν = f t . On the other hand, because t is the first time f ′ − f − m + δt = 0, f ′ t − f t + δ ≤ . Again, we arrive a contradiction. In other words,Ω t ⊂ Ω ′ t f ′ − f − m + δt > t ∈ [0 , T ). The Lemma then follows readily. (cid:3) Remark . If we let m → min { f ′ ( x ) − f ( x ) | x ∈ Ω } δ → m ( t ) = min { f ′ ( x , t ) − f ( x , t ) | x ∈ Ω t } is a non-decreasing function.We prove a slightly improved version of the last lemma. Lemma 7.2. Suppose f and f ′ are solutions up to time T to the problem P( ǫ ) and P( ǫ ′ )respectively for some ǫ ≥ ǫ ′ > . Suppose also that at the time t = 0 , Ω ⊂ Ω ′ f ( x ) ≤ f ′ ( x ) in Ω . Then for any t ∈ [0 , T ) , Ω t ⊂ Ω ′ t f ( x , t ) ≤ f ′ ( x , t ) in Ω t . Proof. Without loss of generality, we can assume that f attains its maximum value at theorigin. For each positive λ , defineΩ λ = { ( x , t ) | ( λ x , λ t ) ∈ Ω } f λ = 1 λ f ( λ x , λ t )Ω λ = { x | λ x ∈ Ω } f λ = 1 λ f ( λ x , λ t ) . It is clear that f λ is a solution to the problem (P( ǫ )) with respect to the initial data f λ .Furthermore, since f is concave, for each λ > λ ⊂ Ω ⊂ Ω ′ f λ < f ≤ f ′ in Ω λ . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 19 The last lemma says that for all t ∈ [0 , T /λ ),Ω λt ⊂ Ω ′ t λ f ( λ x , λ t ) = f λ ( x , t ) < f ′ ( x , t ) in Ω λt . Let λ → t ⊂ Ω ′ t f ( x , t ) ≤ f ′ ( x , t ) in Ω t for all t ∈ [0 , T ). (cid:3) Corollary 7.3. Suppose ( f, Ω) and ( f ′ , Ω ′ ) are two solutions to the problem (P( ǫ )) withrespect to the same initial value f up to time T . Then f = f ′ on R n × [0 , T ) . Long-Time Existence for the Regularized Problem Lemma 8.1. Assume that f satisfies all hypotheses of the Lemma 3.1. Then there existsa unique solution to the problem P( ǫ ) up to some positive time T > where lim t → T f ( x , t ) = 0 for all x ∈ R n .Proof. Let T be the maximal existence time for solutions to the problem P( ǫ ) with theinitial data f . Due to the uniqueness result in section 10, there must be a solution f thatexists up to time T . From the short-time existence result, T must be positive. We will showthat lim t → T f ( x , t ) = 0 for all x ∈ R n . Assuming otherwise, then the same argument in the proof for the Lemma 11.1 can be usedto show that T must be finite. In other words, due to the concavity of f , there exists anumber c < (cid:0) ( | Df | + ǫ ) q − Df (cid:1) < c in Ω and consequently, T ≤ max f | c | . We will prove that we can then extend this solution to a time T ′ > T . From the concavityof f ( ., t ), f is a decreasing function in t . Define f T ( x ) = lim t → T f ( x , t ) . Since | Df | ≤ f T is continuous. Because f T is not identically 0, there exist a ball B r ( x ′ ) and a positive number m such that f T > m in B r ( x ′ ) . From the Lemma 6.1, for all t ∈ [ T / , T ), there exists a positive number d such that f issmooth up to the boundary and time T in the set { ( x , t ) | dist( x , ∂ Ω t ) < d } ∩ Ω [ T/ ,T ) . Combine with the smoothness (depending on ǫ ) of f up to time T in the interior of Ω [ T/ ,T ) from the standard theory of parabolic equation, we obtain the smoothness up to the bound-ary and time T of f in Ω [ T/ ,T ) . Consequently, f T is smooth up to the boundary. Fromthe Lemma 5.1, we know that Ω T is convex and f T is concave in Ω T . However, we needa stronger result that Ω T is strictly convex and f T is strictly concave in Ω T in order toapply the Lemma 3.1. In deed, we can improve the result in the lemma 5.1 by duplicatingthe proof and substituting T ′ by T directly. In that proof, because we did not have thesmoothness of f up to time T , we need to introduce T ′ < T to guarantee the existence ofa finite number C ( T ′ ) such that | f λij Da ij + f ij D ( a ij ) λ | < C for all t ∈ [ t, T ′ ]. But now we have the smoothness of f up to time T , we can derive thefact that there exists a number C ( T ) such that the above inequality holds for all t ∈ [0 , T ).The proof then guarantees that f is strictly concave at the time T .The function f T now satisfies all hypotheses of the Lemma 3.1. By that Lemma, we canthen extend the solution f to some time T ′ > T . It contradicts the maximality of T . So wemust have lim t → T f ( x , t ) = 0for all x ∈ R n . (cid:3) Existence of Solution to the p -Laplacian problem In this section, we will pass ǫ to 0 and obtain a solution to our degenerate problem. Lemma 9.1. Assume that Ω is a bounded and convex domain. The function f is positiveand concave in Ω . Furthermore, on the boundary ∂ Ω , f satisfies f ( x ) = 0 for all x | Df ( x ) | = 1 for a.e. x . Then there exists a solution to the problem (P) up to some time T where lim t → T f ( x , t ) = 0 ∀ x ∈ R n . The free-boundary ∂ Ω t is smooth for all t ∈ (0 , T ) . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 21 Proof. Choose a sequence of functions f ǫ with positive sets Ω ǫ for all ǫ ∈ (0 , 1) such thatΩ ǫ ∈ C ∞ and f ǫ ∈ C ∞ (Ω ǫ ) , Ω ǫ is strictly convex ,f ǫ is strictly concave , Ω ǫ ⊂ Ω ǫ and f ǫ ≤ f ǫ if ǫ > ǫ , Ω = ∪ Ω ǫ and f ( x ) = lim ǫ → f ǫ ( x ) for all x ∈ R n , | Df ǫ | = 1 on ∂ Ω ǫ . In other words, f ǫ is an increasing sequence of smooth and strictly concave function thatconverge to f as ǫ → 0. From the Lemma 8.1, for each ǫ , there exists a unique solution f ǫ to the problem (P( ǫ )) up to some time T ǫ where it vanishes identically. We will prove thatlim ǫ → f ǫ is a solution to the original problem (P).From the lemma 7.2 and our choice of f ǫ , it is clear that if ǫ > ǫ , then T ǫ ≤ T ǫ Ω ǫ ⊂ Ω ǫ f ǫ ≤ f ǫ in Ω ǫ . Define T = lim ǫ → T ǫ Ω = ∪ Ω ǫ f ( x , t ) = lim ǫ → f ǫ ( x , t ) for ( x , t ) ∈ R n × [0 , T ) . Due to the uniform smoothness of f ǫ in a neighborhood of ∂ Ω ǫ , we haveΩ (0 ,T ) ∈ C ∞ f = 0 and | Df | = 1 on ∂ Ω × { < t < T } . If ( x , t ) ∈ Ω, there exists an ǫ such that ( x , t ) ∈ Ω ǫ for all ǫ < ǫ . Since f ǫ ( x , t ) increasesas ǫ decreases to 0, f ( x , t ) = lim ǫ → f ǫ ( x , t ) > . From the bound | Df | ≤ C , / t estimate for f ǫ as functions of t , uniformly in t and ǫ . Together with the fact thatfor every x , lim t → T ǫ f ǫ ( x , t ) = 0we have lim t → T f ( x , t ) = 0 . Because for every ǫ and t f ǫ ( x , t ) ≤ f ǫ ( x , < f ( x , , we have f ( x , t ) = lim ǫ → f ǫ ( x , t ) ≤ f ( x ) . On the other hand, since f ǫ ( x , t ) increases as ǫ decreases to 0,lim t → f ( x , t ) ≥ lim t → f ǫ ( x , t ) = f ǫ ( x )for any ǫ . Consequently, lim t → f ( x , t ) ≥ lim ǫ → f ǫ ( x ) = f ( x ) . Hence lim t → f ( x , t ) = f ( x ) . From | Df ǫ | ≤ 1, we can choose a sequence of ǫ converging to 0 such that Df ǫ ⇀ Df in all compact subsets of Ω. Given any function θ ∈ C ∞ (Ω) and 0 < t < t < T we havefrom the equation f ǫt = div(( | Df ǫ | + ǫ ) q − Df ǫ )that Z Ω ( t ,t f ǫ θ t d x dt − Z f ǫ θ d x (cid:12)(cid:12)(cid:12)(cid:12) Ω t Ω t = Z Ω ( t ,t ( | Df ǫ | + ǫ ) q − Df ǫ · Dθ d x dt. Passing to the limit we then obtain Z Ω ( t ,t f θ t d x dt − Z f θ d x (cid:12)(cid:12)(cid:12)(cid:12) Ω t Ω t = Z Ω ( t ,t | Df | q − Df · Dθ d x dt. (cid:3) Uniqueness Lemma 10.1. Solution obtained in the Lemma 9.1 is the unique solution to the problem(P).Proof. Assume that there exists another solution g to the problem (P). Let Ω ∗ be thepositive set of g and T ∗ its existence time. Also without loss of generality, assuming that f attains it maximum value at 0. For each positive λ , it is clear that g λ ( x , t ) = λ − g ( λ x , λ p +2 t )is a solution to the problem P( ǫ ) with positive setΩ λ = { ( x , t ) | ( λ x , λ p +2 t ) ∈ Ω ∗ } and initial data g λ ( x , t ) = λ − f ( λ x , λ p +2 t ) . FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 23 Clearly, for λ < 1, Ω ⊂ Ω λ f < g λ in Ω . We will show that for all t < min( T, λ − ( p +2) T ∗ ),Ω t ⊂ Ω λt f ( x , t ) < g λ ( x , t ) in Ω t . Assuming it is not the case, then there must be a first time t where at least one of thosetwo inequalities is violated. If the first one is violated at the time t , it means ∂ Ω t and ∂ Ω λt touch at some point x . At that point ( x , t ), | Df | = 1 > λ = | Dg λ | . There must be then a point x ∈ Ω t such that f ( x , t ) > g ǫ ( x , t ) . which implies that the second inequality must be violated at some time before t . So, upto time t , Ω t ⊂ Ω λt . Consequently, on the parabolic boundary of Ω [0 ,t ] , f < g λ . Thus, from the lemma 3.1 inChapter VI of [6], we have f < g in Ω [0 ,t ] which contradicts our choice of t . Hence for all t < min( T, λ − ( p +2) T ∗ ), Ω t ⊂ Ω λt f ( x , t ) < g λ ( x , t ) = λ − g ( λ x , λ p +2 t ) in Ω t . Let λ → t < min( T, T ∗ ),Ω t ⊂ Ω ∗ t f ( x , t ) ≤ g ( x , t ) in Ω t . Arguing similarly for λ > ∗ t ⊂ Ω t g ( x , t ) ≤ f ( x , t ) in Ω ∗ t . Thus for all t < min( T, T ∗ ) Ω t = Ω ∗ t f ( x , t ) = g ( x , t ) . (cid:3) Vanishing in finite time Lemma 11.1. The existence time T of the solution obtained in the Lemma 9.1 is finite.Proof. Clearly from the Comparison Principle in section 7 and scaling that it is enough toprove this lemma for one particular initial function f . We show that if a smooth function f satisfies all hypotheses of the Lemma 9.1 and∆ p f < c for some c < 0, then for any 0 < t < t < T and any x ∈ Ω t , the corresponding solution f satisfies the inequality(11.1) f ( x , t ) − f ( x , t ) ≤ c ( t − t ) . It then readily follows that T ≤ max f | c | . Choose the sequence { f ǫ } so thatdiv(( | Df ǫ | + ǫ ) q − Df ǫ ) < c. We will show that f ǫ satisfies f ǫt ≤ c for all ǫ and (11.1) then follows immediately.Differentiating the equation satisfied by f ǫ with respect to t , it is easy to see that f ǫt satisfies the Maximum Principle. Since f ǫt ≤ c at the time t = 0 from our choice of f ǫ , allwe need to show is that f ǫt cannot attain its maximum value on the free-boundary. Fromthe Hopf’s Lemma, if f ǫt attains its maximum value at a point x on the free-boundary ∂ Ω t , then we must have ( f ǫν ) t ( x , t ) = ( f ǫt ) ν ( x , t ) < ν is the inward unit normal at x with respect to ∂ Ω t . On the other hand, since Ω t shrinks in time, for any t < t x ∈ Ω t and so, f ǫν ( x , t ) < f ǫν ( x , t ) = 1which lead to ( f ǫν ) t ( x , t ) ≥ . (cid:3) Remark . The finiteness for the existence time holds for any initial data with boundedsupport, not just concave ones. FREE-BOUNDARY PROBLEM FOR THE EVOLUTION p -LAPLACIAN 25 References 1. A. Akopyan and G. Shakhgolyan, Uniqueness of solutions of a problem of p -parabolic type with a freeboundary , Izv. Nats. Akad. Nauk Armenii Mat. (2001). MR MR1964578 (2004b:35344)2. Luis A. Caffarelli and Juan L. V´azquez, A free-boundary problem for the heat equation arising in flamepropagation , Trans. Amer. Math. Soc. (1995), no. 2, 411–441. MR MR1260199 (95e:35097)3. D. Danielli, A. Petrosyan, and H. Shahgholian, A singular perturbation problem for the p -Laplace oper-ator , Indiana Univ. Math. J. (2003), no. 2, 457–476. MR MR1976085 (2005d:35066)4. P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation , J.Amer. Math. Soc. (1998), no. 4, 899–965. MR MR1623198 (99d:35182)5. P. Daskalopoulos and Ki-Ahm Lee, Convexity and all-time C ∞ -regularity of the interface in flame prop-agation , Comm. Pure Appl. Math. (2002), no. 5, 633–653. MR MR1880645 (2003c:35164)6. Emmanuele DiBenedetto, Degenerate parabolic equations , Universitext, Springer-Verlag, New York,1993. MR MR1230384 (94h:35130)7. Antoine Henrot and Henrik Shahgholian, Existence of classical solutions to a free boundary problemfor the p -Laplace operator. I. The exterior convex case , J. Reine Angew. Math. (2000), 85–97.MR MR1752296 (2001f:35442)8. , Existence of classical solutions to a free boundary problem for the p -Laplace operator. II. Theinterior convex case , Indiana Univ. Math. J. (2000), no. 1, 311–323. MR MR1777029 (2001m:35326)9. C. Lederman, J. L. V´azquez, and N. Wolanski, Uniqueness of solution to a free boundary problemfrom combustion , Trans. Amer. Math. Soc. (2001), no. 2, 655–692 (electronic). MR MR1804512(2001m:35327)10. Gary M. Lieberman, Second order parabolic differential equations , World Scientific Publishing Co. Inc.,River Edge, NJ, 1996. MR MR1465184 (98k:35003)11. Arshak Petrosyan, Convexity and uniqueness in a free boundary problem arising in combustion theory ,Rev. Mat. Iberoamericana (2001), no. 3, 421–431. MR MR1900891 (2003d:35280)12. , On existence and uniqueness in a free boundary problem from combustion , Comm. PartialDifferential Equations (2002), no. 3-4, 763–789. MR MR1900562 (2003d:35281) Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address ::