BENO Inwon C Kim

AND Î T Perthame

Panagiotis E Souganidis

FREE BOUNDARY PROBLEMS FOR TUMOR GROWTH: A VISCOSITY SOLUTIONS APPROACH

Keywords: Elliptic-Parabolic problems, viscosity solutions, free boundary, Tumor growth, AMS Class. No: 35K55, 35B25, 35D40, 76D27

The mathematical modeling of tumor growth leads to singular "stiff pressure law" limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation.

Introduction

Motivated by models of tumor growth (see for instance the survey papers by Friedman [F], Lowengrub et al [LFJ + ]) and extending Perthame, Quiros and Vazquez [PQV], in a recent paper Perthame, Quiros, Tang and Vauchelet [PQTV] studied the limiting behavior, as m → ∞, of the solution (density) ρ m of the porous medium diffusion equation (pme for short)

ρ m,t -∆ρ m m -ν∆ρ m = ρ m G(p m ) in Q T := Ω × (0, T ), (1.1)
where Ω is either a bounded domain in R n or Ω = R n , with boundary and initial conditions

ρ m = ρ L in ∂ p Q T := ∂Ω × [0, T ) and ρ m = ρ 0,m on Ω × {0}, (1.2) satisfying 0 ≤ ρ L < 1 and 0 ≤ ρ 0 ≤ 1 if Ω is bounded or ρ 0 ∈ L 1 (R n ) if Ω = R n . (1.3) Here p m := m m -1 ρ m-1 m (1.4)
is the pressure, ν > 0, and G : R → R is a smooth function, which describes the cell multiplication, satisfying G(p M ) = 0 for some p M > 0 and G < 0.

(1.5)

Using the pressure variable, (1.1) can be rewritten as

ρ m,t -div(ρ m Dp m -νDρ m ) = ρ m G(p m ),
a form which represents better the mechanical interpretation of the model with v m := -Dp m the tissue bulk velocity according to Darcy's law. Note that, if, as m → ∞, the p m 's and ρ m 's converge respectively to p and ρ, then p will be nonzero only where ρ = 1. This indicates, that, in the limit, a phase transition may take place with an evolving free boundary between the tumor region (the support of p) and the pre-tumor zone (the support of 1ρ). The convergence, as m → ∞, of the p m 's and ρ m 's has already been investigated using a distributional solution approach in [PQTV].

Here we study the asymptotic behavior of p m and ρ m in the limit m → ∞ using viscosity solutions. This yields a different description of the limit problem, allows for more general initial data and yields pointwise information about the free boundary evolution, uniform convergence results as well as some quantified statements about the speed of propagation of the tumor zone.

In order to state the result it is necessary to introduce the limit problem which we derive next formally following [PQTV]. We use the auxiliary variable

u m := -ρ m m + ν(1 -ρ m ), (1.6)
which is close to -p m + ν(1ρ m ) for large m, and, recalling that (1.1) can also be written as

p m,t -(m -1)p m ∆p -|Dp m | 2 -ν∆p m = (m -1)p m G(p m ) -ν m -2 m -1 |Dp m | 2 p , (1.7)
we find that u m satisfies

[b m (u m )] t -ν∆u m = -νρ m G(p m ) with b m (u m ) = ν mρ m-1 m + ν .
(1.8)

Assume next that, as m → ∞, the ρ m 's, p m 's and u m 's converge respectively to ρ, p and u. Then, formally, we find p = u -, ν(1ρ) = u + where u + := max(u, 0) and u -:=min(u, 0).

(1.9)

Letting m → ∞ in (1.7) and noting that {p > 0} = {ρ = 1} yields that p and ρ solve respectively -∆p(•, t) = G(p) (•, t) in Ω(t) := {p(•, t) > 0}, (1.10) and ρ t -ν∆ρ = ρG(0) in {ρ < 1};

this last equation which can be rewritten in terms of ρ := ν(1ρ) as ρt -ν∆ρ = -νρG(0) = (ρν)G(0) in R n -Ω(t).

(1.11)

Combining (1.9), (1.10) and (1.11) the limiting problem can be recast with b(u) := u + as the ellipticparabolic equation b(u) t -ν∆u = (b(u)ν)G(u -) in Q T .

(1.12)

In view of the Lipschitz continuity of b, (1.12) yields the free boundary condition

∂ η u + = ∂ η u -on ∂Ω(t), (1.13)
where η is the (outward) normal at (x, t) ∈ ∂Ω(t), which provides an implicit motion law for the free boundary ∂{u > 0}. This problem but without the growth term has been studied by Alt-Luckhaus [AL] in the weak (duality) setting, by Carrillo [C] in the weak (entropy) setting and also by Kim-Pozar [KP] using viscosity solutions. The right hand side, however, plays an important role because it can generate pressure nucleation, thus generating a change of topology in the free boundary (in a zone where p = 0, ρ can grow and reach ρ = 1 thus generating a new island where p > 0). Viscosity solutions for porous medium equations were introduced by Caffarelli and Vazquez [CV], see also the book by Vazquez [V], and later by Kim [K] for the Hele-Shaw problem, for the Stefan problem and additional references the reader can refer to the book by Caffarelli and Salsa [CS].

Next we introduce the precise assumptions on the initial data ρ m,0 . We assume that there exists (1.14) and, as m → ∞, ρ m,0 → ρ 0 uniformly, where ρ 0 : Ω → [0, 1] is continuous and Ω 0 := {ρ 0 = 1} is bounded domain with locally Lipschitz boundary.

M 0 > 0 such that p m,0 := m m -1 (ρ m,0 ) m-1 ≤ M 0 ,
(1.15)

and, in addition, if Ω = R n that

ρ 0 ∈ L 1 (R n ).
(1.16)

The main result of the paper is:

(x) := -wχ Ω0 + ν(1 -ρ 0 )χ Ω c 0 ,
where w be the unique solution of -∆w = G(w) in Ω 0 and w = 0 on ∂Ω 0 . Then, for all T > 0 and as m → ∞, the ρ m 's converge to 1ν -1 b(u) uniformly in Q T , where u is the unique viscosity solution of (1.12), with the initial and boundary data u 0 and ν(1ρ L ) respectively.

We remark that the p m 's converge uniformly to u -as long as u -is continuous. It turns out, however, that u -may be discontinuous in time. This is due to the fact that (the positive part) b(u) solves a parabolic equation with a sink term, which means that b(u) can decrease to zero in the interior of its positive phase and nucleate a negative phase. Once the negative phase is created, the elliptic equation that needs to be satisfied in the negative phase leads to the jump discontinuity of u -over time. We refer to Section 5 for a discussion about the propagation of the elliptic and parabolic phases. The Figure 1 illustrate the time discontinuities in p m , and Figure 2 shows and additional discontinuity in the ρ m 's when viscosity is neglected (Hele-Shaw problem).

there exists (1.14) and, as m ! 1, ( ⇢ m,0 ! ⇢ 0 uniformly, where ⇢ 0 : ⌦ ! [0, 1] is continuous and ⌦ 0 := {⇢ 0 = 1} is bounded domain with locally Lipschitz boundary.

M 0 > 0 such that p m,0 := m m 1 (⇢ m,0 ) m 1  M 0 ,
(1.15)

and, in addition

, if ⌦ = R n that ⇢ 0 2 L 1 (R n ).
(1.16) The main result of the paper is:

Theorem 1.1. Let ⇢ m solve (1.1) with data satisfying (1.3), (1.14), (1.15), and (1.16), if ⌦ = R n , and define u 0 (x) := w ⌦ 0 + ⌫(1 ⇢ 0 ) ⌦ c 0 , where w be the unique solution of w = G(w) in ⌦ 0 and w = 0 on @⌦ 0 . Then, for all T > 0 and as m ! 1, the ⇢ m 's converge to 1 ⌫ 1 b(u) uniformly in Q T , where u is the unique viscosity solution of (1.12), with the initial and boundary data u 0 and ⌫(1 ⇢ L ) respectively.

We remark that the p m 's converge uniformly to u as long as u is continuous. It turns out, however, that u may be discontinuous in time. This is due to the fact that (the positive part) b(u) solves a parabolic equation with a sink term, which means that b(u) can decrease to zero in the interior of its positive phase and nucleate a negative phase. Once the negative phase is created, the elliptic equation that needs to be satisfied in the negative phase leads to the jump discontinuity of u over time. We refer to Section 5 for a discussion about the propagation of the elliptic and parabolic phases. The Figure 1 illustrate the time discontinuities in p m , and Figure 2 shows and additional discontinuity in the ⇢ m 's when viscosity is neglected (Hele-Shaw problem). The parameters are m = 20, ⌫ = .5. These figures illustrate how the pressure profile is building up when density approaches one. Between the last two pictures, the density has continuously reached ⇢ = 1 in the center, while p has jumped discontinuously.

Theorem 1.1 is proved by showing first that the u m 's converge to u as m ! 1 (see Corollary 4.4). The convergence results for the ⇢ m 's then follow from the definition of u m and the continuity properties of b(u).

As already mentioned above we use viscosity solutions which are based on appropriate choices of test functions. In the problem at hand these will be radial smooth solutions which are defined in Section 2. The main step in the proof of Theorem 1.1 is the following:

Theorem 1.2. Assume that is a classical radial solution of (1.12). Then there exists a family of radial sub-and solutions super-solutions of (1.8) converging, as m ! 1, to . As already mentioned above we use viscosity solutions which are based on appropriate choices of test functions. In the problem at hand these will be radial smooth solutions which are defined in Section 2. The main step in the proof of Theorem 1.1 is the following:

Theorem 1.2. Assume that φ is a classical radial solution of (1.12). Then there exists a family of radial sub-and solutions super-solutions of (1.8) converging, as m → ∞, to φ. Organization of the paper. In Section 2 we introduce the notion of viscosity solution of the limit problem and discuss its existence and uniqueness which rely on the results of [AL] and [KP]. In Section 3 we prove Theorem 1.2. We used the results of Section 2 and Section 3 to prove Theorem 1.1 in Section 4. In Section 5 we present estimates on the speed of propagation of the interface. In the Appendix we touch upon the proof of the comparison for viscosity solutions to the limit problem.

Notation and Terminology

. A nonempty set E ⇢ R n ⇥ R is called a parabolic neighborhood if E = U \ R n ⇥ (0, ⌧ ]
for some open set U ⇢ R n ⇥ R and some ⌧ 2 (0, 1) and @ P E := Ē \ E is the parabolic boundary of E. Whenever we refer to a parabolic neighborhood, we assume U and ⌧ are known. For an open subset A of R n ⇥R and a continuous function v : Ā ! R, where Ā is the closure of A, {v > 0}, {v = 0} and {v < 0} are respectively the sets {y 2 Ā : v(y) > 0}, {y 2 Ā : v(y) = 0} and {y 2 Ā : v(y) < 0}. For a set A, we write A for its characteristic function. If A, B are subsets of R k , A + B := {a + b : a 2 A, b 2 B}. Given a set U ⇢ R k , U SC(U ) and LSC(U ) are respectively the sets of upper and lower semicontinuous functions on U ; if we are dealing with bounded semicontinuous functions we write BU SC(U ) and BLSC(U ). If u : U ! R is bounded, u ?, Ū and u ?, Ū denote respectively its upper and lower-semicontinuous envelope. We write B R (x) for the open ball in R n centered x 2 R n with radius R > 0. The duality map between a Banach space and its dual is denoted by <

•, • >. If A is open subset of R n ⇥ R, C 2,1
x,t (A) denotes the space of functions on A which are continuously twice di↵erentiable in x and continuously di↵erentiable in

t; similarly, if B is an open subset of R k , C l
x (B) denotes the space of l-times continuously di↵erentiable functions on B; when there is no confusion we omit the dependence on A and B. We say that a constant is dimensional if it depends only on the dimension. We denote by D (⇠, ⌧ ) the gradient at (⇠, ⌧ ) of :

U ! R, where U is an open subset of R n ⇥ R and, D ± (⇠, ⌧ ) := lim (x,t)!(⇠,⌧ ) (x,t)2{±'>0} D'(x, t). For a family of functions f m : ⌦ ⇥ (0, 1) ! R, lim inf ⇤ f m and lim sup ⇤ f m denote the lower and upper semi-continuous limits, that is lim inf ⇤ f m (x, t) := lim r!0 inf{f m (x+y, t+s) : (x+y, t+s) 2 ⌦⇥(0, 1), |y|+|s|  r, m r 1 }, and lim sup ⇤ f m (x, t) := lim r!0 sup{f m (x + y, t + s) : (x + y, t + s) 2 ⌦ ⇥ (0, 1), |y| + |s|  r, m r 1 }.

Viscosity and weak solutions of the limit problem

We introduce the notions of viscosity and weak (distributional) solutions to the limit problem

8 > > < > > : b(u) t ⌫ u = (b(u) ⌫)G(u ) in Q T , u = u 0 on ⌦ ⇥ {0}, u = g on @⌦ ⇥ [0, T ],
(2.1) Organization of the paper. In Section 2 we introduce the notion of viscosity solution of the limit problem and discuss its existence and uniqueness which rely on the results of [AL] and [KP]. In Section 3 we prove Theorem 1.2. We used the results of Section 2 and Section 3 to prove Theorem 1.1 in Section 4. In Section 5 we present estimates on the speed of propagation of the interface. In the Appendix we touch upon the proof of the comparison for viscosity solutions to the limit problem.

Notation and Terminology

. A nonempty set E ⊂ R n × R is called a parabolic neighborhood if E = U ∩ R n × (0, τ ] for some open set U ⊂ R n × R
Ā : v(y) > 0}, {y ∈ Ā : v(y) = 0} and {y ∈ Ā : v(y) < 0}. For a set A, we write χ A for its characteristic function. If A, B are subsets of R k , A + B := {a + b : a ∈ A, b ∈ B}. Given a set U ⊂ R k , U SC(U )
and LSC(U ) are respectively the sets of upper and lower semicontinuous functions on U ; if we are dealing with bounded semicontinuous functions we write BU SC(U ) and BLSC(U ). If u : U → R is bounded, u , Ū and u , Ū denote respectively its upper and lower-semicontinuous envelope. We write B R (x) for the open ball in R n centered x ∈ R n with radius R > 0. The duality map between a Banach space and its dual is denoted by <

•, • >. If A is open subset of R n × R, C 2,1
x,t (A) denotes the space of functions on A which are continuously twice differentiable in x and continuously differentiable in t; similarly, if

B is an open subset of R k , C l
x (B) denotes the space of l-times continuously differentiable functions on B; when there is no confusion we omit the dependence on A and B. We say that a constant is dimensional if it depends only on the dimension. We denote by Dφ(ξ, τ ) the gradient at (ξ, τ ) of φ :

U → R, where U is an open subset of R n × R and, Dφ ± (ξ, τ ) := lim (x,t)→(ξ,τ ) (x,t)∈{±ϕ>0}
Dϕ(x, t). For a family of functions

f m : Ω × (0, ∞) → R, lim inf * f m and lim sup * f m denote the lower and upper semi-continuous limits, that is lim inf * f m (x, t) := lim r→0 inf{f m (x+y, t+s) : (x+y, t+s) ∈ Ω×(0, ∞), |y|+|s| ≤ r, m ≥ r -1 }, and lim sup * f m (x, t) := lim r→0 sup{f m (x + y, t + s) : (x + y, t + s) ∈ Ω × (0, ∞), |y| + |s| ≤ r, m ≥ r -1 }.

Viscosity and weak solutions of the limit problem

We introduce the notions of viscosity and weak (distributional) solutions to the limit problem

       b(u) t -ν∆u = (b(u) -ν)G(u -) in Q T , u = u 0 on Ω × {0}, u = g on ∂Ω × [0, T ],
(2.1) and we discuss their existence and uniqueness. While we rely on the theory of weak solutions to obtain uniqueness results, viscosity solutions are used to prove pointwise convergence results in Section 4 as well as to derive information about the evolution of parabolic and elliptic phases in Section 5.

Viscosity solutions: definition and comparison principle. Following [KP] we define viscosity solutions of (2.1). We begin by introducing the class of test functions which are classical sub-and super-solutions of the problem with the specific properties stated in the next definition.

Definition 2.1 (Classical sub-and super-solutions). E be a parabolic neighborhood. Then

ϕ ∈ C( Ē) is a classical subsolution of (1.12) in E if (i) ϕ ∈ C 2,1 x,t ({ϕ > 0}) ∩ C 2,1 x,t ({ϕ < 0}), (ii) {ϕ = 0} ⊂ ∂ {ϕ > 0} ∩ ∂ {ϕ < 0} and |Dϕ ± | > 0 on {ϕ = 0}, (iii) b(ϕ) t -ν∆ϕ ≤ (b(ϕ) -ν)G(ϕ + ) on {ϕ > 0} and {ϕ < 0}, and (iv) |Dϕ + | > |Dϕ -| on {ϕ = 0};
ϕ is a strict classical subsolution if the inequalities in (iii) and (iv) are strict. Supersolutions are defined similarly by reversing the inequalities in (iii) and (iv).

Next we define the viscosity solutions to (1.12). We begin with the notions of (semicontinuous) sub-and super-solutions.

Definition 2.2 (Viscosity sub-and super-solutions). We say

u ∈ BU SC(Q T ) (resp. u ∈ BU SC(Q T ) is a viscosity subsolution (resp. supersolution) to (2.1) in Q T , if (i) u(•, 0) ≤ u 0 (respectively u(•, 0) ≥ u 0 ) on Ω × {0}, u ≤ g (resp. u ≥ g) on ∂Ω × [0, T ], (ii) u < ϕ (resp. u > ϕ) on E for any strict classical subsolution (resp. supersolution) ϕ on any parabolic neighborhood E ⊂ Q T for which u < ϕ (resp. u > ϕ) on ∂ P E.
We say u has initial data u 0 if u converges uniformly to u 0 as t → 0.

Given that we do not expect to have continuous but rather lower semicontinuous solutions to (2.1) (recall the earlier discussion about possible nucleation), we define the notion of viscosity solution for such functions.

Definition 2.3 (Viscosity solutions). A function u ∈ BLSC(Q T ) is a viscosity solution of (2.1) in Q T if its upper semi-continuous envelope u * ,Q T is a viscosity subsolution in Q T and u is a viscosity supersolution in Q T .
Next we introduce conditions on the data of (2.1) that are necessary in order to have a well-posed theory.

For the lateral boundary we assume, for simplicity, that g is continuous and positive.

(2.2)

For the initial condition, if Ω is bounded, we assume that u 0 : Ω → R satisfies

u 0 ∈ C(Ω), u 0 < 0 on ∂Ω and -∆u 0 = G(u 0 ) in {u 0 ≥ 0}, (2.3) and Γ(u 0 ) := ∂{u 0 ≥ 0} is locally a Lipschitz graph. (2.4) When Ω = R n , we require, in addition, that {u(x, 0) < 0} ⊂ B R (x) for some R > 0 and u(•, 0) -ν ∈ L 1 (R n ).
(2.5) Note that, in terms of p = u + and ρ = ν -1 u -+ 1, the above conditions imply that the pressure phase is initially bounded and the density is integrable.

We recall that we use only strict sub-and super-solutions as test functions. As a result it is possible to narrow the choice of the parabolic neighborhood E in Definition 2.2 and to use, instead, parabolic cylinders of the form

Q = Ω × (t 1 , t 2 ] ⊂ Q T , with Ω ⊂ Ω having a smooth boundary. Indeed, suppose that E ⊂ Q T is a parabolic neighborhood, ϕ is a strict classical supersolution on E, u < ϕ on ∂ P E and u ≥ ϕ at some point in E. Define τ := sup {s : u < ϕ on E ∩ {t ≤ s}} ∈ R. The set A := {x : (x, τ ) ∈ E, u ≥ ϕ} is compact and, therefore, d := distance(A × {τ } , ∂ P E) > 0. Consider the parabolic cylinder Q = (A + B d/2 ) × (τ -d, τ ]. Clearly Q ⊂ E and u < ϕ on ∂ P Q ; the boundary of A + B d/2
can be easily regularized.

The proof of the comparison principle follows along the lines of the one in [KP] the only difference being the contribution of the source term (b(u) + ν)G(u), which is not present in the equation considered in [KP]. The needed modifications are presented in the Appendix. Here we note, that, although, as discussed earlier, the source term in the negative (elliptic) phase of u may give rise to nucleations, this does not hinder the argument in [KP]. Indeed nucleations can not occur at the contact point of two "regularized" solutions which are initially strictly ordered (see Appendix). Hence we have the following theorem, which corresponds to Theorem 3.1 of [KP].

Theorem 2.1. Assume (2.2), (2.3), (2.4) and, if Ω = R n , (2.5) and let u ∈ BU SC(Q T ) and v ∈ BLSC(Q T ) be respectively a sub-and a super-solution of (1.12) in Q T for some

T > 0. If u < v on the parabolic boundary of Q T , then u < v in Q T .
Note that in the above theorem we require that u and v are strictly separated on the parabolic boundary of Q T . For this reason, Theorem 2.1 yields, for given data, only the maximal and minimal viscosity solutions. Removing the condition of strict separation remains an interesting (and important) open question for most of free boundary problems. For the problem at hand, we are able to borrow the theory of regular weak solutions of [AL] to study the stability and uniqueness properties of the viscosity solutions.

Weak solutions: existence and comparison principle. We recall the notion of regular weak solution to (2.1) and then state the theorems in [AL] which concern the existence and stability of such solutions. The difference in the proofs due to the source terms are discussed in the Appendix. We note that we do not try to recall the full generality of [AL] but we modify definitions and statements to apply to the specific problem we study here. Finally, we remark that [AL] only considers bounded domains Ω.

We begin with the definitions of weak and regular solutions. The former is a notion of solutions based on duality (integration by parts) while the second, as the name indicates, requires more regularity (in time).

Definition 2.4 (Weak solutions). u ∈ L 2 (0, T ; H 1 (Ω)) with b(u) ∈ L ∞ (0, T ; L 1 (Ω)) and ∂ t b(u) ∈ L 2 (0, t; H -1 (Ω)) is a weak solution to to (1.12) in Q T if, for every test function ζ ∈ L 2 (0, T ; H 1 0 (Ω)), T 0 < ∂ t b(u), ζ > dt + T 0 Ω Du • Dζdxdt = T 0 Ω (b(u) -ν)G(u -)ζdxdt. (2.6)
Weak sub-and super-solutions are defined with the corresponding inequalities replacing the equality above.

Definition 2.5 (Regular weak solutions). A weak solution u is regular if

∂ t b(u) ∈ L 2 (Q T ).
The following corresponds to Theorem 2.2 of [AL].

Theorem 2.2. Assume that the bounded domain Ω has Lipschitz boundary. If u and v are respectively a weak subsolution and supersolution of (1.12) in Q T and, in addition,

∂ t (b(u) -b(v)) ∈ L 1 (Q T ) and u ≤ v on the parabolic boundary of Q T , then u ≤ v in Q T .
The proof parallels the one of Theorem 2.2 in [AL]. It is rather simple, and, for the reader's convenience, we present it next.

Proof. Fix δ > 0 and let ψ(z) := min[1, max(z/δ, 0)]. Applying the definition to u and v with ζ := ψ(uv) and using the fact that G is decreasing and u -< v -if v < u, we find, for some C > 0,

t 0 Ω ∂ t (b(u) -b(v))ψ(v -u)dxdt + C δ t 0 Ω χ {0<u-v<δ} |D(u -v)| 2 dxdt ≤ t 0 Ω [χ {0<u-v<δ} (b(u) -ν)G(u -) -(b(v) -ν)G(v -)]dxdt = t 0 Ω [χ {0<u-v<δ} (b(u) -b(v))G(0) + ν(-G(u -) + G(v -))]dxdt ≤ t 0 Ω (b(u) -b(v)) + G(0)dxdt.
Next we let δ → 0. The parabolic term on the left converges to Ω (b(u)b(v)) + (•, t)dx, and, thus, Gronwall's lemma yields that b(u) ≤ b(v).

Using this last information in the previous inequality, we find that, for any δ > 0, b(uv) = 0 in {0 < uv < δ}, which, in view of the fact that u ≤ v on the parabolic boundary of Q T , yields the claimed comparison.

The next results assert the existence and uniqueness of regular weak solutions.

Theorem 2.3. Assume that the bounded domain Ω has Lipschitz boundary. Then there exists a unique regular weak solution u in Q T for a given boundary data

u D ∈ H 1 (0, T ; H 1 (Ω)). Moreover, b(u) is continuous.
The uniqueness follows from Theorem 2.2 while the continuity of b(u) is a consequence of the main result of Di Benedetto and Gariepy [DG]. The existence is based on the Galerkin approximation and parallels that of Theorem 2.3 in [AL] where we refer for the details.

Weak solutions to (1.12) are more flexible than viscosity solutions with respect to the range of the boundary data. On the other hand, we do not see how to prove directly the pointwise convergence of solutions of (1.8) to the solutions of (1.12) without going through the viscosity solutions framework. Indeed the convergence proof depends on the facts that we allow semi-continuity for viscosity suband super-solutions, and that, even with such low regularity, there is a comparison principle.

Uniqueness and existence of viscosity solutions for bounded domains. We discuss now the uniqueness for the viscosity solutions and we argue separately depending on whether Ω is bounded or not.. Bounded domain. The first step to establish the uniqueness is that viscosity solutions evolve continuously from the initial data. This is quantified in terms of the growth in time of the distance between the parabolic phase and ∂Ω \ Ω 0 , the initial position, where ∂Ω 0 := {u 0 < 0}.

Lemma 2.4. Assume (1.5), (2.2), (2.3) and (2.4) and let u be a viscosity solution of (2.1). There exists some sufficiently small t 0 such that, for all 0 < t ≤ t 0 , d(x, ∂{u 0 > 0}) < t 1/3 for any x ∈ ∂{u(•, t) > 0}.

Proof. To prove the claim we construct suitable super-and sub-solution barriers. We begin with the former. Fix ε > 0 and let Σ(t) be the

Σ(t) := {x : d(x, Ω \ Ω 0 ) ≤ ε + t 1/3 }.
Let w + and w -be respectively the solutions to

             w + = ∆w + in {(x, t) : x ∈ Σ(t), 0 ≤ t ≤ 1}, w + = ρ L on ∂Ω × [0, 1], w + = 0 on {(x, t) : x ∈ ∂Σ(t), 0 ≤ t ≤ 1}, w + (•, 0) = (u 0 ) + on Σ(0),
and to the elliptic equation in (1.12) set in the complement of Σ(t) with zero boundary data, and define w := w +w -. Since Σ(t) has the exterior ball condition, for small t > 0, we have

|Dw -|(•, t) ≥ ε + t 1/3 on ∂Σ(t).
On the other hand, it follows, using as barriers, if necessary, shifted versions of heat kernel as barriers, that the restriction of |Dw + | on ∂Σ(t) converges exponentially fast to zero as t → 0. Hence, we can choose a sufficiently small t 0 > 0 such that, for all 0 < t ≤ t 0 ,

|Dw + |(•, t) < |Dw -|(•, t) on ∂Σ(t).
It follows that w is as a supersolution for (1.12). This yields that ∂{u(•, t) > 0} cannot expand faster than order of t 1/3 . Similarly one can construct a subsolution barrier, based on the fast-decreasing set Σ(t) := {x : d(x, Ω 0 ) ≥ t 1/3 }.

Since the arguments are similar we omit them.

Using the weak theory described above, we can prove, using arguments similar to the ones in Section 5.1 of [KP], the following theorem.

Theorem 2.5. Assume (1.5), (2.2), (2.3) and (2.4). There exists a unique viscosity solution u to (2.1). Moreover, u = (u * ) * , and u coincide a.e. with the regular weak solution u of (1.12) in Q T . In particular, b(u) is continuous.

Unbounded domain Ω = R n . It can be checked easily, using a suitable radial barrier, that the negative phase of any viscosity solution of (1.12) with initial data u 0 satisfying (2.3), (2.4) and (2.5) is contained, for 0 ≤ t ≤ T , in B R1 (0) with R 1 = R 1 (T ). Let u n be the viscosity solution of (2.1) in B R1+n (0) × [0, T ] with initial data u 0 χ {|x|≤R1+n} and boundary data -ν -1 n on ∂B R1+n (0) × [0, T ] respectively. It is immediate that the u n 's are monotonically increasing with respect to n, and, thus, converge pointwise to some w : R n × [0, T ] → R. Furthermore, the u n 's solve a uniformly parabolic equation outside of B R1 , and, thus, are smooth in

{R 1 ≤ |x| ≤ R 1 + n} × (0, T ].
We may, therefore, conclude that the convergence of the u n 's to w is locally uniform in {|x| ≥ R 1 + 1} × [0, T ) and, moreover, that w is smooth , since it solves a parabolic equation in {|x| ≥ R 1 + 1} × [0, T ). Now we can use standard stability arguments about viscosity solutions to show that w is the unique viscosity solution of (2.1) in {|x| ≤ R 1 + 1} × [0, T ] with the initial and boundary data u 0 and w on {|x| = R 1 + 1} × [0, T ] respectively. It follows that w is a viscosity solution of (1.12) in R n × (0, T ].

Theorem 2.6. Assume that u 0 satisfies (1.5), (2.3), (2.4) and (2.5). Then (1.12) has a unique viscosity solution in R n × (0, T ] with initial datum u 0 .

Proof. The barrier argument described above yields that any viscosity solution u to (1.12) in R n × (0, T ], with initial condition satisfying the assumptions in the statement and has, in addition, compact non-positive phase, stays positive outside of a compact set, and, thus, solves the parabolic equation in (1.12) in {|x| ≥ R} × (0, T ] for some R > 0.

In particular, it follows that u and b(u) are smooth when |x| is large, and b(u)ν ∈ L ∞ (0, T ; L 1 (R n )). Moreover, u is a viscosity solution of (1.12) in {|x| ≤ R 1 + 1} × (0, T ] with smooth lateral boundary data. It follows that u is also the unique regular weak solution of (1.12) given by Theorem 2.3. This last claim and the decay of u for large |x| imply that ∂ t b(u) ∈ L 1 (R n × (0, T )). The proof of Theorem 2.2 yields that u is the unique viscosity solution of (1.12) with initial data u 0 .

3. The construction of radial barriers and the proof of Theorem 1.2

We prove here Theorem 1.2. Let φ be a radial classical subsolution to (1.12), as defined in Definition 2.1 in the domain {r 1 ≤ |x| ≤ r 2 } × [0, T ], where 0 < r 1 < r 2 ≤ ∞, and, to simplify, we assume that the elliptic phase is contained in B r1 . We recall that this means that φ is smooth in its positive and negative phase and, for some a ∈

C 1 ([0, T ]; R),      -∆φ -≤ G(φ -) in {φ < 0} = {(x, t) : r 1 < |x| < a(t)}, φ t -ν∆φ ≥ (φ -ν)G(0) in {φ > 0} = {(x, t) : a(t) < |x| < r 2 }, 0 < |Dφ + | < |Dφ -| on {(x, t) : |x| = a(t)}.
We will perturb φ to construct subsolutions to (1.8). It is, however, important to remark that these subsolutions do not have to be, actually will not be, smooth across their interface. Discontinous subsolutions to the equation (1.1) can be defined in the viscosity sense in a fashion similar to Definition 2.2

To keep things simple, we assume that (i) we have equality instead of inequality in the elliptic and parabolic equation in each phase of φ, and (ii) the domain is R n × [0, ∞) and the positive (elliptic) phase of φ is {|x| < a(t)}, that is we assume that there is no r 1 and r 2 = ∞. A minor modification of the arguments presented below yields the general case.

We introduce the notation p 0 := φ -and ρ 0 := 1 + ν -1 φ + , and note that, for each t ∈ (0, T ], p 0 (•, t) solves

-∆p 0 = G(p 0 ) in {|x| < a(t)}, p 0 = 0 on {|x| = a(t)}, (3.1) while ρ 0 satisfies ρ t -ν∆ρ = ρG(0) in {(x, t) : |x| > a(t)} (3.
2) with the free boundary condition

|Dφ -| = ν|Dρ 0 | > |Dp 0 | = |Dφ + | on {(x, t) : |x| = a(t)}. (3.3)
Our first claim is:

Lemma 3.1. There exist subsolutions u m to (1.8) which converge, as m → ∞, uniformly to φ, and, hence, there exist viscosity subsolutions ρ m of (1.1) such that, as m → ∞, the ρ m 's and the associated pressure variables p m 's converge uniformly to ρ 0 and p 0 respectively.

Proof. Consider the one-to one functions

Φ(ρ) := -ρ m + ν(1 -ρ m ) and Ψ(p) := Φ(( m -1 m ρ) 1/m-1 ) (3.4)
and recall from the introduction that

u m = Φ(ρ m ) = Ψ(p m ). (3.5)
It is then immediate that, if p m is bounded, then

u m = -p m + O(m -1 ln m) if p m ≥ m -1 ln m. (3.6)
The heuristics in the introduction suggest as a possible way to prove the claim to perturb p 0 and ν(ρ 0 -1) from the regions {p 0 > 0} = {|x| < a(t)} and {ρ 0 < 1} = {|x| > a(t)} and patch them together to construct a supersolution u m of (1.8), which then would imply that ρ := Φ -1 (u) is a supersolution of (1.1). The barrier will be of the form

u m (x, t) = u m (|x|, t) = -u m,1 (|x|, t) in {|x| < a(t)}, u m,2 (|x|, t) in {|x| > a(t)},
where -u m,1 and u m,2 are classical subsolutions to (1.8) in their respective regions and at r = a(t)

satisfy |Du 1 | < |Du 2 |.
This inequality, which follows from (3.3), will prevent any smooth functions from crossing u m from below at {r = a(t)}. It follows that u m is a viscosity supersolution of (1.8) in the entire domain.

We begin with u 1,m . Let ũm (•, t) solve

-∆ũ m (•, t) = G(p m ) + f m (|x|) in {|x| < a(t)} and ũm (a(t), t) = 0,
where f m (r) := A 0 ν -1 χ {p0≤m -1/3 } (r) + m -1/3 and pm := Ψ -1 (-ũ m ), (3.7) with A 0 > 0 an independent of m sufficiently large constant to be determined below, and observe that, in view of the form of the equation above, ũm is spatially radial and

ũm (x, t) = ũm ( x a(t) , 1). Note that, as m → ∞, f → 0 in L 1 ([0, a(t)]), since |Dp 0 |(a(t)) = 0. Using that p is radial, we find that, as m → ∞, ũ → p 0 in the C 1 -norm in {|x| ≤ a(t)}. Since |Dp 0 | > 0 and ũ = p 0 = 0 at |x| = a(t), it then follows that p 0 (x) = ũ(x) + o(|x -a(t)|) near |x| = a(t).
In particular, in view of (3.6), we have

f (r) ≥ A 0 ν -1 χ {ũ≤ 2 3 m -1/3 } (r) + m -1/3 ≥ A 0 ν -1 χ { p≤ 1 2 m -1/3 } (r) + m -1/3 . (3.8)
Next we define u m,1 := ũmc m with c m := Ψ(ν/m 3 ), and remark that, for any

k > 1, 1 m ln m ≤ c m ≤ 1 m k . (3.9)
The aim is to show that, for u m,1 , ρ m,1 := Φ -1 (-u m,1 ) and p m,1 := Ψ -1 (-u m,1 ),

(u m,1 ) t -(mp m,1 + ν)∆u m,1 ≥ (mp m,1 + ν)ρ 1 G(p m,1 ) in {(x, t) : |x| < a(t)}, (3.10)
which yields that u m = -u m,1 is a subsolution of (1.8) in the region {(x, t) : |x| < a(t)}.

To show (3.10), note that, since u m,1 (x, t) = u m,1 ( x a(t) , 1), it follows that, for sufficiently large m,

|(u m,1 ) t | = |(ũ m ) t | ≤ |(x/a(t)) Dũ m | ≤ C|Dp 0 | = O(1) in {(x, t) : |x| < a(t)}.
Also note that, in view of (3.9), the difference pmp m,1 is of order of c m if p m ≥ m -1/2 . In particular if p m,1 ≤ 1 10 m -1/3 , then pm ≤ 1 2 m -1/3 and, therefore, in view of (3.8), we find

-(mp m,1 + ν)∆u m,1 = -(mp m,1 + ν)∆ũ m ≥ (mp m,1 + ν)[G(p m ) + m -1/3 ] + (mp m,1 + ν)(f (r) -m -1/3 ) ≥ (mp m,1 + ν)G(p m ) + m 2/3 p 1 + (mp m,1 + ν)(f (r) -m -1/3 ), ≥ (mp m,1 + ν)p 1/m-1 m,1 G(p m,1 ) -(u m,1 ) t ,
provided that m and A 0 are sufficienty large. In particular the second inequality follows from the fact that, in view of (3.8), νf m (p m ) > A 0 if pm ≤ 1 3 m -1/3 , while, otherwise, p m,1 ≥ 1 10 m -1/3 and, thus, mp m,1 ≥ 1 10 m 2/3 → ∞ as m → ∞. It follows that (3.10) holds in {(x, t) : |x| < a(t)}. We next define u m,2 := Φ(ρ m ), where ρm is a perturbation of ρ 0 solving ρm,t -

ν∆ρ m = ρm G(0) -m -1/2 in Σ := {(x, t) : |x| > a(t), t > 0} with ρm (•, 0) = ρ 0 (•, 0) -1 + Φ -1 (c m ) and Φ(ρ m ) = c m on {|x| = a(t)}. Note that Φ -1 (c m ) ∼ ( 1 m ln m) 1/m-1 → 1 -as m → ∞, therefore, ρm = ( ν m 3 ) 1/m-1 ∼ 1 on {|x| = a(t)}. Thus the ρm 's converge to ρ 0 in the C 2,1
x,tnorm in Σ and in C 1 x up to the boundary. Hence, since |Dρ 0 |(a(t), t) > 0 and ρ 0 is strictly away from 1 in Σ, for sufficiently large m we have ρm-1 m ≤ ν/m 3 , and, thus,

∆(ρ m m ) = mρ m-1 m ∆ρ m + m(m -1)ρ m-2 m |D ρm | 2 ∼ 1 m 2 ∆ρ m + ν m |D ρm | 2 = O(m -1 ). (3.11)
Then, for sufficiently large m,

ρm,t -∆(ρ m m ) -ν∆ρ m ≤ ρm,t -ν∆ρ m + O(m -1 ) ≤ ρm G(0) in Σ, and, therefore, u m,2 is a supersolution to (1.8) in Σ.
To conclude we check that the spatial gradients of u m,1 and u m,2 are ordered in the right order at the patching location |x| = a(t). This follows since, in view of (3.3) and the C 1 -convergence of ũm to p 0 in {|x| ≤ a(t)},

|Du m,1 | = |Dũ m | = (1 + o(1))|Dp 0 | < (1 -o(1))ν|Dρ 0 | < |Du m,2 | on {|x| = a(t)},
where the second inequality is due to (3.3) and the last inequality follows from the fact that

|Dρ m m | ≤ m -2 |Dρ m | when ρ m-1 = ν/m 3 .
Next we choose a radially monotone classical supersolution φ of (1.12) and define ρ 0 and p 0 as before. Note that in this case we have

|Du + | = ν|Dρ 0 | < |Dp 0 | = |Du -| on {(x, t) : |x| = a(t)}.
(3.12) Lemma 3.2. There exists a supersolution ρ m of (1.1) such that, as m → ∞, ρ m and p m converge uniformly to ρ 0 and p 0 respectively.

Proof. The argument parallels that of the proof of the previous lemma. We define the barrier u m (x, t) = u m (|x|, t) as This makes u m a (nonsmooth) supersolution of (1.8).

u m (r, t) = -u m,1 (r, t) in {r < a(t)}, u m,2 (r, t),
We define u m,1 = ũmc m , where c m = Ψ(ν/m 3 ) is as before and ũm solves

-∆ũ m = G(p m ) -f m (x) in {|x| < a(t)}, with u m,1 (a(t)) = 0,
with f m as in (3.7) in {|x| < a(t)}.

It follows that ũm (•, t) converges to p 0 (•, t) in the C 1 x -norm in {|x| < a(t)}. Moreover, since u m,1,t = O(1), a straightforward computation, similar to the supersolution case, yields that

u m,1,t -(mp m,1 + ν)∆u m,1 ≤ ρ m,1 (mp m,1 + ν)G(p m,1 ),
and thus (3.13) is satisfied. Next let u m,2 := Φ(ρ m ), where ρm is a perturbation of ρ 0 solving ρm,t -

ν∆ρ m = ρm G(0) + m -1/2 in Σ := {(x, t) : |x| > a(t), t > 0}, with ρm (•, 0) = ρ 0 (•, 0) -1 + Φ -1 (c m ) and Φ(ρ m ) = c m on {|x| = a(t)}.
Observe that, as m → ∞, ρm converges to ρ 0 in the C 2,1

x,t -norm in Σ and in the C 1 x -norm up to the boundary, and, since |Dρ 0 |(a(t), t) > 0 and ρ 0 < 1 in Σ, for sufficiently large m, we find ρm-1

m ≤ ρm-1 m (a(t)) ≤ Ψ -1 (c m ) = ν/m 3 in Σ. Thus it follows that ∆(ρ m m ) = mρ m-1 m ∆ρ m + m(m -1)ρ m-2 m |D ρm | 2 ∼ 1 m 2 ∆ρ m + ν m |D ρm | 2 = O(m -1 ), (3.14)
and, for sufficiently large m, u m,2 is a supersolution of (1.8), in view of the fact, that in this case ρm,t -∆(ρ m m ) -ν∆ρ m ≥ ρm,t -ν∆ρ m -O(m -1 ) ≥ ρG(0) in Σ. Lastly, in view of (3.12), we have on {(x, t)

: |x| = a(t)} |Du m,1 |(a(t), t) ≥ (1 -o(1))|Dp 0 |(a(t), t) > (1 + o(1))ν|Dρ 0 |(a(t), t) ≥ |Du m,2 |(a(t), t),
which is the correct order for ũm to be a subsolution to (1.8).

Combining the two lemmas above yields the proof of Theorem 1.2.

Convergence in the general case

We use the results of the previous section to show the general convergence result, that is Theorem 1.1. We only consider the case Ω = R n , since the arguments for bounded Ω are similar. We have:

Theorem 4.1. u is a subsolution and v is a supersolution to (1.12).

Proof. We will only show that u is a subsolution of (1.12). The proof of the other claim is similar. If u is not a subsolution, there must exist a classical strict supersolution φ of (1.12) in a parabolic cylinder C := B r (x 0 ) × [t 1 , t 0 ] such that u < φ on the parabolic boundary of C and u crosses φ from below for the first time at t = t 0 . Let (x 0 , t 0 ) be where uφ take its nonnegative maximum.

Next we perturb φ so that it becomes slightly smaller and thus uφ achieves a positive maximum in C. For instance, we may replace φ by the inf-convolution φ(x, t) := inf {y∈ Ω:|x-y|≤r} φ(y, t)

for sufficiently small r > 0, and solve the corresponding elliptic and parabolic problem in the positive and negative phase of φ. Note that this perturbation preserves the supersolution condition |Dφ + | < |Dφ -| on the zero level set.

The definition of u yields a subsequence, that we still denote by m, along which, for sufficiently large m, u mφ has a positive maximum in C with u m < φ on the parabolic boundary of C. Since u m is continuous, this means that u m = φ at (x m , t m ) for the first time in C with t 1 < t m ≤ t 0 . Let (y 0 , s 0 ) be a limit point of (x m , t m ). We claim that (y 0 , s 0 ) ∈ ∂{φ > 0}.

(4.2)

To see this, suppose φ(y 0 , s 0 ) > δ > 0. Then the same holds at (x m , t m ) for large enough m. It follows from the definition of u m that, as m → ∞, mp m → 0 uniformly. This and the fact that φ is a supersolution of the parabolic equation in (1.12) yield a contradiction by a maximum principle argument, since, in view of (1.6), at (x m , t m )

φ t -ν∆φ ≤ u m,t -ν∆u m ≤ mp m ∆u m -mp m ρ m G(p m )νρ m G(p m ) ≤ -νρ m G(0) + o(1) ≤ (φ -ν)G(0) + o(1).
Note that in the third inequality of above computation we have used that mp m ∆u m (x m , t m ) ≤ mp m ∆φ(x m , t m ) and, in view of the regularity of φ in the positive phase, lim m→∞ mp m ∆φ(x m , t m ) = 0. In the last inequality we have used that -νρ m = u mν. Similar arguments yield that the case (y 0 , s 0 ) ∈ ∂{φ > 0}.

For the rest of the proof we consider the subsequence of m's along which the (x m , t m )'s converges to (y 0 , s 0 ). Since φ has C 2,1 -free boundary, there exists a spatial ball B r (z 0 ) ⊂ {φ(•, s 0 ) < 0} which touches the free boundary at y 0 . Now we approximate φ by a radial supersolution ψ of (1.12) in the space-time domain

Σ := {x : r 1 < |x -z 0 | < r 2 } × [s 0 -τ, s 0 ],
for some r 1 < r < r 2 . Roughly speaking the construction amounts to taking Taylor's expansion of φ at (y 0 , s 0 ) in each phase, up to the first order in space and time, and constructing a radial function with them. Let s be the outward normal velocity of the free boundary ∂{φ > 0} at (y 0 , s 0 ), pick ε > 0 small, let r(t) := r + a(ts 0 ) for a = s + ε so that B r(t) (z 0 ) ⊂ {φ(•, t) < 0} for t 1 < t < s 0 with t 1 sufficiently close to s 0 , and, moreover,

φ(x, t 1 ) ≤ -Cε(s 0 -t 1 ) in B r(t1) (z 0 ). (4.3) Now consider ψ = ψ(x, t) in Σ such that            ψ(•, t) = 0 on {|x -z 0 | = r(t)}, -∆ψ -= G(ψ -) in {r 1 ≤ |x -z 0 | ≤ r(t)}, ψ +,t -ν∆ψ + = (ψ + -ν)G(0) in {r(t) ≤ |x -z 0 | ≤ r 2 }, ψ = φ + (z 0 + r 2 ν, t) + ε(r 2 -r(t)) on {|x -z 0 | = r 2 }, ψ = φ -(z 0 + r 1 ν, t) -ε(r(t) -r 1 ) on {|x -z 0 | = r 1 }.
Note that (3.8) and the maximum principle for the elliptic equation in (1.12) yield that φ(•, t 1 ) < ψ(•, t 1 ) in B r(t1) (z 0 ) \ B r1 (z 0 ). The initial datum for ψ in the parabolic phase at t = t 1 is given, for r = |xz 0 |, by (φ(z 0 + r 2 ν, t 1 ) + ε(r 2r(t)))(r 2r(t 1 )) -1 (rr(t 1 )), so that, if r 2 is sufficiently close to r(t 1 ), φ(x, t 1 ) ≤ ψ(x, t 1 ) on B r2 (z 0 ) \ B r(t1) (z 0 ). Also note that, since φ is a smooth strict supersolution of (1.12) with nonzero gradient at (x 0 , t 0 ), if r 1 and r 2 are sufficiently close to r, then |Dψ + | and |Dψ -| are close to φ at (y 0 , s 0 ) up to order ε. Thus we can conclude that ψ is a supersolution of (1.12) in Σ if (t 1 , r 1 , r 2 , ε) is sufficiently close to (s 0 , r, r, 0). Finally note that u m < φ < ψ on the parabolic boundary of Σ with φ = ψ at (y 0 , s 0 ). The construction in the previous section yields a supersolution ψ m of (1.8) which converges uniformly to ψ as m → ∞. By shifting, if needed, ψ down by a small amount, yields that u m crosses ψ from below in the interior of Σ, a contradiction to the comparison principle of (1.1).

Next we show that u and v coincide initiallly. For this we need (1.14) and (1.15). Lemma 4.2. Assume (1.14) and (1.15). Then, as t → 0 + , (a) the sets ∂{u(•, t) > 0} and ∂{v(•, t) > 0} converge uniformly to Ω 0 in the Hausdorff distance, and (b) both u(•, t) and v(•, t) converge in Ω 0 to the harmonic function w in Ω 0 with the zero boundary data, while outside of Ω 0 , they converge to the original initial data (u 0 ) + := ν(1ρ 0 ).

Proof. Assumptions (1.14) and (1.15) imply that, as m → ∞, there exist a sequence of positive numbers ε m → 0 such that with c 0 is chosen small enough so that φ(•, 0) sits below u m (•, 0) in B ε/2 (y 0 ); note that the support of φ(•, 0) equals B ε/2 (y 0 ). It follows from by Lemma A.1 that |Dφ + |(•, a(t)) is sufficiently large for some large M and for small enough t. Choosing such M and t we have |Dφ -| < |Dφ + | on x = a(t), and, thus, φ a subsolution of (1.12) for a small time interval [0, t 0 ]. Using the approximating solutions for the u m -equation constructed in the previous section as well as (4.4), we conclude that v ≥ φ, and, hence, {v(•, t) < 0} is outside of B a(t) (y 0 ) for 0 ≤ t ≤ t 0 . Since x 0 was arbitrarily and Ω ε is compact, we find {v(•, t) < 0} ⊂ Ω ε for t ≤ t 0 = t 0 (ε).

Ω 0 = {ρ 0 = 1} ⊂ {u m (•, 0) ≤ 0} ⊂ {ρ 0 > 1 -ε m }, (4.4
Letting ε → 0 gives lim sup where h(x, t) solves a heat equation in (R n -B r (x 0 )) × [0, 1] with initial datum (u 0 ) + and zero boundary condition on ∂B r (x 0 ). Since (u 0 ) + = 0 in B 2r (x 0 ), it follows from the heat kernel estimate that, for small time, |Dh|(•, t) ≤ t -3/2 e -r 2 /t ≤ |Df |(•, t) on ∂B r (x 0 ).

Thus g is a supersolution of (1.8) in R n × (0, t 0 ) for sufficiently small t 0 depending on r, and, hence, for 0 ≤ t ≤ t 0 , B r (x 0 ) ⊂ {u(•, t) < 0}. Since r is arbitrarily, (4.6) follows.

In view of the facts that v ≤ u and {u(•, t) < 0} ⊂ {v(•, t) < 0}, the two inclusions above yield (a). In view of (a), there exist δ = δ(ε) > 0, t 0 = t 0 (ε) > 0 and M such that for m > M and 0 ≤ t ≤ t 0 the following holds: u m ≥ δ on ∂Ω g , u m ≤ -δ in Ω f , and moreover Letting m → ∞ and using arbitrarily small ε > 0, we conclude that the (u m ) -'s converge uniformly to the solution of the elliptic equation Ω 0 with zero boundary data. Similar arguments apply to (u m ) + .

To derive the convergence result for u m 's using the previous lemmata, we need to first show that, as m → ∞, the ν(1ρ m )'s stay nonnegative. The claim follows after sending ε to zero.

Figure 1 .

 1 Figure1. Snapshots of the evolution of the density ⇢ m (blue solid line) and pressure p m (black discontinuous line). The parameters are m = 20, ⌫ = .5. These figures illustrate how the pressure profile is building up when density approaches one. Between the last two pictures, the density has continuously reached ⇢ = 1 in the center, while p has jumped discontinuously.

Figure 1 .

 1 Figure1. Snapshots of the evolution of the density ρ m (blue solid line) and pressure p m (black discontinuous line). The parameters are m = 20, ν = .5. These figures illustrate how the pressure profile is building up when density approaches one. Between the last two pictures, the density has continuously reached ρ = 1 in the center, while p has jumped discontinuously.

Figure 2 .

 2 Figure 2. Snapshots of the evolution of the density ⇢ m (blue solid line) and pressure p m (black discontinuous line). The parameters are m = 20, ⌫ = 0 (not treated in this paper). Compared to Figure 1, we observe also a discontinuous behaviour of the density.

Figure 2 .

 2 Figure 2. Snapshots of the evolution of the density ρ m (blue solid line) and pressure p m (black discontinuous line). The parameters are m = 20, ν = 0 (not treated in this paper). Compared to Figure 1, we observe also a discontinuous behaviour of the density.

  and some τ ∈ (0, ∞) and ∂ P E := Ē \ E is the parabolic boundary of E. Whenever we refer to a parabolic neighborhood, we assume U and τ are known. For an open subset A of R n ×R and a continuous function v : Ā → R, where Ā is the closure of A, {v > 0}, {v = 0} and {v < 0} are respectively the sets {y ∈

  in {r > a(t)}, with -u m,1 and u m,2 solving, in the respective region, u m,t -(mp m + ν)∆u m ≥ -ρ(mp m + ν)G(p m ), (3.13) and |Du m,1 | > |Du m,2 | on r = a(t).

  Let u := lim sup * u m and v := lim inf * u m .(4.1)

  ) and, in particular, the sets {u m (•, 0) ≤ 0}'s converge to Ω 0 in the Hausdorff distance. We prove (a) first. Fix ε > 0, x 0 ∈ Ω ε := {x : d(x, Ω 0 ) ≤ ε} and let B ε (y 0 ) be the ball which touches ∂Ω 0 at x 0 from outside of Ω 0 . With a(t) := ε/2 -M t, we consider the radial function φ given by           -∆φ -= G(φ -) in {x : a(t) ≤ |xy 0 | ≤ (C + 1)ε}, φ -(•, t) = M 0 on {|xy 0 | = (C + 1)ε}, φ(•, t) = 0 on {|xy 0 | = a(t)}, φ t -ν∆φ = (φν)G(0) in {x : |xy 0 | ≤ a(t)}, φ(•, 0) = c 0 |xy 0 | 2c 0 ε 2 /4 in the parabolic phase,

Fix

  B 2r (x 0 ) ⊂ Ω 0 , let f (x, t) := t(|xx 0 | 2r 2) and denote by p f the pressure function corresponding to f . Then, as long as r is sufficiently small,f t -(mp f + ν)∆f = (|xx 0 | 2r 2 ) + 2nA(mp f + ν)t|x| 2 ≥ -νG(p f ) in B r (x 0 ) × [0, 1], Let g(x, t) := f (x, t)χ |x-x0|≤r + h(x, t)χ |x-x0|≥r ,

  Now we prove (b). Fix ε > 0 and defineΩ f := {x : d(x, R n \ Ω 0 ) > ε} and Ω g := {x : d(x, Ω 0 ) ≤ ε}.

-

  ∆f = G(f )ε in Ω f and f = δ on ∂Ω f , and -∆g = G(g) + ε in Ω g and g = δ on ∂Ω g . Let φ(x, t) := -a(t)f (x) and ψ(x, t) := -b(t)g(x), where a(t) := min[-δe mεt , 1] and b(t) := max[δ -1 e -mεδt , 1]. Direct calculations then yield that, for sufficiently large m and any choice of t 0 > 0, φ and ψ are respectively supersolution and subsoluton to the u m -equation in Ω f × (0, t 0 ] and Ω g × (0, t 0 ]. Thus the comparison principle for the u m -equation and the choice of δ and t 0 yield ψ ≤ u m in Ω g × [0, t 0 ] and u m ≤ φ in Ω f × [0, t 0 ].

  Lemma 4.3. Assume (1.2), (1.3) and (1.5). Thenlim sup m→∞ p m (x, t) ≤ p M in Q T ,and, hence, lim supm→∞ ρ m ≤ 1 in Q T . Proof. Fix ε > 0 and let C := max[p m (•, 0), p M + ε]. Then G(p M + ε) = -δ < 0 for some δ > 0. Thus f (x, t) := max[C -(m -1)Cδt, p M + ε] is a supersolution of (1.7) and p m ≤ f in Q T , if p m ≤ f on the parabolic boundary of Q T . At t =0 this is guaranteed from the definition of C, and on the lateral boundary it follows from (1.3). Since f = p M + ε after t ≥ O( 1 m ), we conclude that lim sup m→∞ p m (x, t) ≤ p M + ε for all (x, t) ∈ Q T .

  Theorem 1.1. Let ρ m solve (1.1) with data satisfying (1.3), (1.14), (1.15), and (1.16), if Ω = R n , and define u 0

We are now ready to show the main convergence result.

Corollary 4.4. Let u and v be as in (4.1). Then:

(a) u = v * is the unique viscosity solution w of (1.12) with the initial data u 0 .

(b) As m → ∞, the ν(1ρ m )'s and ρ m 's converge uniformly respectively to b(w) and 1ν -1 b(w) in Q T .

Proof. It follows from Theorem 4.1 and Lemma 4.2 that u is a subsolution and v is a supersolution of (1.12) with initial data u 0 . Let w be the unique viscosity solution of (1.12) with initial data u 0 given in Theorem 2.5. Then u ≤ w and w * ≤ v, and, hence, To prove the first inequality in the claim above, we observe that

To show the second inequality, suppose that, for some (x 0 , t 0 ) ∈ Q T , lim sup * ν(1ρ m )(x 0 , t 0 ) > 0, since otherwise there is nothing to show. In this case, along a subsequence of m → ∞, we have lim inf * ρ m (x 0 , t 0 ) < c 0 < 1 and, thus, lim inf * p m (x 0 , t 0 ) = 0. This yields that

and the conclusion follows.

The uniform convergence of ρ m , or Corollary 4.4 seem to be the best one can do without further restriction on the initial data, even for the short time interval. In particular, when u 0 is a (strict) supersolution of the elliptic equation in (1.12), then the u m 's converge to the solution of (1.12) with the initial data strictly smaller than u 0 , that is one should expect, for sufficiently large m, almost discontinuous decrease of the pressure profile at the initial time for p m .

Finite speed of propagation

We derive some properties of the speed of propagation for both the negative and positive phases under certain assumptions. We know that the negative phase can nucleate in the interior of the positive phase, when u decreases from a positive value to become zero in finite time. We show here a quantitative version of this phenomena. Indeed when u becomes very small in an open neighborhood in the positive phase, then it turns into the negative phase in short amount of time.

Lemma 5.1. Suppose u(•, t 0 ) ≤ ε in Br (x 0 ) with 0 ≤ ε < r 2 . There exists a dimensional constant

In particular, the interior of {u * (•, t) ≤ 0} coincides with {u * (•, t) < 0}.

Proof. Note that any solution to the parabolic equation

which stays positive, serves as a supersolution to (1.12). This fact and the parabolic Harnack inequality yield

. It is immediate that f is a supersolution to (1.12) and thus u ≤ f . In particular,

The following two local estimates quantify the finite propagation property. First we show that the negative phase (the tumor) does not shrink too fast over time. Note that it can shrink at least temporarily, when the positive part of u has steep growth near the interface. In terms of the original density variable ρ this means that its is close to zero near the tumor boundary.

Lemma 5.2. [Finite-speed Shrinkage] Suppose u ≤ 1 in Q T . The negative phase {u(•, t) < 0} shrinks at most by t 2/5 over a time period t. More precisely, there exits c 0 > 0 depending only on n, such that, whenever B r (x 0 ) ⊂ {u(•, t 0 ) < 0} with 0 < r < c 0 and t 0 ≥ 0, then

(5.2)

Let ψ be the solution to

Using the properties of the heat kernel and scaling arguments we find, for some dimensional C > 0,

and it follows that |Dψ| ≤ 1 r e -1/ √ r on the lateral boundary of Σ.

Hence, if r is small, φ = φ +φ -defined by

is a supersolution of (1.12) in R n × [t 0 , t 0 + r 5/2 ] and the conclusion follows by comparing u with φ.

Due to the possible nucleation of the negative phase, it is not feasible to estimate the speed of expansion of the negative phase without knowing the positive density distribution near a given point.

The following lemma states that, if (x 0 , t 0 ) is on the interface and u + is nondegenerate near it, then the negative phase cannot expand too fast.

Lemma 5.3. [Expansion bound] Suppose u(•, t 0 ) > 0 in B 3r (x 0 ) with u(•, t 0 ) > r in B r (x 0 ). Then, for r ≤ r 0 where r 0 depends only on n, we have u > 0 in B r/2 (x 0 ) × [t 0 , t 0 + r 3 ].

Proof. Without loss of generality we set x 0 = 0. Comparing u with a fundamental solution of the heat equation with initial data -χ |x|>3r as a subsolution to (1.12), we find, for some C > 0 which again depends only on n, u -(•, t) ≤ C 0 r on ∂B 2r (0) × [t 0 , t 0 + r 3 ]. Next we construct a subsolution to (1.12) in B 2r × [t 0 , t 0 + r 3 ] by solving, for r(t) := r -√ tt 0 , the initial-boundary value problem

The parabolic Harnack inequality for the heat equation gives

We can now conclude by comparing u with φ in B 2r (0) × [t 0 , t 0 + r 3 ].

Appendix A. Discussion on the proof of the comparison principle in [KP].

For fixed r > 0, we consider the open balls

Finally, we define the domain Ξ r (x, t) that is used in the definition of regularizations of solutions by

where + is the Minkowski sum. Note that near its top portion at t = r, Ξ(x, t) shrinks its spatial radius with the same rate. Finally, when (x, t) = (0, 0) we omit and we simply write Ξ r .

We write next a simple version of the results stated in Lemma 3.13 and Lemma 3.16 of [KP] .

Lemma A.1. (a) Let u ≥ 0 solve u t -ν∆u ≤ 0 in a parabolic neighborhood of Ξ r for some r ∈ (0, 1] and assume that u = 0 on Ξ r . There exists ε > 0 and g(s

The first part of the lemma states that, if the support of v is Ξ, then the spatial gradient of v(•, t) grows to infinity on the lateral boundary of Ξ as t increases to r. The second part implies that, if the support of v is outside of Ξ, then the spatial gradient of v(•, t) vanishes on the lateral boundary of Ξ as t increases to r. Now we touch on the proof of the comparison principle. Since it parallels that of Theorem 3.1 in [KP], we only sketch it pointing out differences. We consider the regularized sub-and supersolutions

and argue by contradiction assuming that there is a finite first crossing time t 0 defined by

It follows that there is a contact point at t = t 0 between the free boundary of Z and W .

Lemma A.2. We have:

Proof. First note that, in view of Lemma 5.2, {u(•, t 0 ) ≥ 0} can not expand discontinuously. The definition of Z then yields that, if x 0 lies in the interior of {Z(•, t 0 ) ≥ 0}, then Z ≥ 0 in B r (x 0 ) × [t 0t, t 0 ] for some sufficiently small r and t. It then follows from the definition of t 0 and the strong maximum principle for the heat equation that {W (•, t 0 ) > 0} in the interior of {Z(•, t 0 ) ≥ 0} and the claim follows.

The next lemma corresponds to Lemma 3.19 in [KP]. Once it is shown, the rest of the proof is the same as in [KP]. To state it and sketch the proof, it is necessary to remind the reader the geometric properties of Z and W as in [KP]. By definition, for any y 0 ∈ ∂{Z(•, t 0 ) ≥ 0} there exists a "cylinder" Ξ Z = Ξ r (z 1 , s 1 ) in {Z ≥ 0} such that (y 0 , t 0 ) lies on the lateral boundary of Ξ Z . On the other hand, for any y 0 ∂{W (•, t 0 ) < 0} there exists an exterior "cylinder" Ξ W = Ξ r (z 2 , s 2 ) in {W < 0} such that (y 0 , t 0 ) lies on the lateral boundary of Ξ W .

Lemma A.3. Fix y 0 ∈ ∂{Z(•, t 0 ) ≥ 0} ∩ ∂{W (•, t 0 ) ≥ 0}, and let Ξ Z and Ξ W be as given above. Then t 0r < s 1 , s 2 < t 0 + r.

Proof. It follows from the definitions that {W (•, t 0 ) ≤ 0} has an interior ball of radius r which touches y 0 on its boundary. This fact and Lemma 5.2 yield that s 1 < t 0 + r. Since Z ≤ W prior to t = t 0 , we also have s 2 > t 0r. If we show that s 2 < t 0 + r, then the ordering between Z and W implies that s 1 > t 0r and, hence, the conclusion. Since s 1 < t 0 + r, Σ := Ξ Z ∩ {t < t 0 } is nonempty and Z ≥ 0 in Σ. Let B := Ξ Z ∩ {t = t 0 }. The strong maximum principle as well as Hopf's lemma imply that W (•, t 0 ) > 0 in B with nonzero gradient on the boundary of B. It then follows from Lemma 5.3 that the negative phase of W cannot expand too fast, that is s 2 < t 0 + r.


Jean-Gabriel Luque
email: jean-gabriel.luque@univ-mlv.fr

Hyperdeterminants on semilattices

Keywords: Hyperdeterminants, meet semilattices, GCD-matrices, multiplicative functions AMS: 15A15, 15A69, 06A12

We compute hyperdeterminants of hypermatrices whose indices belongs in a meet-semilattice and whose entries depend only of the greatest lower bound of the indices. One shows that an elementary expansion of such a polynomial allows to generalize a theorem of Lindström to higher-dimensional determinants. And we gave as an application generalizations of some results due to Lehmer, Li and Haukkanen.

Introduction

Since the end of the nineteen century, it is known that some determinants, with entries depending only of the gcd of the indices, factorize. Readers interested in the story of the problem can refer to [START_REF] Krattenthaler | Advanced determinant calculus: a complement[END_REF] and [START_REF] Altinisik | GCD matrices, posets and nonintersecting paths[END_REF]. In 1876, Smith [START_REF] Smith | On the value of certain arithmetical determinant[END_REF] evaluated the determinant of a GCD matrix whose entries belong to a factor closed set (i.e., a set which contains all the factors of its elements) as a product of Euler's totient. The interest of this computation lies in its links with arithmetic functions [START_REF] Apostol | Introduction to Analytic Number Theory[END_REF] and in particular multiplicative functions (see [START_REF] Lascoux | Addition of 1, Séminaire Lotharingien[END_REF][START_REF] Lascoux | Multiplicative functions[END_REF] for interesting remarks about the last notion). During the last century, many generalizations of Smith's theorem have been investigated. One of the 1 ways to extend his result consists in changing the set of the indices of the matrices. In 1990, Li [START_REF] Li | The determinants of gcd matrices[END_REF] gave the value of GCD determinant for an arbitrary set of indices. Beslin and Ligh [START_REF] Beslin | Another generalization of Smith's determinant[END_REF] shown that such a determinant factorizes when the indices belongs to a gcd-closed set (i.e., a set which contains the gcd of any pairs of its elements) as a product of certain functions evaluated in terms of Euler's totient. The fact that these determinants factorize can be seen as a corollary of a very elegant theorem due to Lindström [START_REF] Lindström | Determinants on semilattices[END_REF] which evaluated the determinant of the GCD-matrix whose indices are chosen in a meet semilattice, i.e., a poset such that each pair admits a greatest lower bound. Another way to generalize Smith's result consists in computing multidimensional analogous. In 1930 Lehmer gave [START_REF] Lehmer | The p dimensional analogue of Smith's determinant[END_REF] the first multi-indexed version of Smith's determinant. Other related computation are collected in [START_REF] Sokolov | Spatial matrices and their applications[END_REF][START_REF] Sokolov | Introduction to the theory of multidimensional matrices[END_REF]. More recently, Haukkanen [START_REF] Haukkanen | Higher-Dimensionnal GCD matrices[END_REF] generalized the results of Beslin and Ligh [START_REF] Beslin | Another generalization of Smith's determinant[END_REF] and Li [START_REF] Li | The determinants of gcd matrices[END_REF]) to hyperdeterminants.

We will see in Section 2 that the main trick for computing these multidimensional determinants consists in expanding it as a sum of (classical 2-way) determinants. In the aim to highlight this method, we apply it to a more general object Det F . In Section 3, we recall shortly a classical technic and give a slight generalization of Lindström's Theorem. As a consequence, we give a multidimensional analogue of Lindström's Theorem. In Section 4, one investigates minors of meet hypermatrices and generalizes two theorems due to Haukkanen [START_REF] Haukkanen | Higher-Dimensionnal GCD matrices[END_REF].

Hyperdeterminants and F-determinants

The question of extending the notion of determinant to higher dimensional arrays has been raised by Cayley [START_REF] Cayley | Mémoire sur les hyperdéterminants[END_REF][START_REF] Cayley | On the theory of permutants[END_REF] few after he introduced the modern notation as square arrays [START_REF] Cayley | On the theory of determinants[END_REF]. The simplest generalization is defined for a kth order tensor on an n-dimensional space M

= (M i 1 ,••• ,i k ) 1≤i 1 ,••• ,i k ≤n by the alternated sum DetM = 1 n! σ=(σ 1 ,••• ,σ k )∈S k n sign(σ)M σ ,
where sign(σ) = sign(σ 1 )

• • • sign(σ k ), M σ = M σ 1 (1)...σ k (1) • • • M σ 1 (n)...σ k (n) and
S n is the symmetric group. A straightforward computation gives DetM = 0 if k is odd.

For any k (even if k is odd), one defines the polynomial

Det 1 M = σ=(Id,σ 2 ,••• ,σ k )∈S k n sign(σ)M σ .
When k is even the two notions coincide but for k odd, only Det 1 does not vanish. This is a special case of the "less-than-full-sign" determinant theory due to Rice [START_REF] Rice | P-way determinants with an application to transvectants[END_REF].

Let us denote by F a map from S k-2 n to a commutative ring. One defines a more general object, which will be called F-determinant of M by

Det F (M) = σ=(σ 2 ,••• ,σ k )∈S k-1 n sign(σ 2 )F(σ 3 , . . . , σ k ) i M iσ 2 (i)...σ k (i) .
It exists an elementary identity which consists in expanding the F-determinant as a sum of (n!) k-2 classical (2-way) determinants.

Lemma 2.1 (Determinantal expansion) One has

Det

F M = σ 3 ,...,σ k F(σ 3 , . . . , σ k ) det(M σ 3 ,...,σ k ),
where M σ 3 ,...,σ k denotes the n×n matrix such that M σ 3 ,...,σ k i,j

= M i,j,σ 3 (i),...,σ k (i) .
Proof -It suffices to remark that

Det F M = σ 3 ,...,σ k F(σ 3 , . . . , σ k ) σ 2 sign(σ 2 ) i M iσ 2 (i)σ 3 (i)•••σ k (i)
One of the most important property of hyperdeterminants is the invariance under the action of k copies of the special linear group. It is a very classical result which can be recover as a straightforward consequence of the following proposition.

Proposition 2.2

The polynomial Det F M is invariant under the action of linear group on M in the following sense

Det F g.M = det gDet F M, (1)
where

g.M = 1≤j 2 ≤n g i 2 j 2 M i 1 ,j 2 ,i 3 ••• ,i k 1≤i 1 ,••• ,i k ≤n .
Proof -By applying Lemma 2.1 to g.M, one gets

Det F g.M = σ 3 ,...,σ k F(σ 3 , . . . , σ k ) det(g.M σ 3 ,...,σ k ) = σ 3 ,...,σ k F(σ 3 , . . . , σ k ) det g det(M σ 3 ,...,σ k ) = det gDet F M.
3 Hyperdeterminants of meet hypermatrices

Meet semilattice

Consider a partially ordered finite set L so that every pairs (x, y) ∈ L 2 has a greatest lower bound denoted by x ∧ y. Such a poset is called a meet semilattice. One defines classically its ζ function by

ζ(x, y) = 1 if x ≤ y, 0 otherwise.
Its Möbius function is the inverse of the zeta function and can be computed by the induction

µ(x, y) =    1 if x = y, -x≤z<y µ(z, y) if x < y, 0
in the other cases.

If F and f verify the equality

F (x) = y≤x f (y) = y∈L ζ(y, x)f (y), (2)
then, one has

f (x) = y∈L µ(y, x)F (y) = (F ⋆ µ)(x) (3)
where the symbol ⋆ means the convolution product.

Lindström Theorem

The factorization properties of the GCD determinants are the consequence of the semilattice structure of the integers with respect to divisibility and can be stated in a more general way. The manipulations of the identities ( 2) and ( 3) are the keys of the proof of Lindström's Theorem [START_REF] Lindström | Determinants on semilattices[END_REF]. We recall its proof in a very slightly more general version.

For each x ∈ L, one considers a fixed element z x ≤ x. Let F x be a function from L to C (or more generally to a commutative ring). Let M be the matrix defined by M = (F x (z x ∧ y)) x,y∈L . Remark that it suffices to define F x (z) only when z ≤ x. In particular, one can suppose that F x (z) = F (z, x) is an incidence function, i.e., F (x, z) = 0 unless x ≤ z. One has

F x (z x ∧ y) = z∈L ζ(z, z x )ζ(z, y)f x (z), where f x (z) = y∈L µ(y, z)F x (y).
Φ = x ζ(x, z x )f x (x) =    x f x (y) if z x = x for each x, 0 otherwise. ( 4
)
Then one obtains Lindström's Theorem.

Theorem 3.1 (Lindström [START_REF] Lindström | Determinants on semilattices[END_REF])

det (F x (z x ∧ y)) x,y∈L = x f x (x) if z x = x for each x, 0 otherwise. (5)
Note that, the original Lindstöm 's Theorem deals with the case where z x = x for each x. Furthermore, equality (4) generalizes a lemma of Cesaro.

Lemma 3.2 (Cesaro) Denote by gcd m (n) = gcd(m, n). One has, (µ * (f • gcd m ))(n) = (f * µ)(n) if m = n, 0 otherwise, ( 6
)
where * is the Dirichlet convolution and • is the composition of functions.

Linström's Theorem for F-determinants

Lindstöm's Theorem can be extended to F-determinants.

Theorem 3.3 (Lindström's theorem for F-determinants) If L = {x 1 , . . . , x n } denotes a meet semilattice, one has

Det F F x i 1 (z x i 1 ∧ • • • ∧ x i k ) =    F(Id, • • • , Id) x f x (x) if z x = x for each x 0 otherwise Proof -Lemma 2.1 gives Det F (F x 1 (z x 1 ∧ • • • ∧ x k )) x 1 ,••• ,x k ∈L = σ 3 ,...,σ k F(σ 3 , . . . , σ k ) det F x i (z x i ∧ x j ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) i,j
From Linström's Theorem (Theorem 3.1), one has det

F x i (z x i ∧ x j ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) = 0
if and only if for each x i one has

z x i ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) = x i .
Equivalently,

z x i ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) ≥ x i ,
for eacu i. Hence, σ 3 = • • • = σ k = Id and z x i = x i for each i. The result follows.

Example 3.4 Consider the semilattice constituted with two elements 2 ≥ 1.

The expansion of the F-determinant gives

Det F (F i (i ∧ j ∧ k ∧ l)) = F(12, 12) F 1 (1) F 1 (1) F 2 (1) F 2 (2) + (F(12, 21) + F(21, 12) + F(21, 21)) F 1 (1) F 1 (1) F 2 (1) F 2 (1) = F(12, 12)F 1 (1)(F 2 (2) -F 2 (1)) = F(12, 12)f 1 (2)f 2 (2)
4 Minors of meet Hypermatrices

Meet closed subsets

Consider a meet closed subset S of L (i.e., a subset closed under the operation ∧) and fix a linear extension l = y 1 . . . y n of S. Following the notations of [START_REF] Altinisik | GCD matrices, posets and nonintersecting paths[END_REF], we denote by x y i the relation x ≤ y i and x ≤ y j for each j < i. Consider a pair of functions f and F verifying

F (y i ) = x≤y i x∈L f (x) (7)
and set f(y i ) = x y i f (x). One has the following lemma.

Lemma 4.1 F (y i ) = y k ≤y i f (y k ). (8)
Proof -Remarking that for each x ∈ L, it exists i such that x y i (it suffices to set y i = min{j|x ≤ y j } and that x y i and x y j implies i = j, we have

F (y i ) = x≤y i x∈L f (x) = y k ≤y i x y k f (x) = y k ≤y i f (y k ).
Note that this identity appears in [START_REF] Altinisik | GCD matrices, posets and nonintersecting paths[END_REF] (Theorem 4.1, p 7). Hence, using Theorem 3.

Det F (F y i 1 (y i 1 ∧ • • • ∧ y i k )) = F(Id, • • • , Id) n i=1 x 1 y i x 2 ∈L µ(x 1 , x 2 )F y i (x 2 ) .
Example 4.3 Consider the semilattice L given by its Hasse diagram

L = 4 5 ↑ ր ↑ 2 3 ↑ ր 1
where i → j means i ≤ j. The sublattice generated by 2, 4 and 5,

S = 4 5 ↑ ր 2 is meet closed and f2 (2) = f 2 (2) + f 2 (1) f4 (4) = f 4 (4) f5 (5) = f 5 (5) + f 5 (3).
Hence,

Det F (F i (i ∧ j ∧ k)) i,j,k∈S = F(123) F 2 (2) F 2 (2) F 2 (2) F 4 (2) F 4 (4) F 4 (2) F 5 (2) F 5 (2) F 5 (5) +F(213) F 2 (2) F 2 (2) F 2 (2) F 4 (2) F 4 (2) F 4 (2) F 5 (2) F 5 (2) F 5 (5) +F(321) F 2 (2) F 2 (2) F 2 (2) F 4 (2) F 4 (2) F 4 (2) F 5 (2) F 5 (2) F 5 (2) +(F(132) + F(231) + F(312)) F 2 (2) F 2 (2) F 2 (2) F 4 (2) F 4 (2) F 4 (2) F 5 (2) F 5 (2) F 4 (2)
The permutation 123 is the only one having a non zero contribution in this sum. Hence,

Det F (F i (i ∧ j ∧ k)) i,j,k∈S = F(123)F 2 (2)(F 4 (4) -F 4 (2))(F 5 (5) -F 5 (2)) = F(123)(f 2 (2) + f 2 (1))f 4 (4)(f 5 (5) + f 5 (3)) = F(123) f2 (2) f4 (4) f5 (5)
Remark 4.4 If L is the semilattice structure of the integers with respect to divisibility. By the specialization F(σ 3 , . . . , σ k ) = sign(σ 3 ) . . . sign(σ k ), one recovers the computations of Lehmer [START_REF] Lehmer | The p dimensional analogue of Smith's determinant[END_REF] as a special case of Theorem 3.3 and the result of Haukkanen ([12] Theorem 1. p 56) from Corollary 4.2.

Factor closed subsets

Let S be a factor closed subset of L. Then, f = f and

Det F (F x 1 (x 1 ∧ • • • ∧ x k )) x 1 ,...,x k ∈S = F(Id, • • • , Id) x f x (x). (9)
As special cases of equality [START_REF] Lehmer | The p dimensional analogue of Smith's determinant[END_REF], one recovers Lehmer's identities [START_REF] Lehmer | The p dimensional analogue of Smith's determinant[END_REF] and the original result of Smith [START_REF] Smith | On the value of certain arithmetical determinant[END_REF].

General case

Let X = {x 1 , . . . , x n } be a subposet of a meet semi-lattice L. We will denote by X = {x 1 , . . . , x n , x n+1 , . . . , x m } the smallest factor-closed subset of L containing X. The aim of this section consists in investigating the Fdeterminant

DF (X) := Det F F x i 1 (z x i 1 ∧ x i 2 ∧ • • • ∧ x i k ) 1≤i 1 ,...,i k ≤n (10)
where for each x ∈ X, z x denotes a fixed element of X such that z ≤ x. As in the previous section, the result follows from the case k = 2 and Proposition 2.1.

Let us consider first the determinant det(F x i (z x i ∧ x j )) 1≤i,j≤n . The functions F x i can be chosen such that F (x, y) is an incidence function. The set X being closed by factors, the functions f x and fx are equal. Hence,

F x i (z x i ∧ x j ) = n+m k=1 C x i ,x k ζ(x k , x j ) (11)
where

C x,y = f x (y)ζ(y, z x ) = f x (y) if y ≤ z x 0 otherwise. ( 12
)
One has Proposition 4.5

det(F x i (x i ∧ x j )) = 1≤k 1 <•••<kn≤n+m det C x i ,x k j 1≤i,j≤n det (ζ(x k i , x j )) 1≤i,j≤n . (13)
Proof -By multi-linearity, one obtains

det(F x i (x i ∧ x j )) = det n+m k=1 C x i ,x k ζ(x k , x j ) = 1≤k 1 ,...,kn≤n+m det C x i ,x k j ζ(x k j , x j ) = 1≤k 1 ,...,kn≤n+m det C x i ,x k j i ζ(x k i , x i ) = 1≤k 1 <•••<kn≤n+m det C x i ,x k j 1≤i,j≤n det (ζ(x k i , x j )) 1≤i,j≤n
O O ] ] X X X X X X X X 3 ] ] X X X X X X X X O O 0 A A Ô Ô Ô Ô Ô Ô Ô Ô ] ] X X X X X X X X O O
and X = {4, 5, 6}. We set z 4 = 1, z 5 = 2 and z 6 = 6 . Consider the determinant

det(F i (z i ∧ j ∧ k)) i,j,k∈X = f 4 (1) + f 4 (0) f 4 (0) f 4 (1) + f 4 (0) f 5 (2) + f 5 (0) f 5 (2) + f 5 (0) f 5 (0) f 6 (1) + f 6 (0) f 6 (3) + f 6 (0) f 6 (6) + f 6 (3) + f 6 (1) + f 6 (0) = -f 4 (0)f 6 (1)f 5 (2) + f 4 (1)f 6 0 f 5 (2) + f 4 (1)f 5 (0)f 6 (3) +f 4 1 f 5 (0)f 6 (6) + f 4 (0)f 5 (2)f 6 3 + 2 f 4 (1)f 5 (2)f 6 (3) +f 4 (1)f 5 (2)f 6 (6)
Using the multilinearity of det, one recovers the expression given by Propo-sition 4.5

det(F i (z i ∧ j)) = f 4 (0) f 4 (1) 0 f 5 (0) 0 f 5 (2) f 6 (0) f 6 (1) 0 • 1 1 0 1 0 1 1 1 0 + f 4 (0) f 4 (1 ) 0 f 5 (0) 0 0 f 6 (0) f 6 (1) f 6 (3) • 1 1 0 1 0 0 1 1 1 + f 4 (0) f 4 (1) 0 f 5 (0) 0 0 f 6 (0) f 6 (1) f 6 (6) • 1 1 0 1 0 0 1 1 1 + f 4 (0) 0 0 f 5 (0) f 5 (2) 0 f 6 (0) 0 f 6 (3) • 1 0 0 1 1 0 1 0 1 + f 4 (1) 0 0 0 f 5 (2) 0 f 6 (1) 0 f 6 (3) • 1 0 0 0 1 0 1 0 1 + f 4 (1) 0 0 0 f 5 (2) 0 f 6 (1) 0 f 6 (6) • 1 0 0 0 1 0 1 0 1
More generally, one has a multi-indexed version of Proposition 4.5.

Theorem 4.7

DF (X) = 1≤k 1 <•••<kn≤n+m Det F f x i 1 (x k i 2 )ζ(x k i 2 , z x i 1 ∧ x i 3 ∧ • • • ∧ x i k ) × × det (ζ(x k i , x j )) (14)
Proof -We use Lemma 2.1 to expand DF (X) and we obtain

DF (X) = σ 1 ,...,σ k F(σ 1 , . . . , σ k ) det F x i (z x i ∧ x j ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) ) .
Now, by Proposition 4.5, one gets

DF (X) = σ 1 ,...,σ k F(σ 1 , . . . , σ k ) 1≤k 1 <•••<kn≤n+m det (ζ(x k i , x j )) × × det f x i (x k j ζ(x k j , z x i ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) )) = 1≤k 1 <•••<kn≤n+m det (ζ(x k i , x j )) σ 1 ,...,σ k F(σ 1 , . . . , σ k )× × det f x i (x k j ζ(x k j , z x i ∧ x σ 3 (i) ∧ • • • ∧ x σ k (i) )) = 1≤k 1 <•••<kn≤n+m Det F f x i 1 (x k i 2 )ζ(x k i 2 , z x i 1 ∧ x i 3 ∧ • • • ∧ x i k ) × × det (ζ(x k i , x j ))
This ends the proof.

Det F F (z x i 1 ∧ x i 2 ∧ • • • ∧ x i k ) = 1≤k 1 <•••<kn≤n+m i f (x k i )× ×Det F ζ(x k i 2 , z x i 1 ∧ x i 3 ∧ • • • ∧ x i k ) det (ζ(x k i , x j ))
Proof -By applying the equality,

Det F f (x k i 1 )ζ(x k i 2 , z x i 1 ∧ x i 3 ∧ • • • ∧ x i k ) = σ 3 ,...,σ k F(σ 3 , • • • , σ k ) det f (x k i )ζ(x k j , z x i ∧ x σ 1 (i) ∧ • • • ∧ x σ k (i) ) = i f (x k i )Det F ζ(x k i 2 , z x i 1 ∧ x i 3 ∧ • • • ∧ x i k ) .
to identity 14, one obtains the result. Remark 4.9 Assume that L is the integer lattice. Then, if z x = x for each x ∈ X and F(σ 3 , • • • , σ k ) = sign(σ 3 ) . . . sign(σ k ) in Corollary 4.8, one recovers Theorem 2 in [START_REF] Haukkanen | Higher-Dimensionnal GCD matrices[END_REF]. Moreover, Proposition 4.5 generalizes the result of Li [START_REF] Li | The determinants of gcd matrices[END_REF].

  Hence, det M factorizes as the product det M = det Φ. det Z, where Φ = (ζ(y, z x )f x (y)) x,y∈L and Z = (ζ(x, y)) x,y∈L . As Φ and Z are triangular, det Z = 1 and det

  3 and Lemma 4.1, one generalizes a result by Altinisik, Sagan and Tuglu ([2], Theorem 4.1 p 7). Corollary 4.2

Example 4 . 6

 46 Let us consider the semilattice L whose Hasse diagram is

Let set F x 1 =

 1 F x 2 = • • • = F xn = F then for 1 ≤ k 1 , k 2 , . . . , k n ≤ n, we getCorollary 4.8