Parabolic quasi-concavity for solutions to parabolic problems in convex rings

Transcription

1 Math. Nachr. 283, No. 11, (2010) / DOI /mana Editor s Choice Parabolic quasi-concavity for solutions to parabolic problems in convex rings Kazuhiro Ishige 1 and Paolo Salani 2 1 Mathematical Institute, Tohoku University, Aoba, Sendai , Japan 2 Dipartimento di Matematica U. Dini, Università di Firenze, V. le Morgagni 67/A, Firenze, Italy Received 9 November 2009, revised 16 April 2010, accepted 4 May 2010 Published online 23 September 2010 Key words Parabolic equations, convexity, quasi-concavity MSC (2000) 35K05, 52A01, 35B05, 35B30 We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi-concavity, which involves time and space jointly and is a stronger property than the spatial quasi-concavity, and study the convexity of superlevel sets of the solutions. 1 Introduction Let u be a solution of the following parabolic problem t γ(u) =F (x, u, u, 2 u) in D (0, ), u(x, 0) = ϕ(x) 0 in D, u(x, t) =0 on Ω (0, ), u(x, t) =g(t) on K [0, ), (1.1) where t = / t, N 1, Ω and K are an open convex set and a closed convex set in R N, respectively, such that K Ω, D =Ω\ K, γ is a nonnegative convex function in [0, ) such that γ(0) = 0, g C 1 ([0, )) is a nonnegative bounded function with g 0 and g(0) ϕ(x) for x D, andf is a proper elliptic operator. Throughout the paper, by a solution u of (1.1), we mean a function u C 2,1 (D (0, )) C(D [0, ) \ D {0}) pointwise satisfying all of the conditions in (1.1). In this paper we investigate some convexity properties of the solutions of (1.1). The study of geometric properties (with special regard to concavity and quasi-concavity) of solutions of partial differential equations is a classical subject, especially in the field of elliptic equations, where it is hardly possible to compile a complete bibliography; then we just recall the classical monograph [19] and very recent related results like [1] and [2], and we refer to these for more references. For parabolic equations, fewer results are disposable, as far as we know, even if it is well-known (see for instance [5] and [12]) that log-concavity is preserved by the heat flow and it also seems to rule the large time behavior of solutions to parabolic equations in R N (see [25]). On the other hand, negative results are contained in [15] and [16], showing that in general one cannot expect the quasi-concavity of the initial datum to be preserved, even by the heat flow. Concavity properties of solutions to parabolic problems have been considered also in [3], [4], [8], [11], [21], [22], [23], [18], [26], for instance. See also references therein. Corresponding author: ishige@math.tohoku.ac.jp, Phone: , Fax: salani@math.unifi.it, Phone: , Fax:

2 Math. Nachr. 283, No. 11 (2010) / 1527 We recall that, roughly speaking, a function v : R N R {± }is said to be quasi-concave in R N if all its superlevel sets {x R N : v(x) l} are convex. More precisely, v is quasi-concave if v((1 λ)x 0 + λx 1 ) min{v(x 0 ),v(x 1 )} for every x 0,x 1 R N, λ (0, 1). If v is defined only in a proper subset Ω of R N,we extend it as outside Ω and we say that v is quasi-concave in Ω if such an extension is quasi-concave in R N. A solution u of (1.1) is said to be spatially quasi-concave if the function x u(x, t) is quasi-concave for every t 0,thatis, u((1 λ)x 0 + λx 1,t) min{u(x 0,t),u(x 1,t)} (1.2) for every x 0,x 1 Ω, t 0, λ (0, 1). A classical approach to study the spatial quasi-concavity of the solution u is based on the spatial quasi-concavity function Φ(x, y, t) =u((1 λ)x + λy, t) min{u(x, t),u(y, t)}, (x, y, t) Ω 2 [0, ). (1.3) Then u is spatially quasi-concave if and only if inf Φ 0 (1.4) Ω 2 [0, ) for any fixed λ (0, 1). Here we introduce a stronger quasi-concavity property, the parabolic quasi-concavity, which involves time and space jointly and seems to be more suitable for the parabolic framework. Precisely, inspired by Borell [4], who introduced the notion of parabolic convexity for subsets of R N [0, ), we give the following definition. Definition 1.1 A function u C(Ω [0, )) is called parabolically quasi-concave if u ((1 λ)x 0 + λx 1, ( (1 λ) t 0 + λ ) ) 2 t 1 min{u(x 0,t 0 ),u(x 1,t 1 )} (1.5) for every x 0,x 1 Ω, t 0,t 1 0, λ (0, 1). Clearly, if a function is parabolically quasi-concave, then it is spatially quasi-concave at every time; indeed, if we fix a time t 0, takingt 0 = t 1 = t, (1.5) coincides with (1.2). Roughly speaking, a function u is parabolically quasi-concave if all its superlevel sets {(x, t) : u(x, t) l} are parabolically convex in the sense of [4] (see Section 2.3 hereafter). An equivalent geometric characterization of parabolic quasi-concavity in terms of level sets is the following: u is parabolically quasi-concave if, for every l R, λ (0, 1), andt 0,t 1 0, there holds { x R N : u(x, t λ ) l } (1 λ) { x R N : u(x, t 0 ) l } + λ { x R N : u(x, t 1 ) l }, where t λ = ( (1 λ) t 0 + λ ) 2. t 1 Analogously to spatial quasi-concavity, to study the parabolic quasi-concavity of u, we can define a parabolic quasi-concavity function Π:Ω 2 [0, ) 2 R as follows ( Π(x, y, r, s) =u (1 λ)x + λy, ( (1 λ) r + λ s ) ) 2 min{u(x, r),u(y, s)}. (1.6) Then u is parabolically quasi-concave if and only if inf Π 0 (1.7) Ω 2 [0, ) 2 for any fixed λ (0, 1). The strategy here adopted is exactly to find suitable conditions such that (1.7) holds. Theorem 3.1 is a general theorem about minimum points of Π and it can be considered as the main result of this paper. As a consequence, we obtain Theorem 3.2, where we prove that, for the case of special initial and boundary data ϕ 0 in D and g 1 in [0, ), if the operator F satisfies a suitable convexity property, then u is parabolically quasi-concave. However, in order to state Theorems 3.1 and 3.2 in a clear and precise way, we need to introduce some notation. Therefore we postpone explicit statements of our main results to Section 3. Here we give a result on parabolic quasi-concavity for the heat equation with absorption, which is a typical application of our main theorems.

3 1528 K. Ishige and P. Salani: Parabolic quasi-concavity Proposition 1.2 Let D =Ω\ K, whereω is a bounded convex domain and K a compact convex set with nonempty interior, such that K Ω R N and Ω, K C 2,α for some α (0, 1). Letu be a solution of t u =Δu f(x, u) in D (0, ), u(x, 0) = 0 in D, u(x, t) =0 on Ω (0, ), u(x, t) =1 on K [0, ), where f = f(x, v) C 1 (Ω R) is a function such that Assume the following: f(x, 0) = 0 for x Ω and f(x, v) f(x, w) whenever v w. (1.9) the function x f(x, v) 1/2 is concave for every v (0, 1). (1.10) Then u is parabolically quasi-concave in Ω [0, ). If Ω is unbounded (and possibly also K is unbounded), by approximation with bounded convex sets, we see that there holds the conclusion of Proposition 1.2. Therefore, as a corollary, we also give the following result about the parabolic quasi-concavity of solutions to the heat equation with absorption in the exterior of a convex closed set. Corollary 1.3 Let D = R N \ K and K beaclosedconvexsetinr N with nonempty interior and K C 2,α for some α (0, 1). Letu be a bounded solution of t u =Δu f(x, u) in D (0, ), u(x, 0) = 0 in D, (1.11) u(x, t) =1 on K [0, ). If f C 1 (D R) satisfies (1.9) (1.10) with Ω replaced by R N,thenuisparabolically quasi-concave in R N [0, ). Notice that Theorem 3.2, Proposition 1.2 and Corollary 1.3 are the first results treating parabolic quasiconcavity in PDE s approach and they are new results even for the heat equation. In Section 3.3 we also give a general result about spatial quasi-concavity of solutions of problem (1.1), with a quasi-concave initial datum ϕ. We address the reader to Sections 3.3, 5 and 7 for more details and comparison with existing results. The paper is organized as follows. In Section 2 we introduce some notation and some basic results about quasi-concavity and parabolic quasi-concavity. In Section 3 we state and discuss our main results, Theorems 3.1 and 3.2, about parabolic quasi-concavity. Furthermore we state and discuss Theorems 3.3 and 3.4 about spatial quasi-concavity. Sections 4 and 5 are devoted to the proofs of Theorems 3.1 and 3.2 and Theorems 3.3 and 3.4, respectively. In Section 6 we prove Proposition 1.2 and we present some other explicit applications to relevant examples of parabolic equations. Section 7 contains some final remarks and comments. The paper ends with an Appendix, where we prove an interesting technical lemma regarding power concave functions, which is useful to simplify the application of our theorems. 2 Preliminaries In this section we introduce some notation and give some preliminary results about quasi-concavity and parabolic quasi-concavity. 2.1 Basic notation We denote by, the classical Euclidean scalar product and by the Euclidean norm in R N, N 1. Throughout this paper, if not otherwise specified, Ω is an open convex sets and K a nonempty compact convex set such that K Ω R N, while D denotes the convex ring Ω \ K. (1.8)

4 Math. Nachr. 283, No. 11 (2010) / 1529 Given a function v of class C 2,1, the time derivative of v is denoted by t v, while the i subscript means differentiation with respect to the space variable x i,i.e. v i = xi v, v ij = xi xj v, while v =(v 1,...,v N ) and 2 v =(v ij ) N denote the (spatial) gradient and the (spatial) Hessian matrix of v, respectively. The set of real symmetric N N matrices is denoted by M N,andM + N is the subset of M N of positive semidefinite matrices. For A, B M N,byA 0, we mean A M + N, and by A B, we mean B A 0, i.e. B A M + N. The transpose of the matrix A is denoted by AT. An operator F :Ω R R N M N R is said to be proper if F (x, u, q, A) F (x, v, q, A) whenever u v. (2.1) Furthermore the operator F is said to be strictly proper if F (x, u, q, A) <F(x, v, q, A) whenever u>v. (2.2) Let Γ be a convex cone in M N containing M + N and with vertex at the origin. Then F is degenerate elliptic in Γ if F (x, v, q, A) F (x, v, q, B), for every A, B Γ such that A B. (2.3) F is elliptic if the above inequality is strict whenever A>B. We put Γ F = Γ, where the union is extended to every cone Γ such that F is elliptic in Γ; whenwe say that F is elliptic, we mean that F is elliptic in Γ F. IfF is an elliptic operator, we say that a function v C 2 (D (0,T)) is admissible for F if 2 v(x, t) Γ F for every x D and for every t (0,T); equivalently, if F is sufficiently regular, v is admissible if the N N matrix with entries F ij (x, v(x, t), v(x, t), 2 v(x, t)) = F (x, v(x, t), v(x, t), 2 v(x, t)) v ij is positive definite for every x D and every t (0,T). A function u C 2,1 (D (0, )) C(D [0, )\ D {0}) is a subsolution of (1.1) if it satisfies (pointwise) t γ(u) F (x, u, u, 2 u) in D (0, ), u(x, 0) ϕ(x) in Ω, u(x, t) 0 on Ω (0, ), u(x, t) 1 on K (0, ). Analogously, u is a supersolution if the same holds with all the inequality signs reversed. We say that the Comparison Principle holds for (1.1) if, whenever u and v are a subsolution and a supersolution, then u v in D [0, ). 2.2 Power-concave and quasi-concave functions Given a 0,a 1 > 0, λ (0, 1),andα [, ], the quantity [(1 λ)a α 0 + λaα 1 ]1/α for α, 0,, a 1 λ 0 a λ 1 for α =0, M α (a 0,a 1,λ)= max{a 0,a 1 } for α =, min{a 0,a 1 } for α = (2.4) is the α-(weighted) mean of a 0,a 1, with ratio λ (for more details we refer to [13]). For a 0,a 1 0, wedefine M α (a 0,a 1,λ) as above if α 0 and M α (a 0,a 1,λ)=0 if α<0 and a 0 a 1 =0. It is easily seen that, for every λ (0, 1), lim α M α(a 0,a 1,λ)=max{a 0,a 1 }, lim M α(a 0,a 1,λ)=min{a 0,a 1 }. α

5 1530 K. Ishige and P. Salani: Parabolic quasi-concavity Definition 2.1 Let α [, ]. A nonnegative function v is called α-concave if v((1 λ)x + λy) M α (v(x),v(y),λ) for every x, y sprt(v), λ (0, 1). We say that v is strictly α-concave, if the inequality sign is strict whenever x y. Roughly speaking, v 0 is called α-concave if it has convex support sprt(v) andinsprt(v) it holds: (i) v α is concave, if α>0; (ii) log v is concave, if α =0; (iii) v α is convex, if α<0. Quasi-concavity corresponds to the case α = and it is the weakest concavity property one can imagine. Usual concavity corresponds to α =1, while log-concavity clearly corresponds to α =0. A simple consequence of Jensen s inequality is that M α (a 0,a 1,λ) M β (a 0,a 1,λ) if α β. (2.5) Therefore α-concavity is monotone with respect to α, thatis:ifv is α-concave, then it is β-concave for every β α (and in particular for β = ). 2.3 Parabolically quasi-concave functions In [4], Borell introduces the notion of parabolic convexity for sets. Definition 2.2 A subset E of R N [0, ) is said parabolically convex if, for every (x 0,t 0 ), (x 1,t 1 ) E, and λ (0, 1), there holds (x λ,t λ ) E, where x λ =(1 λ)x 0 + λx 1 and t λ = ( (1 λ) t 0 + λ ) 2. t 1 Borell [4] proves that the heat balls of a parabolically convex domain P R N [0, ) (i.e. the superlevel sets of the Green function of the heat operator in P, with the homogeneous Dirichlet boundary condition) are parabolically convex, while it is unknown whether heat balls are convex when P is convex. This suggests that parabolic convexity may be more suitable than usual convexity when dealing with solutions to parabolic equations. Then we introduce in Definition 1.1 the notion of parabolic quasi-concavity for functions. Of course, as expected, a function u C(Ω [0, )) is parabolically quasi-concave if and only if its superlevel sets {(x, t) Ω [0, ) : u(x, t) l} are parabolically convex, in the sense of Definition 2.2. This parallels the analogous characterization of quasi-concave functions, depending only on the space variable. By the way, it is not hard to see that parabolic quasi-concavity of a function u is also equivalent to the quasiconcavity with respect to (x, t) of v(x, t) =u(x, t 2 ) or of v(x, t) =u(x/t, 1/t 2 ). Notice that Definition 1.1 can also be expressed in the following way: u C(Ω [0, )) is parabolically quasi-concave if u(m 1 (x 0,x 1,λ),M 1/2 (t 0,t 1,λ)) M (u(x 0,t 0 ),u(x 1,t 1 ),λ) (2.6) for every x 0,x 1 Ω, t 0,t 1 0, λ (0, 1). Of course, one can easily imagine how to define analogous notions of parabolic power (and logarithmic) concavity by substituting M with M p in the right-hand side of (2.6). Other possible variations can be obtained by suitably changing the means M 1 and/or M 1/2 in (2.6) (see [17]). 3 Mainresults Throughout the rest of the paper we assume the following condition: γ = γ(v) C 1 ([0, )) is a convex function such that γ(0) = 0 and γ(v) > 0 for v>0; F = F (x, v, p, A) C 1 (Ω R R N M N : R) is a proper degenerate elliptic operator and the Comparison Principle holds for problem (1.1).

6 Math. Nachr. 283, No. 11 (2010) / t t=(x+1) 2 t=(x 1) x Fig. 1 (online colour at ) An example of parabolically convex set: the parabolically convex envelope of the segments [ 1, 1] {0} and [ 2, 2] {1}. We remark that the first condition implies γ (v) > 0 for v>0. (3.1) Moreover, for any fixed v (0, ) and θ S N 1, we define the functions F v,θ (x, p, A) and F + v,θ (x, p, A) on Ω (0, ) Γ F as follows: F v,θ (x, p, A) =F (x, v, p 1 θ, p 3 A), F + v,θ (x, p, A) =max{f v,θ(x, p, A), 0}. 3.1 Parabolic quasi-concavity Now we are ready to state our main result regarding the minimum points of the parabolical quasi-concavity function Π defined by (1.6). Theorem 3.1 Let D be an open subset of R N and let u C 2 (D (0, )) be an admissible nonnegative solution of such that t γ(u) =F (x, u, u, 2 u) in D (0, ) (3.2) t γ(u) 0 in D (0, ), (3.3) u > 0 in D (0, ). (3.4)

7 1532 K. Ishige and P. Salani: Parabolic quasi-concavity Assume that and (F 1 ) (F 1 ) { the function F + v,θ is ( 1/2)-concave in Ω (0, ) Γ F for any v (0, ) and θ S N 1 at least one of the following conditions holds: (i) γ is strictly convex and t γ(u) > 0 in D (0, ); (ii) F is strictly proper; (iii) F + v,θ is strictly ( 1/2)-concave with respect to the x variable (in its support). Then, for any λ (0, 1), the parabolic quasi-concavity function Π associated to u cannot attain a negative local minimum at any point (x, y, r, s) D 2 (0, ) 2 such that (1 λ)x + λy D. As a consequence of the above theorem, we can prove parabolic quasi-concavity of solutions to problems (1.1), but only in the case of vanishing initial datum. What we have to do is just to give conditions which exclude that Π becomes negative on the boundary of D 2 (0, ). In particular, to avoid that Π become negative as t, we need to link the large time behavior of u to the solution of the associated stationary problem, whose quasi-concavity has been studied in [2]. Moreover it is useful to introduce the following condition: (F ) for every x 0 K there exists δ>0such that F (x, v, w,a) ρ 2 F (ρ(x x 0 )+x 0,v,ρ 1 w,ρ 2 A) for ρ (1 δ, 1] for every fixed (x, v, w,a) D [0, 1] R n Γ F. Now we can state our main result about parabolic quasi-concavity. Theorem 3.2 Let D = Ω \ K, letg 1 in [0, ) and let ϕ 0 in D. Assume that Ω is a bounded convex domain in R N and that K is a convex compact set with nonempty interior such that K Ω. Letu be a nonnegative solution of problem (1.1) such that u C 2+β,1+β/2 (D [T, )) < for some β (0, 1) and T>0. (3.5) Assume that (3.3), (F 1 ), (F 1 ) and (F ) hold; moreover, assume that, either the strict inequality sign holds in (3.3) or (3.4) holds. Then u is parabolically quasi-concave in Ω [0, ). We conjecture that Theorem 3.2 holds not only for vanishing initial datum, but for every quasi-concave ϕ, satisfying suitable compatibility conditions. We explain in Remark 4.7 the technical difficulty to obtain the full result. Theorems 3.1 and 3.2 are proved in Section Assumptions of Theorems 3.1 and 3.2 In this section we discuss in detail the assumptions of Theorems 3.1 and 3.2. Some of them can be indeed weakened, substituted by explicit assumptions on the boundary and initial data or even neglected in many cases. We first discuss assumption (3.3). We easily see that it is equivalent to t u 0 in D (0, ). (3.6) Indeed, clearly t u(x, t) =0if u(x, t) =0,sinceu 0, while (3.6) is equivalent to (3.3) when u>0, thanks to (3.1). It is then worth to notice that, in many cases, for the problem (1.1), (3.6) can be easily obtained if the initial datum satisfies F (x, ϕ(x), ϕ(x), 2 ϕ(x)) 0. (3.7)

8 Math. Nachr. 283, No. 11 (2010) / 1533 Indeed, consider for instance the case γ(u) =u: if the operator F is regular enough and the boundary and initial data satisfy suitable compatibility conditions, the time derivative v = t u solves the following parabolic Dirichlet-Cauchy problem t v = F ij (x, u, u, 2 u)v ij + F ui v i + F u v in D (0, ), v(x, 0) = F (x, ϕ(x), ϕ(x), 2 ϕ(x)) in D, (3.8) v(x, t) =0 on Ω (0, ), v(x, t) =g (t) on K (0, ). Then (3.6) easily follows from the maximum principle and (3.7). Furthermore, if v(x, 0) 0 in D (or g (t) > 0 near t =0), by applying the strong maximum principle to the problem (3.8), we have t u>0, (x, t) D (0, ). (3.9) See also [8]. Regarding (3.4), we notice that (F ) yields the parabolic starshapedness of superlevel sets of u, see Lemma 4.4, which in turn implies (3.4), jointly with (3.9), see Remark 4.7. Then (3.4) can be reduced in Theorem 3.2, if we have a strict inequality sign in (3.3). About the possibility to neglect (3.4), see also Section 2 in [8] and [9], where the author proved a minimum principle for the starshapedness of the level sets of solutions to problem (1.1), which obviously implies u > 0. We also notice that (F ) can be in general removed thanks to (F 1 ), see the argument in the proof of Proposition 1.2, which could be generalized with some technical effort. Next we discuss assumption (F 1). If{u n } is a sequence of parabolically quasi-concave functions in the convex cylinder Ω (0, ), converging pointwise to the upper semicontinuous function u,thenu is also a parabolically quasi-concave function in Ω (0, ). Then parabolic quasi-concavity of solutions can be obtained by the use of approximate solutions, without the assumption (F 1). In fact, for any n =1, 2,...,letu n be a solution of the problem (1.1) with γ replaced by γ n (v) =γ(v)+n 1 v 2 or with the operator F replaced by F n (x, u, p, A) =F (x, u, p, A)+n 1 u or F n (x, u, p, A) =F (x, u, p, A)+n 1 f(x), where f(x) 0 is a strictly concave function in D, sothatγ n and F n satisfy (F 1 ). Then, if Theorem 3.1 (or Theorem 3.2) is applicable to the solutions u n and {u n } converges pointwise to u as n, we can prove the same conclusion as in Theorem 3.1 (or Theorem 3.2), without assumption (F 1 ). This consideration holds true in many relevant examples (see Proposition 1.2 for instance). Finally, we could say that the most important assumption of our theorems is the structure condition (F 1 ).We notice here that, by the properties of power concave functions recalled in Section 2.2, (F 1 ) is satisfied if F + v,θ is α-concave for some α 1/2, in particular, if it is concave or log-concave. Moreover, (F 1 ) could be also weakened, arguing as in Theorem 3.10 of [2] (see Remark 4.2). 3.3 Spatial quasi-concavity Arguing in a way similar to the one that takes us to the results of the previous section, we can state two theorems: one is related to the minimum points of the spatial quasi-concavity function Φ, defined by (1.3), and the other is related to spatial quasi-concavity of solutions of (1.1). Theorem 3.3 Let D be an open subset of R N and u C 2 (D (0, )) be an admissible nonnegative solution of (3.2), satisfying (3.3) and (3.4). Assume the following: (F 2 ) { the function (x, p, A) p F + v,θ (x, p, A) is concave in Ω (0, ) Γ F for any v (0, ) and θ S N 1 ;

9 1534 K. Ishige and P. Salani: Parabolic quasi-concavity and (F 2) at least one of the following conditions holds: (i) γ is strictly convex and t γ(u) > 0 in D (0, ); (ii) F is strictly proper; (iii) p F + v,θ is strictly concave with respect to the x variable (in its support). Then, for any λ (0, 1), the spatial quasi-concavity function Φ associated to u cannot attain a negative local minimum at any point (x, y, t) D 2 (0, ) such that (1 λ)x + λy D. As for the case of spatial quasi-concavity, the way from Theorem 3.3 to the spatial quasi-concavity of solutions to problem (1.1) is not very long. We have only to give conditions which exclude that Φ becomes negative on the boundary of D 2 (0, ). To this aim, we assume that (B) (i) g C 1 ([0, )), g(0) 0, g 0 and g = lim t g(t) < ; (ii) ϕ is a nonnegative, quasi-concave, continuous function in Ω such that ϕ =0on Ω and ϕ g(0) on K. As a consequence of Theorem 3.3, we can now state the following. Theorem 3.4 Let D =Ω\ K, whereω is a bounded convex domain in R N and K is a convex compact set with nonempty interior such that K Ω. Letu be a solution of problem (1.1) satisfying (3.5) and assume (B) and all the assumptions of Theorem 3.3 hold; then u(,t) is quasi-concave in Ω for every t 0. Theorems 3.3 and 3.4 will be proved in Section 4. Notice that analogous considerations as in Section 3.2 about the assumptions of Theorems 3.3 and 3.4 can be obviously done. The spatial quasi-concavity of solutions to problem (1.1) has been already considered in [8, Theorem 3]. Unfortunately, the proof of such theorem seems to be incomplete. Here we revisit and slightly improve the arguments in [8], obtaining Theorem 3.3. We must notice that Theorem 3.3 is valid for a general class of operators F, but it does not apply to the heat equation or to the operators considered in [8]. Then, while we conjecture that the claim of [8, Theorem 3] is true, it is very likely still unproved. See Section 7 for further details and comments. 4 Proof of Theorems 3.1 and 3.2 For any v [0,g ] and θ S N 1, we put H v,θ (x, t, p, A) =tf v,θ (x, p, A) ( = tf(x, v, p 1 θ, p 3 A) ), H + v,θ (x, p, A) =max{h v,θ(x, t, p, A), 0} = tf + v,θ (x, p, A), for any x Ω, t [0, ), p (0, ),anda Γ F. We first give the following lemma on the condition (F 1 ), which is a direct consequence of Lemma A.1 in Appendix. Lemma 4.1 The condition (F 1 ) is equivalent to the condition (H) { the function H + v,θ (x, t, p, D) is ( 1)-concave in Ω [0, ) (0, ) Γ F for any v (0, ) and θ S N 1. P r o o f. Applying Lemma A.1 with α = 1 and g(x, p, A) =F + v,θ (x, p, A) 1, we can easily obtain Lemma 4.1 (see also the definition of the power-concavity given in Section 1 and Section 2.2). Now we are ready to prove Theorem 3.1.

10 Math. Nachr. 283, No. 11 (2010) / 1535 Proof of Theorem 3.1. Letλ (0, 1) and assume that (x, y, r, s) is a local negative minimum point for Π in {(x, y, r, s) D 2 (0, ) 2 :(1 λ)x+λy D}, that is, there exist a neighborhood I = I x I y I r I s D 2 (0, ) 2 of (x, y, r, s) such that (1 λ)i x + λi y D and m := Π(x, y, r, s) =u(z, t ) min{u(x, r),u(y, s)} =infπ(x, y, r, s) < 0, (4.1) I where z =(1 λ)x + λy and t = ((1 λ) r + λ s ) 2.Thenwehave u(x, r) =u(y, s). (4.2) Indeed, assume by contradiction that u(x, r) <u(y, s); then there exists a neighborhood U I y of y such that u(x, r) <u(y, s) for y U, and we have Then (4.1) implies Π(x, y, r, s) =u((1 λ)x + λy, t ) u(x, r), y U. 0=( y Π)(x, y, r, s) =λ( u)(z,t ), which contradicts (3.4); hence we have u(x, r) u(y, s). Similarly we see u(x, r) u(y, s), and obtain (4.2). Next we consider the following minimum problem: minimize in I the function subject to Π(x, y, r, s) =u ( (1 λ)x + λy, ((1 λ) r + λ s) 2) u(x, r) u(x, r) =u(y, s). By (4.1) and (4.2), the minimum is attained at (x, y, r, s), and by the Lagrange multiplier theorem, it should be a free critical point of the Lagrangian function Hence we obtain w(x, y, r, s) = Π(x, y, r, s)+μ(u(x, r) u(y, s)). u(z, t )= 1 μ 1 λ u(x, r) = μ u(y, s), (4.3) λ so that either u(z,t ), u(x, r) and u(y, s) are all null or they are parallel. The first possibility is avoided by (3.4) and it is not hard to see that they must be also equally oriented (i.e. μ (0, 1)), otherwise we can move (x, r) and (y, s) so that u(x, r) and u(y, s) both increase, maintaining (z,t ) fixed, which contradicts the optimality of (x, y, r, s) for the function Π. Then we can set θ := Furthermore, by setting the equality (4.3) gives u(x, r) u(y, s) = u(x, r) u(y, s) = u(z, t ) u(z, t ). a 1 = u(x, r), b 1 = u(y, s), and c 1 = u(z, t ), c =(1 λ)a + λb. (4.4) Similarly we put A 1 = t u(x, r), B 1 = t u(y, s) and C 1 = t u(z,t ),

11 1536 K. Ishige and P. Salani: Parabolic quasi-concavity and obtain C = [ (1 λ) r + λ ] ( 1 λ s A + λ ) B, (4.5) r s if t u(x, r), t u(y, s),and t u(z, t ) do not vanish (simultaneously). Now let us consider the function u (z) =max { min{u(x, r),u(y, s)} : x Ω, y Ω, (1 λ)x + λy = z }, which corresponds to the function whose level sets are the λ-linear combination of the corresponding level sets of the functions v 0 (x) =u(x, r) and v 1 (y) =u(y, s). Then, by (4.1), the point z is also a minimum point for the function u(z,t ) u (z), hence, by [2] and [27], there holds 2 u(z, t ) u(z, t ) 3 (1 λ) 2 u(x, t ) u(x, t ) 3 + λ 2 u(y, s) u(y, s) 3. (4.6) We assume the condition (F 1)-(i), and continue the proof of Theorem 3.1. We put l = u(z, r), = u(x, t )= u(y, s), D 0 = 2 u(x, r), D 1 = 2 u(y, s), andd λ = 2 u(z, t ). Thenwehave 0 < rγ ( )A 1 = r t γ(u(x, r)) = rf(x,, a 1 θ, D 0 ) = H +,θ (x, r, a, a 3 D 0 ), 0 < sγ ( )B 1 = s t γ(u(y, s)) = sf (y,, b 1 θ, D 1 ) = H +,θ (y, s, b, b 3 D 1 ). (4.7) By (H) and (4.7), we have H +,θ ((1 λ)x + λy, (1 λ) r + λ s, λa +(1 λ)b, (1 λ)a 3 D 1 + λb 3 D 1 ) M 1 (H +,θ (x, r, a, a 3 D 0 ), H +,θ (y, ) s, b, b 3 D 1 ),λ > 0, and obtain ( F,θ (1 λ)x + λy, λa +(1 λ)b, (1 λ)a 3 D 1 + λb 3 ) D 1 > 0, (4.8) ((1 λ)x + λy, (1 λ) r + λ s, λa +(1 λ)b, (1 λ)a 3 D 1 + λb 3 D 1 ) 1 H +,θ (1 λ)h +,θ (x, r, a, a 3 D 0 ) 1 + λh +,θ (y, s, b, b 3 D 1 ) 1. (4.9) Furthermore, by (4.1), l<, and by (F 1)-(i), (1.1), (4.6), and (4.8), we have γ ( )C 1 >γ (l)c 1 = t γ(u(z, t )) = F (z,l,c 1 θ, D λ ) F ( z,, c 1 θ, c 3 ((1 λ)a 3 D 0 + λb 3 D 1 ) ) ( = F,θ (1 λ)x + λy,λa +(1 λ)b, (1 λ)a 3 D 1 + λb 3 ) D 1 > 0. This together with (4.5) and (4.7) implies

12 Math. Nachr. 283, No. 11 (2010) / 1537 H +,θ ((1 λ)x + λy, (1 λ) r + λ s, λa +(1 λ)b, (1 λ)a 3 D 1 + λb 3 D 1 ) 1 [ > (1 λ) r + λ ] 1 s γ ( ) 1 C = 1 λ γ ( ) 1 A + λ γ ( ) 1 B r s =(1 λ)h +,θ (x, r, a, a 3 D 0 ) 1 + λh +,θ (y, s, b, b 3 D 1 ) 1. This contradicts (4.9). Similarly we have a contradiction if there holds either (F 1 )-(ii) or (iii); thus the proof of Theorem 3.1 is complete. Remark 4.2 Inequality (4.6) can be substituted by an exact relation between the Hessian matrices 2 u(z, t ), 2 u(x, t ) and 2 u(y, t ), according to [2] and [27], in order to obtain a sharper theorem, where assumption (F 1 ) is replaced by a weaker one involving F ( x, t, θ/p, JBJ T /p ), like in Theorem 3.10 of [2]. On the other hand, such a theorem would be more involved and less immediate to apply than Theorem 3.1, hence we are not going to write here the explicit statement and we trust the interested reader can obtain it by following Section 3.2 of [2]. Next we prove Theorem 3.2. For this aim, we first prove the following lemmas. Lemma 4.3 Assume the same conditions as in Theorem 3.2. Then there exists a classical solution ū of such that F (x, u, u, 2 u)=0 in D, u =1 on K, u =0 on Ω (4.10) lim u(x, t) =ū(x), x Ω. (4.11) t P r o o f. By the comparison principle, we have 0 u(x, t) 1 (4.12) for all (x, t) D (0, ). Then, by (3.3), (3.6), and (4.12), for any fixed x Ω, u(x, t) is a bounded and monotone increasing function in the variable t, and there exists a function u on Ω such that lim u(x, t) =u (x), x Ω. t Let {t j } j=1 be such that t 1 > 0 and t j+1 >t j +2for j =1, 2,... Put u j (x, t) =u(x, t j + t) and Q = D ( 1, 1). By (3.5), the Ascoli-Arzelà Theorem, and the diagonal argument, taking a subsequence if necessary, we see that there exists a function ū =ū(x, t) C 2+β,1+β/2 (Q) such that lim u j ū C j 2+β,1+β/2 (R) =0 for any compact set R Q. Thenū(x, t) =u (x) for any (x, t) Q,andū is independent of the time variable t. Therefore, by (1.1), we see that ū is a classical solution of F (x, u, u, 2 u)=0 in D, ū =1 on K, ū =0 on Ω, and the proof of Lemma 4.3 is complete. Lemma 4.4 Assume that the origin 0 of R N belongs to K and let Ω and K be starshaped sets (with respect to 0). Let u be a solution of (1.1) with g 1 in (0, ) and assume that the initial datum ϕ has starshaped superlevel sets, i.e. ϕ(ρx) ϕ(x) if ρ 1, x Ω. Moreover, assume that there exists ɛ>0suchthat, for every fixed (x, v, w,a) Ω (0, 1) R N Γ F, it holds ρ 2 F (ρx, v, wρ, Aρ ) 2 F (x, v, w,a) for ρ (1 ɛ, 1). (4.13) Then it holds u(ρx, ρ 2 t) u(x, t) for 0 ρ 1, t > 0, x Ω. (4.14)

13 1538 K. Ishige and P. Salani: Parabolic quasi-concavity P r o o f. Following a similar argument as in [8], we consider the function v(x, t) =u(ρx, ρ 2 t), which satisfies t γ(v) =ρ 2 F (ρx,v,ρ 1 v, ρ 2 2 v) in ρ 1 D (0, ), v(x, 0) = ϕ(ρx) in ρ 1 Ω, v(x, t) =0 on ρ 1 Ω (0, ), v(x, t) =1 in ρ 1 K (0, ). Then, by applying the comparison principle between u and v in Ω \ (ρ 1 K) (0, ), wehave This yields (4.14). u(ρx, ρ 2 t) u(x, t) for ρ (1 ɛ, 1), t > 0, x Ω. Remark 4.5 When u satisfies (4.14), we say that u has parabolically starshaped level sets. Notice that (4.14) implies u(x, t),x +2tu t (x, t) 0. (4.15) As a consequence, if the strict sign holds in (3.3), then (3.4) is easily obtained whenever u>0, and hence it can be neglected in the assumptions of Theorem 3.2. Remark 4.6 Notice that if F is differentiable and we set g(ρ) =ρ 2 F (ρx,v,ρ 1 w,ρ 2 A), then (4.13) implies g (1) = 2F (x, v, w,a)+ x F (x, v, w,b),x w F (x, v, w,a),w 2F ij (x, v, w,a)a ij 0, (4.16) for every fixed (x, v, w,a) Ω (0, 1) R n Γ F,whereA =(a ij ) N. Vice versa, a strict inequality sign in (4.16) gives (4.13). We notice that (F ) coincides with (4.16) when the origin is in x 0 and that (F ) could be obtained, in general, from (F 1 ) with an argument similar to the one we will use in the proof of Proposition 1.2 for the special case F (x, v, w,a)=trace (A) f(x, v). We are ready to prove Theorem 3.2. Proof of Theorem 3.2. Itsufficestoprove { inf Π(x, y, r, s) :(x, y, r, s) D 2 [0, ) 2} 0 for any λ (0, 1). Let us argue by contradiction and assume that { m := inf Π(x, y, r, s) :(x, y, r, s) D 2 [0, ) 2} < 0 for some λ (0, 1) and let {(x j,y j,r j,s j )} j=1 be a minimizing sequence, i.e. lim j Π(x j,y j,r j,s j )= m. (4.17) By the compactness of Ω, we can obviously assume lim j x j = x and lim j y j = y with x, y Ω. First, let us consider the case Π approaches its infimum as time tends to infinity, that is assume that r j s j and lim j s j =. On the other hand, by [2], ū is quasi-concave in Ω, and by (3.3), (3.6), and (4.11), we see that ū(x) u(x, t), (x, t) Ω (0, ). Then we have

14 Math. Nachr. 283, No. 11 (2010) / 1539 lim Π(x j,y j,r j,s j ) lim [u((1 λ)x j + λy j,t j ) min{ū(x j ), ū(y j )}] j j =ū((1 λ)x + λy) min{ū(x), ū(y)} 0, where t j = ((1 λ) r j + λ s j ) 2. This contradicts (4.17). Therefore r j and s j must be bounded and there exist r, s [0, ) such that lim j r j = r and lim j s j = s. Since the case r =0and x K (or s =0 and y K ) is avoided thanks to Lemma 4.4, by the regularity of u,wehave Π(x, y, r, s) =u(z, t ) min{u(x, r),u(y, s)} = m <0, where z =(1 λ)x + λy and t = ((1 λ) r + λ s ) 2.Ifmin{r, s} =0, min{u(x, r),u(y, s)} =0,since ϕ 0 in D, andπ(x, y, r, s) 0. Thenwehave r>0, s>0. (4.18) Furthermore we easily see the following: (i) (ii) since u =0on Ω, by (3.3), Π(x, y, r, s) 0 if x Ω or y Ω (then we have x, y Ω); by the convexity of K, ifx K and y K, thenz K and Φ(x, y, r, s) =0(then we have x K or y K). Finally, if x K and y K, thenu(y, s) <u(x, r) =1, and we can exclude this case again due to (3.4). Therefore we have x, y K, and obtain (x, y, r, s) D 2 (0, ) 2, z D. This is excluded by Theorem 3.1, Lemma 4.4, and Remark 4.5. Thus we have a contradiction, and the proof of Theorem 3.2 is complete. Remark 4.7 In the case of a general (not identically vanishing) initial datum, we were not able to prove (4.18). Once one is able to prove (4.18) for the solution u of (1.1) with some initial datum ϕ, the proof of Theorem 3.2 remains true and the solution u is parabolicallyquasi-concavein Ω [0, ). 5 Proof of Theorems 3.3 and 3.4 Proof of Theorem 3.3. Let λ (0, 1) and assume that (x, y, t ) D 2 (0, ) is a local negative minimum point for Φ in {(x, y, t) D 2 (0, ) :(1 λ)x + λy D}, that is, there exist a neighborhood I = I x I y I t D 2 (0, ) of (x, y, t ) such that (1 λ)i x + λi y D and m := Φ(x, y, t )=u(z, t ) min{u(x, t ),u(y, t )} =infφ(x, y, t) < 0, (5.1) I where z =(1 λ)x + λy. First we prove u(x, t )=u(y, t ). (5.2) In order to prove (5.2) by contradiction, we assume u(x, t ) <u(y, t ). Then there exists a neighborhood U I y of y such that u(x, t ) <u(y, t ) for y U, and we have Then (5.1) implies Φ(x, y, t) =u((1 λ)x + λy, t ) u(x, t ), y U. 0=( y Φ)(x, y, t )=λ( u)(z,t ), which contradicts (3.4); hence we have u(x, t ) u(y, t ). Similarly we see u(x, t ) u(y, t ), and obtain (5.2).

15 1540 K. Ishige and P. Salani: Parabolic quasi-concavity Next we consider the following minimum problem: minimize in I the function subject to the constraint Φ(x, y, t) =u((1 λ)x + λy, t) u(x, t) u(x, t) =u(y, t). By (5.1) and (5.2), the minimum is attained at (x, y, t ), and by the Lagrange multiplier theorem, (x, y, t ) should be a free critical point of the Lagrangian function Hence we obtain w(x, y, t) = Φ(x, y, t)+μ(u(x, t) u(y, t)). u(z, t )= 1 μ 1 λ u(x, t )= μ u(y, t ), (5.3) λ so that either u(z, t ), u(x, t ) and u(y, t ) are all null or they are parallel. The first possibility is avoided by (3.3) and it is not hard to see that they must be also equally oriented (i.e. μ (0, 1)), otherwise we can move (x, t ) and (y, t ) so that u(x, t ) and u(y, t ) both increase, maintaining (z,t ) fixed, which contradicts the optimality of (x, y, t ) for the function v. Then we can set θ := u(x, t ) u(x, t ) = u(y, t ) u(y, t ) = u(z, t ) u(z, t ). Furthermore, by setting equality (5.3) gives a 1 = u(x, t ), b 1 = u(y, t ), and c 1 = u(z, t ), c =(1 λ)a + λb. (5.4) In the same way as above, we obtain that C =(1 μ)a + μb =(1 λ) aa c + λbb c, (5.5) where A = t u(x, t ), B = t u(y, t ) and C = t u(z, t ). Now let us consider the function u : Ω [0, ) R defined by ū (z) =max { min{u(x, t ),u(y, t )} : x D, y D, (1 λ)x + λy = z }, which corresponds to the function whose level sets are the λ-linear combination of the corresponding level sets of the functions v 0 (x) =u(x, t ) and v 1 (y) =u(y, t ). Then, by (5.1), the point z is also a minimum point for the function u(z,t ) u (z), hence 2 u(z, t ) 2 u (z) 0. On the other hand, we have (see [2], [27]) that is, 2 u(z, t ) u(z, t ) 3 (1 λ) 2 u(x, t ) u(x, t ) 3 + λ 2 u(y, t ) u(y, t ) 3, (5.6) c 3 2 u(z,t ) (1 λ)a 3 2 u(x, t )+λb 3 2 u(y, t ). (5.7)

16 Math. Nachr. 283, No. 11 (2010) / 1541 We assume the condition (F 1)-(i), and continue the proof of Theorem 3.4. Put l = u(z, t ), = u(x, t )= u(y, t ), D 0 = 2 u(x, t ), D 1 = 2 u(y, t ),andd λ = 2 u(z, t ), for simplicity. Then, by (F 1 )-(i), we have 0 <γ ( )A = t γ(u(x, t )) = F (x,, a 1 θ, D 0 )=F +,θ (x, a, a3 D 0 ), 0 <γ ( )B = t γ(u(y, t )) = F (y,, b 1 θ, D 1 )=F +,θ (y, b, b3 D 1 ). Furthermore, by (F 2 ) and (5.8), we have (λa +(1 λ)b)f + (,θ (1 λ)x + λy, (1 λ)a + λb, (1 λ)a 3 D 1 + λb 3 ) D 1 (1 λ)f +,θ (x, a, a3 D 0 )+λf +,θ (y, b, b3 D 1 ) > 0. (5.8) (5.9) On the other hand, by (5.1), l<, and by (F 1 )-(i), (1.1), and (5.7), we have γ ( )C >γ (l)c = t u(z, t ) = F (z, l, c 1 θ, D λ ) F ( z,, c 1 θ, c 3 ((1 λ)a 3 D 0 + λb 3 D 1 ) ) ( = F,θ (1 λ)x + λy, (1 λ)a + λb, (1 λ)a 3 D 1 + λb 3 ) D 1. Therefore, by (5.5)and (5.8) (5.10), we have (1 λ)af +,θ (x, a, a3 D 0 )+λbf +,θ (y, b, b3 D 1 ) = γ ( )cc ( (1 λ)x + λy, (1 λ)a + λb, (1 λ)a 3 D 1 + λb 3 ) D 1 >cf +,θ > 0. (5.10) This contradicts (F 2 ). Similarly we have a contradiction if there holds either (F 1 )-(ii) or (iii), and the proof of Theorem 3.3 is complete. Now we are ready to prove Theorem 3.4. Proof of Theorem 3.4. Itsufficestoprove { } inf Φ(x, y, t) :(x, y, t) Ω 2 [0, ) 0 for any λ (0, 1). Let us argue by contradiction and assume that { } m := inf Φ(x, y, t) :(x, y, t) Ω 2 [0, ) < 0 for some λ (0, 1). Then, thanks to Lemma 4.3, by the similar argument as in the proof of Theorem 3.2, we see that the infimum is attained at some point (x, y, t ) Ω Ω [0, ),thatis, Φ(x, y, t )=u(z, t ) min{u(x, t ),u(y, t )} = m <0, where z =(1 λ)x + λy. Then we easily see the following: (i) by the condition (B), the initial datum ϕ(x) is quasi-concavein Ω and Φ(x, y, 0) 0, hence we have t>0; (ii) since u =0on Ω, by (3.3), Φ(x, y, t) 0 if x Ω or y Ω, hence we have x, y Ω; (iii) by the convexity of K,ifx K and y K,thenz K and Φ(x, y, t) =0, hence x K or y K. Furthermore, if x K and y K, thenu(y, t ) <u(x, t )=g(t), and this is avoided thanks to (3.4). Therefore we have x, y K, and obtain (x, y, t ) D 2 (0, ) with (1 λ)x + λy D. This is excluded by Theorem 3.3, and the proof of Theorem 3.4 is complete.

17 1542 K. Ishige and P. Salani: Parabolic quasi-concavity 6 Selected examples The results of the previous sections can be applied to a very large class of parabolic equations. In this section we prove Proposition 1.2 and Corollary 1.3, regarding the heat equation with absorption, and we analyze some other relevant cases. 6.1 Proofs of Proposition 1.2 and Corollary 1.3 Proof of Proposition 1.2. Let n =1, 2,... Then, under the condition (1.9), there exists a classical solution u n of t u =Δu f n (x, u) in D (0, ), u(x, 0) = 0 in Ω, (6.1) u(x, t) =0 on Ω (0, ), u(x, t) 1 on K (0, ), where f n (x, u) =f(x, u)+n 1 u. (See for instance [24].) By the comparison principle and (1.9), we have 0 u n (x, t) u n+1 (x, t) 1, (x, t) D (0, ). (6.2) Then, by [24, Chapters III and IV], for any compact set R D [0, ) \ Ω {0},wehave sup u n C 2+β,1+β/2 (R) < for some β (0, 1). (6.3) n On the other hand, by (1.9), we apply similar arguments as in Section 3.2 to see that t u n > 0 in D (0, ). (6.4) By (1.9), the operator F (x, v, q, A) =tracea f n (x, v) is strictly proper and by (1.10), it satisfies the condition (F 1 )(see Section 6.2, use condition (F 3 ) and Lemma A.1). Furthermore, it is easily seen that (1.10) implies (4.13) in this case. Indeed, for any fixed value v (0, 1),leth(x) =fn 1/2 (x, v); then h(x) = f n(x, v) 2fn 3/2 (x, v), and, by a classical characterization of differentiable convex function (see for instance [29, Theorem (b)]), we have h(x 0 ) h(x)+ h(x),x 0 x for every x, x 0 Ω. The latter, thanks also to (1.9) which gives f n (x, v) > 0 in Ω, implies 2f n (x, v)+ f n (x, v),x x 0 > 0, x D, for every x 0 K, which coincides with (4.16) with the strict inequality for the case at hand. Then, by Lemmas 4.4 and Remark 4.6, we have x, u n +2t t u n 0 in D (0, ), and by (6.4), we obtain u n > 0 in D (0, ). Therefore, by Theorem 3.2, we see that u n is parabolically quasi-concave in Ω [0, ). On the other hand, by (6.2) and (6.3), we apply the Ascoli-Arzelá theorem to {u n }, and see that the limit function ũ(x, t) = lim n u n (x, t) is a solution of (1.8). Then, by the uniqueness property for parabolic equations (see [24]) and (6.2), we have ũ(x, t) =u(x, t) in Ω [0, ). Therefore, since u n is parabolically quasi-concave in Ω [0, ), u is also parabolically quasi-concave in Ω [0, ); thus the proof of Proposition 1.2 is complete.

18 Math. Nachr. 283, No. 11 (2010) / 1543 Proof of Corollary 1.3. Letn =1, 2,... and let K n be a compact convex set such that K n B(0,n)=K B(0,n), K n B(0,n+1), K n K n+1, K n C 2,α. Let u 0 0 in B(0,n+1)and let u n be a solution of (1.8) with Ω and K replaced by B(0,n+1)and K n, respectively. By the comparison principle, we have 0 <u n (x, t) u n+1 (x, t) 1, (x, t) B(0,n+1) (0, ). Then, by the same argument as in the proof of Proposition 1.2, there exists a solution ũ of (1.11) such that ũ(x, t) = lim n u n(x, t), (x, t) R N (0, ). Furthermore we see that ũ is parabolically quasi-concave in R N (0, ). Put w = u ũ. Thenw is a bounded solution of t w =Δw a(x, t)w in D (0, ) such that w(x, t) =0on D (0, ) and w(x, 0) = 0 in D, where f(x, u) f(x, ũ) if u ũ, a(x, t) = u ũ f u (x, u) if u =ũ. Since a L (D (0, )), by the uniqueness property for parabolic equations with the homogeneous Dirichlet boundary condition (see, for example, [14] and [28]), we have w 0 in D [0, ), thatis,u(x, t) =ũ(x, t) in R N [0, ); thus u is parabolically quasi-concave in R N [0, ), and the proof of Corollary 1.3 is complete. 6.2 Other examples Interesting generalizations of (1.8) are for instance obtained by replacing the Laplace operator with the q-laplacian (q >1), the mean curvature operator, or a Finsler Laplacian. Let us write explicitly what condition (F 1 ) becomes in these cases. We first consider the case of the q-laplacian. Let F (x, u, u, 2 u)=δ q u f(x, u, u), where Δ q u = div( u q 2 u) and q>1. Then the following condition (F 3 ) the function (x, p) p q+1 f(x, u, θ/p) is convex for every u (0, 1), θ S N 1 implies (F 1 ) if q 2. Indeed, since Δ q u = u q 2 ( Δu +(q 2) u 2 u u T u 2 ), we have F v,θ (x, p, A) = 1 p q+1 ( trace(a)+θaθ T ) f(x, v, θ/p). Let Λ= { (x, p, A) Ω (0, ) M N : trace(a)+θ T Aθ p q+1 f(x, v, θ/p) > 0 }.Then(F 1 ) requires the function p (q+1)/2 if (x, p, A) Λ, (x, p, A) trace(a)+θt Aθ p q+1 f(x, v, θ/p) + otherwise

19 1544 K. Ishige and P. Salani: Parabolic quasi-concavity to be convex. By Lemma A.1 with α =(q +1)/2, this happens if and only if the function G(x, A) =trace(a)+θ T Aθ p q+1 f(x, v, θ/p) is 1/(q 1)-concave in Λ. Since trace(a) +θ T Aθ is linear, if (F 3 ) holds, G is concave in Λ, sothatitis α-concave for every α 1, in particular, for α =(q 1) 1,ifq 2. For the possibility to substitute (3.3) with an explicit assumption on f, we refer again to [8] and [9]. Next we consider the case of the mean curvature operator. Let F (x, u, u, 2 u)=mu f(x, u, u), where ( ) Mu =(1+ u 2 ) 3/2 u div =(1+ u 2 )Δu u i u ij u j. 1+ u 2 Then, following [6, Section 4.2.2], we can see that the condition (F 3 ) with q 2 implies (F 1 ). Next we let F (x, u, u, 2 u)=δ H u f(x, u, u), where H is a norm on R N and Δ H u is the Finsler Laplacian associated to H,thatis Δ H u = div(h( u) ξ H( u)), where H is a positively homogeneous (of degree one) nonnegative convex function, which vanishes only at ξ =0. In this case the equation t u = F (x, u, u, 2 u) models the anisotropic diffusion dictated by the norm H. Then we can write Δ H u = ( ) u V ξiξ j u i,j, u ij where V (ξ) = 1 2 H2 (ξ), whichisa2-homogeneous convex function, and the condition (F 3 ) with q 2 can take the place of (F 1 ) again. Other examples can be obtained with Pucci extremal operators or any other operator F which can be treated with the methods of [2] (as the ones considered above). 7 Final remarks and comments In [8], Diaz and Kawohl investigated the starshapedness and spatial convexity of level sets of solutions to the following problem t γ(u) =Δ q u f(u) in (0, ) D, u =0 on (0, ) Ω, (7.1) u =1 on (0, ) K, u(x, 0) = ϕ(x) in Ω, where γ is continuous convex and nondecreasing, f is continuous and nondecreasing, γ(0) = f(0) = 0, q>1, D =Ω\ K a convex ring as usual. In particular, in [8, Theorem 3] they state that (under some reasonable regularity assumptions on the domain and u) if the initial datum ϕ is quasi-concave and Δ q ϕ 0, then the solution u is spatially quasi-concave for every t > 0. That would coincide with our Theorem 3.4 when F (x, u, u, 2 u)=δ q u f(u). Unfortunately their proof seems to contain a gap at the very last step. Indeed, at page 279 in [8], the authors proved the inequality 1 a q > [ α b q + 1 α c q ], (7.2)

20 Math. Nachr. 283, No. 11 (2010) / 1545 where α (0, 1),anda, b,andcare positive constants such that 1 a = 1 ( 1 2 b + 1 ). c Then they claimed that the inequality (7.2) contradicts the convexity of the map d d q.ifα =1/2, this claim would be true; however α is not necessarily equal to 1/2 in their argument. We remark that the proof of our Theorem 3.4 essentially follows the same scheme of the proof of [8, Theorem 3], up to the last step, where we had to recover the above explained gap. However, we regret to notice that Theorem 3.4 does not apply to problem (7.1), even when q =2and f 0. Indeed, assumption (F 2 ) would require (p, A) p 2 trace(a) to be convex, which is not satisfied. Although we conjecture that the statement of [8, Theorem 3] is actually true, we point out that the spatial quasi-concavity of solutions to (7.1) with quasiconcave initial datum is still an open problem, even in the case of the heat equation. We also remark here that some results strongly related to Proposition 1.2 are contained in [3] and in [7], an unpublished earlier version of [8], which we received during the completion of this paper from the authors through a personal communication. In both of these papers, the authors study the quasi-concavity with respect to (x, t) of solutions to parabolic problems, that is equivalent to the nonnegativity of the quasi-concavity function Υ(x, y, r, s) =u((1 λ)x + λy, (1 λ)r + λs) min{u(x, r),u(y, s)}. In [3] Borell considers problem (1.8) with f 0, that is the case of heat equation. In [7], the authors prove Υ 0 when u solves (7.1) with q =2and ϕ 0. We notice that, due to (2.5) and to u t 0, we trivially have Υ(x, y, r, s) Π(x, y, r, s) for every (x, y, r, s). Therefore parabolic quasi-concavity is a stronger property than the usual quasi-concavity and the results of both papers [3] and [7] are recovered by Theorem 3.2. A Appendix Here follows an interesting lemma regarding power concave functions. Lemma A.1 Let α 1, β =(1 α) 1, m 1. LetG : R m (0, ) R be defined by G(x, t) =t α g(x), where g C 2 (R m ) is a positive function. Then (i) in the case α 0, G is convex if and only if g β is convex; (ii) in the case 0 <α<1, G is concave if and only if g β is concave; (iii) in the case α>1, G is convex if and only if g β is concave. Proof. Set f(x) =g(x) β. For clarity of our proof, we will treat separately and first the one-dimensional case m =1. A straightforward calculation gives ( ) α(α 1)t 2 G = α 2 g(x) αt α 1 g (x) αt α 1 g (x) t α g (x) and f (x) =βg(x) β 2 [g(x)g (x)+(β 1)g (x) 2 ] [ g(x)g (x)+ = 1 2α 1 g(x) 1 α 1 α The matrix 2 G is semidefinite if and only if α 1 α g (x) 2 ]. det 2 G = αt 2α 2 [ (α 1)g(x)g (x) αg (x) 2] 0,

21 1546 K. Ishige and P. Salani: Parabolic quasi-concavity that is equivalent to g(x)g (x)+ α 1 α g (x) 2 0 for the cases (i) and (iii); g(x)g (x)+ α 1 α g (x) 2 0 for the case (ii). Comparing this with the sign of f and taking in account that case (i) if f is convex, then g is convex; case (ii) if f is concave, then g is concave; case (iii) if f is concave, then g is convex, we have Lemma A.1 for the case m =1. Let m 2 and let ξ =(ξ 0,ξ 1,...,ξ m ), ξ =(ξ 1,...,ξ m ).Thenf is convex if and only if 2 f(x) =βg(x) β 1 2 g(x)+β(β 1)g(x) β 2 g(x) g(x) 0, x R m. (A.1) We treat only the case (i) α 0, which implies β>0, and leave the other cases to the reader. Then (A.1) is equivalent to g(x) g ij (x)ξ i ξ j + α 1 α g i (x)g j (x)ξ i ξ j 0 (A.2) for all ξ R m and x R m. Analogously, the function G(x, t) is convex if and only if α(α 1)ξ 2 0 g(x)tα 2 +2αξ 0 t α 1 for all ξ R m and x R m ; this is equivalent to m g i (x)ξ i + t α m i=1 g ij (x)ξ i ξ j 0 α(α 1) ξ2 0 g(x) t 2 +2α ξ 0 t g i (x)ξ i + i=1 g ij (x)ξ i ξ j 0 (A.3) for all ξ R m and x R m. Therefore it suffices to prove that (A.2) is equivalent to (A.3). Assume g(x) satisfies (A.2). Then, since m 2 g i (x) ξ i ξ 0 = t 2 ξ0 g(x)(1 α) 1 m g i (x)ξ i t i=1 g(x)(1 α) i=1 (1 α)g(x)ξ2 0 1 t 2 + g i (x)g j (x)ξ i ξ j, (1 α)g(x) we obtain α(α 1) ξ2 0 g(x) t 2 +2α ξ 0 t α 1 α g(x) 1 0, g i (x)ξ i + i=1 g i (x)g j (x)ξ i ξ j + g ij (x)ξ i ξ j g ij (x)ξ i ξ j where the last inequality is due to (A.2). Thus g(x) satisfies (A.3).

22 Math. Nachr. 283, No. 11 (2010) / 1547 On the other hand, if g(x) satisfies (A.3), for any ξ R m, we put and obtain X = i=1 g i (x)ξ i, ξ 0 = 1 1 α tg(x) 1 X, α(α 1) ξ2 0 g(x) t 2 +2α ξ 0 t g i (x)ξ i + i=1 = α α 1 g(x) +2 α 1 α = 0. X 2 α (1 α)g(x) X g(x) g ij (x)ξ i ξ j g i (x)ξ i + i=1 g i (x)g j (x)ξ i ξ j + g ij (x)ξ i ξ j g ij (x)ξ i ξ j This implies (A.2). Therefore we see that (A.2) is equivalent to (A.3), and the proof of Lemma A.1 is complete. References [1] B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math. 177, (2009). [2] C. Bianchini, M. Longinetti, and P. Salani, Quasiconcave solutions to elliptic problems in convex rings, Indiana Univ. Math J. 58, (2009). [3] C. Borell, Brownian motion in a convex ring and quasiconcavity, Comm. Math. Phys. 86, (1982). [4] C. Borell, A note on parabolic convexity and heat conduction, Ann. Inst. H. Poincaré Probab. Statist. 32, (1996). [5] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22, (1976). [6] A. Colesanti and P. Salani, Quasi-concave envelope of a function and convexity of level sets of Solutions to elliptic equations, Math. Nachr. 258, 3 15 (2003). [7] J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, Preprint No. 393, Sonderforschungsbereich Vol. 123 (Universität Heidelberg, 1986). Available at: kawohl. [8] J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl. 177, (1993). [9] E. Francini, Starshapedness of level sets for solutions of nonlinear parabolic equations, Rend. Ist. Mat. Univ. Trieste 28, (1996). [10] M. Gabriel, A result concerning convex level surfaces of three dimensional harmonic functions, London Math. Soc. J. 32, (1957). [11] Y. Giga, S. Goto, H. Ishi, and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40, (1991). [12] A. Greco and B. Kawohl, Log-concavity in some parabolic problems, Electron. J. Differential Equations 1999, 1 12 (1999). [13] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1934). [14] K. Ishige, An intrinsic metric approach to uniqueness of the positive Dirichlet problem for parabolic equations in cylinders, J. Differential Equations 158, (1999). [15] K. Ishige and P. Salani, Is quasi-concavity preserved by heat flow?, Arch. Math. (Basel) 90, (2008). [16] K. Ishige and P. Salani, Convexity breaking of free boundary in porous medium equation, Interfaces Free Bound. 12, (2010). [17] K. Ishige and P. Salani, On a new kind of convexity for solutions of parabolic problems, Discrete Contin. Dyn. Syst. Ser. S, to appear. [18] S. Janson and J. Tysk, Preservation of convexity of solutions to parabolic equations, J. Differential Equations 206, (2004).