(1)Weak solutions of equations of the type of non-stationary filtration D

Weak solutions of equations of the type of non-stationary filtration

D. Andreucci¹ E. DiBenedetto²³

Appeared in Nonlinear Analysis TMA, 19 (1992), pp.29–41.

Copyright Elsevier, Oxford UK.

1Dipartimento di Matematica U. Dini, Universit`a di Firenze, v.le Morgagni 67/a 50134 Firenze, Italy.

2Department of Mathematics, Northwestern University, Evanston, Ill. 60208 USA.

3Partially supported by NSF Grant DMS 8802883.

1. Introduction.

Consider the filtration equation

(1.1) u_t = ∆ϕ(u), in D⁰(G) ,

where G is a domain in R^N+1 and ϕ : R⁺ → R⁺ is locally absolutely continuous, non decreasing, ϕ(0) = 0, ϕ(1) = 1 and

(1.2) ∃Λ > 2, such that 1 + 1

Λ ≤ sϕ⁰(s)

ϕ(s) ≤ Λ, a.e. s ≥ 0.

Theorem. Every local non-negative distributional solution u ∈ L¹_loc(G) such that ϕ(u) ∈ L¹_loc(G), is locally H¨older continuous in G.

The proof consists in approximating u by locally bounded solutions of (1.1) to which the results of [4], [5] can be applied.

When ϕ(s) = s^m, m > 1, the result is due to Dahlberg–Kenig [3]. Far more than extending Dahlberg–Kenig theorem to general non-linearities ϕ(·), we are interested in giving a different proof of their result which we believe is simpler. Our approach avoids the concept of “traces”, does not require energy estimates and dispenses with the boundedness of Pierre potentials [9].

If f ∈ C_o^∞(R^N), we have

(1.3) Z

R^N

∆f (y)

|x − y|^N−1dy = − lim

σ→0

Z

|x−y|>σ

(N − 1)fyi(y)(xi− yi)

|x − y|^N+1 dy≡ −

N

X

i=1

R_i[f_yi].

Here Ri[fxi] are the Riesz transforms of fxi . These are operators of strong type 2,2 ; i.e.

there exists a constant A depending only upon N such that

(1.4) kRi[fxi]k_2,R^N ≤ Akfxik_2,R^N ; i = 1, 2, . . . , N.

A crucial fact in the proof is to exploit the connection between the integral in (1.3) and the harmonic extension of f in R^N × (0, ∞) for which we refer to (Stein [11], pp. 60–66).

This is the observation of Dahlberg–Kenig that makes the result possible.

Here we propose a different version of this fact that uses only elementary calculus, thereby avoiding the abstract harmonic analysis employed in [3].

2. Structure of the proof.

First we write (1.1) in a classical sense as

(2.1) ∂

∂tu_ε− ∆ϕ_ε(u) = 0 , in Gε,

where, denoting with z ≡ (x, t) points of G,

Gε ≡ {z ∈ G | dist (z, ∂G) > ε} and ε ∈ (0, 1) is so small that Gε 6≡ ∅;

(i)

uε = u ∗ kε ∈ C^∞(Gε); ϕε(u) = ϕ(u) ∗ kε∈ C^∞(Gε);

(ii)

k(z) = e

|z|2

|z|2 −1 if |z| < 1 and k(z) = 0 otherwise;

(iii)

kε(z) = 1

k_oε^N+1kz ε

∈ C_o^∞(|z| < ε), ko = Z

R^N+1

k(z)dz ; (iv)

z 7→ u_ε(z) = Z

R^N+1

k_ε(ζ)u(z − ζ)dζ, (v)

z 7→ (ϕ(u) ∗ kε)(z) = Z

R^N+1

kε(ζ)ϕ(u)(z − ζ)dζ.

Inside Gε consider cylinders

Q_r(zo) ≡ {|x − xo| < r} × (t_o, t_o+ r); r >0, z_o ≡ (x_o, t_o) ∈ Gε.

Modulo a translation, we may assume zo ≡ (0, 0) and will be working with balls Br ≡ {|x| < r} and cylinders Q_r ≡ B_r× (0, r). We also let

Kr ≡ Br× (−r, r).

Step 1.

Modulo a dilation we may assume K₂ ⊂ Gε and consider the box Q≡ Q₁ ⊂ K₂ ⊂ G_ε; Q≡ B × (0, 1); B ≡ {|x| < 1}

(in the dilation ϕ(·) is modified by a multiplicative constant depending upon r). Let vε

denote the unique weak solution of

∂

∂tvε− ∆ϕ(vε) = 0 , in Q , (2.2)

vε ∈ C(0, 1; L²(B)), ϕ(vε) ∈ L²(0, 1; W^1,2(B)), (vε− uε)(x, 0) = 0 , |x| < 1 ,

(ϕ(vε) − ϕε(u))(x, t) = 0 , |x| = 1 , t ∈ (0, 1).

The notion of weak solution is standard and we refer to [3], [4], [5].

Subtract (2.1) from (2.2) and write the difference in the weak form (here D ≡ (_∂x^∂

1,_∂x^∂

2, . . . ,_∂x^∂

N)) Z

Q

 − (vε− uε)ψt+ (D(ϕ(vε) − ϕε(u))) · Dψ dz = 0, (2.3)

ψ∈ W^1,2(0, 1; L²(B)) ∩ L²(0, 1; W_o^1,2(B)); ψ(·, 1) ≡ 0.

In (2.3) take

(2.4) ψ=

Z1

t

(ϕ(vε) − ϕε(u))dτ ≡ Z1

t

W(τ )dτ.

Then since

Z

Q

D(ϕ(vε) − ϕε(u))Dψdz = Z

Q

DW Z1

t

DW(τ )dτ
dz = −1 2

Z

Q

d dt

  Z1

t

DW(τ )dτ 

2dz ≥ 0

we deduce (2.5)

Z

Q

(vε− u_ε)(ϕ(vε) − ϕε(u))dz ≤ 0.

Remark 2.1. Testing functions like (2.4) were first introduced by Oleinik–Kalashnikov–

Chzhou-Yui-Lin [8].

Step 2.

Lemma 2.1. Every nonnegative local distributional solution of (1.1) such that ϕ(u) ∈ L¹_loc(G) satisfies

(2.6) uϕ(u) ∈ L¹_loc(G).

Lemma 2.2. There holds

(2.7)

(θ^Λϕ(s) ≤ ϕ(θs) ≤ θ^Λ+1Λ ϕ(s) , ∀θ ∈ (0, 1), ∀s ≥ 0 ; θ^Λ+1Λ ϕ(s) ≤ ϕ(θs) ≤ θ^Λϕ(s) , ∀θ ≥ 1, ∀s ≥ 0 . Assuming these two facts for the moment, it follows from (2.5) that (2.8)

Z

Q

v_εϕ(vε)dz ≤ Z

Q

v_εϕ_ε(u)dz + Z

Q

u_εϕ(vε)dz.

We write

Z

Q

v_εϕ_ε(u)dz = Z

RN+1

nZ

Q

v_ε(z)ϕ(u(z − ζ))dzo

k_ε(ζ)dζ,

and for ζ fixed in the (N + 1)-dimensional ball {|ζ| < ε} we let

E⁽¹⁾ ≡ {z ∈ Q|vε(z) ≤ γu(z − ζ)}; E⁽²⁾ ≡ {z ∈ Q|vε(z) > γu(z − ζ)},

where γ > 1 has to be chosen. Then Z

Q

v_εϕ_ε(u)dz ≤ γ Z

R^N+1

n Z

E⁽¹⁾

[uϕ(u)](z − ζ)dzo

k_ε(ζ)dζ + γ^−Λ+1Λ Z

R^N+1

n Z

E⁽²⁾

v_εϕ(vε)dzo

k_ε(ζ)dζ

≤ γ Z

Q

[uϕ(u)] ∗ kεdz+ γ^−Λ+1Λ Z

Q

v_εϕ(vε)dz.

By the properties of mollifiers and Lemma 2.2 we conclude Z

Q

vεϕε(u)dz ≤ γkuϕ(u)k_1,K2+ γ^−Λ+1Λ Z

Q

vεϕ(vε)dz.

The second integral on the right hand side of (2.8) is estimated analogously by Z

Q

uεϕ(vε)dz ≤ γ^Λkuϕ(u)k_1,K2 + γ⁻¹ Z

Q

vεϕ(vε)dz.

Step 3.

Choosing γ = 4, these estimates have two implications. The first is that by the results of [1], for every compact set K ⊂ Q

kv_εk_∞,K ≤ C_okuϕ(u)k^C_1,K¹ ₂ + Co,

where Ci, i = 0, 1 are independent of ε, depend only upon N, Λ and Co depends also upon dist{K, ∂pQ}, where ∂pQ is the parabolic boundary of Q.

It is easily seen that the results of [5] continue to hold for ϕ(·) satisfying (1.2). It follows (see [4], [5]) that ∃α = α(N, Λ) ∈ (0, 1) such that

v_ε∈ C_loc^α (Q) uniformly in ε,

and a subnet (indexed again with ε) can be selected so that vε→ w ∈ C_loc^α (Q), uniformly on compact subsets of Q, and weakly in L^p(Q), p = ^Λ+1_Λ (see (2.7)).

The second consequence is about the convergence of the “mixed terms” in (2.5).

Namely

Lemma 2.3. There holds

ε→0lim Z

Q

vεϕε(u)dz = Z

Q

wϕ(u)dz, (2.9)

ε→0lim Z

Q

u_εϕ(vε)dz = Z

Q

uϕ(w)dz.

(2.10)

Now we let ε → 0 in (2.5) and obtain, with the aid of Fatou theorem and Lemma 2.3 Z

Q

(w − u)(ϕ(w) − ϕ(u))dz ≤ 0.

Hence u = w ∈ C_loc^α (Q) and the theorem follows.

The proof of Lemma 2.1 will be given in the next sections. To prove Lemma 2.2 we observe that if θ ∈ (0, 1), by (1.2), ∀s > 0

lnϕ(θs) ϕ(s) = −

Z1

θ

d

dτ ln ϕ(τ s)dτ = − Z1

θ

1 θ

τ sϕ⁰(τ s) ϕ(τ s)



dτ ≤ Λ + 1 Λ ln θ ,

implying ϕ(θs) ≤ θ^Λ+1Λ ϕ(s). The proof of the other inequalities is similar.

Proof of Lemma 2.3: If σ ∈ (0, 1) let Qσ ≡ B_1−σ × (σ, 1); D_σ ≡ Q − Q_σ, so that

|Dσ| ≤ ωN(N + 1)σ, where ωN is the measure of the unit ball in R^N. Then 

Z

Q

v_εϕ_ε(u)dz − Z

Q

wϕ(u)dz   ≤

Z

Qσ

v_εϕ_ε(u) − wϕ(u)  dz+

(2.11)

+ Z

Dσ

wϕ(u)dz + Z

Dσ

v_εϕ_ε(u)dz = I_ε⁽¹⁾(σ) + I_ε⁽²⁾(σ) + I_ε⁽³⁾(σ).

Fix δ ∈ (0, 1) arbitrarily small. It follows from the estimates of Step 2 and Fatou theorem that wϕ(u) ∈ L¹(Q); then ∃σo ∈ (0, 1) such that

(2.12)

Z

Dσ

wϕ(u)dz ≤ δ, ∀0 < σ ≤ σ_o.

By the properties of convolutions I_ε⁽³⁾(σ) =

Z

Dσ

 Z

|ζ|<ε

kε(ζ)vε(z)ϕ(u(z − ζ))dζ dz=

Z

|ζ|<ε

kε(ζ)Z

Dσ

vε(z)ϕ(u(z − ζ))dz dζ.

For ζ ∈ Bε fixed, introduce two sets

E₁(ζ) ≡ {z ∈ Dσ|vε(z) ≤ γu(z − ζ)} , E₂(ζ) ≡ {z ∈ Dσ|vε(z) > γu(z − ζ)}, where γ > 1 has to be chosen. Then

Z

Dσ

vε(z)ϕ(u(z − ζ))dz = Z

E1(ζ)

vεϕ(u(z − ζ))dz + Z

E2(ζ)

vε(z)ϕ(u(z − ζ))dz

≤ γ Z

Dσ

[uϕ(u)](z − ζ)dz + γ^−Λ+1Λ Z

Dσ

v_εϕ(vε)dz.

Therefore

I_ε⁽³⁾(σ) ≤ γ Z

|ζ|<ε

kε(ζ)Z

Dσ

[uϕ(u)](z − ζ)dz

dζ+ γ^−Λ+1Λ Z

Q

vεϕ(vε)dz

≤ γ Z

Dσ

[uϕ(u)]ε(z)dz + γ^−Λ+1Λ Ckuϕ(u)k_1,K2,

where C = C(N, Λ) depends only upon N and Λ and [uϕ(u)]_ε= [uϕ(u)] ∗ k_ε. Choose first γ so large that

(2.13) γ^−Λ+1Λ Ckuϕ(u)k_1,K2 ≤ δ , and, such a γ being fixed, write

γ Z

Dσ

[uϕ(u)]_ε(z)dz ≤ γ Z

Q

[uϕ(u)]_ε− [uϕ(u)]

 dz+ γ

Z

Dσ

uϕ(u)dz.

Since uϕ(u) ∈ L¹_loc(G), ∃σ₁ ∈ (0, 1) such that

(2.14) γ

Z

Dσ

uϕ(u)dz ≤ δ, ∀0 < σ ≤ σ₁.

By the properties of convolutions ∃ε∈ (0, 1) such that

(2.15) γ

Z

Q

[uϕ(u)]ε− [uϕ(u)]

dz≤ δ, ∀0 < ε ≤ ε.

Set σ∗ = min{σ1, σ_o}. Then we deduce from (2.11) Z

Q

v_εϕ_ε(u) − wϕ(u) 

dz≤ 4δ + Z

Q_σ∗

v_εϕ_ε(u) − wϕ(u) 

dz , ∀0 < ε ≤ ε.

Since vε → w uniformly in Q_σ∗ and ϕε(u) → ϕ(u) in L¹(Q), ∃ε_∗ ∈ (0, 1) such that Z

Q_σ∗

v_εϕ_ε(u) − wϕ(u) 

dz ≤ δ , ∀0 < ε ≤ ε_∗.

This proves (2.9) in view of the arbitrariness of δ ∈ (0, 1). The proof of (2.10) is analogous.

3. Preliminary estimates.

Let zo ∈ G, n ∈ N and let R > 0 be so small that {|x − x_o| < nR} × (t_o, t_o+ R) ⊂ Gε. The number n ∈ N will be determined later only in terms of N, Λ. We may assume z_o ≡ (0, 0), R = 1 and set

(3.1) K ≡ {|x| < n} × (0, 1); B_s ≡ {|x| < s}; s= 1, 2, . . . , n.

Let η ∈ C_o^∞(0, 1) be such that

(3.2) η(t) = 1, t ∈ (1

4,3

4); |ηt| < 8; η≥ 0.

By using the cutoff function η, we derive two basic equations. First set for (x, t) ∈ K

(x, t) 7→ w(x, t) =

t

Z

0

η^Λ+1(τ )ϕε(u(x, τ ))dτ, (3.3)

(x, t) 7→ v(x, t) = (Λ + 1) Zt

0

ητ(τ )η^Λ(τ )uε(x, τ )dτ.

(3.4)

Then multiplying (2.1) by η^Λ+1 and integrating over (0, t), t ∈ (0, 1) we obtain

(3.5) η^Λ+1u_ε− ∆w = v , in K.

Next define

x 7→ w(x) = w(x, 1) =

1

Z

0

η^Λ+1(τ )ϕε(u(x, τ ))dτ, (3.6)

x7→ v(x) = v(x, 1) = (Λ + 1)

1

Z

0

η_t(τ )η^Λ(τ )uε(x, τ )dτ, (3.7)

and obtain from (2.1), upon multiplication by η^Λ+1 and integration over (0, 1)

(3.8) −∆w = v , in B_n.

The local information contained in these two equations suffices to prove Lemma 2.1.

In particular no trace information is needed.

In the estimates to follow we denote with C = C(N, Λ) a generic positive constant that can be determined quantitatively a priori, only in terms of N and Λ and is independent on ε.

Lemma 3.1. Suppose that w ∈ L^q(Bs) and

kwkq,Bs ≤ C 1 + kϕ(u)k_1,K
,

for some q ∈ [1, ∞], some C = C(N, Λ) and some s = 1, 2, . . . , n. Then v∈ L^qm(Bs) , m= Λ + 1

Λ , and kvkqm,Bs ≤ C 1 + kϕ(u)k_1,K
m¹ .

Proof: Assume first 1 ≤ q < +∞ . We use (1.2), (3.7) and the properties of the mollifiers;

here z = (x, τ ).

Z

Bs

|v|^qmdx≤ C Z

Bs

Z1

0

η^Λuε(x, τ )dτ  

qm

dx

≤ C Z

Bs

Z1

0

η^Λ+1(τ )h Z

R^N+1

k_ε(z − ζ)u(ζ)dζi^Λ+1_Λ dτ

q

dx

≤ C Z

Bs

1

Z

0

η^Λ+1(τ ) Z

RN+1

kε(z − ζ) max{1, u(ζ)}^Λ+1Λ dζ dτ

q

dx

≤ C 1 +

Z

Bs

Z1

0

η^Λ+1(τ ) Z

R^N+1

kε(z − ζ)ϕ(u(ζ))dζ dτ

q

dx

≤ C 1 + kϕ(u)k_1,K
q

. The proof for q = ∞ is analogous.

Proposition 3.1. ∃C = C(N, Λ) such that ∀t ∈ (0, 1) kw(·, t)k_∞,B4 ≤ C 1 + kϕ(u)k_1,K
;

(3.9)

kv(·, t)k_∞,B4 ≤ C 1 + kϕ(u)k_1,K
m¹

; (3.10)

kDw(·, t)k_2,B4 ≤ C 1 + kϕ(u)k_1,K
.

(3.11)

Proof: It suffices to consider the case N ≥ 3. Multiply (3.8) by y7→ c_N|x − y|^2−Nξ(y) , x∈ B_n−2; c_N = 1

(2 − N )σ_N⁻¹,

where σN is the area of the unit sphere in R^N and ξ ∈ C_o^∞(Bn) is such that ξ ≡ 1 on Bn−1; |Dξ| ≤ 2; 0 ≤ ξ ≤ 1. Standard calculations give ∀x ∈ Bn−2

w(x) ≤ C Z

n−1<|y|<n

w(y)|x − y|^1−Ndy+ C Z

Bn

|v(y)||x − y|^2−Ndy.

This implies ∀x ∈ Bn−2

(3.12) w(x) ≤ C 1 + kϕ(u)k_1,K
+ C Z

Bn

|v(y)||x − y|^2−Ndy.

Since ϕ(u) ∈ L¹_loc(G), by (1.2), u ∈ L^m_loc(G), m = ^Λ+1_Λ , and kvkm,Bn ≤ C 1 + kϕ(u)k_1,K
m¹

. Assuming 2m < N , by elliptic theory (see for example [6]),

w ∈ L^q1(Bn−2); q₁ = mN

N − 2m; kwk_q1,Bn−2 ≤ C 1 + kϕ(u)k_1,K
.

By Lemma 3.1

kvk_q1m,Bn−2 ≤ C 1 + kϕ(u)k_1,K
m¹ ,

and we repeat the process with m replaced by m₁ = q₁mand Bn replaced by B_n−2 as long as 2m₁ < N. Proceeding in this fashion we find numbers

m_o = m; q_i+1 = N m_i

N − 2m_i; m_i+1 = qi+1m_o; i= 0, 1, 2, . . . ,

such that if 2mi < N, w ∈ L^qi+1(B_n−2(i+1)) and kvkqi+1m,B_n−2(i+1) ≤ C 1 + kϕ(u)k_1,K
m¹ . If (j − 1) is the largest non-negative integer such that 2m_j−1 < N, we will have 2m_j ≥ N and again by elliptic theory (see [6]), ∀q ∈ (1, ∞)

kwkq,B_n−2(j+1) ≤ C 1 + kϕ(u)k_1,K
; kvkqm,B_n−2(j+1) ≤ C 1 + kϕ(u)k_1,K
m¹ . The number j being quantitatively determined, we take n − 2(j + 1) = 6. Then (3.8) holds in B₆ with v ∈ L^p(B₆), ∀p ∈ (1, ∞) and estimates (3.9)–(3.10) follow from standard elliptic theory, the definitions of w and v and Lemma 3.1.

To prove (3.11), multiply (3.5) by wξ² and integrate over B₆. Here ξ ∈ C_o^∞(B₆), ξ = 1 on B4, |Dξ| ≤ 1 and 0 ≤ ξ ≤ 1. Estimate (3.11) follows by standard calculations, after we drop the non-negative integral resulting from the first term on the left hand side of (3.5).

Letting

Q_∗ ≡ {|x| < 1

2} × (1 4,3

4),

our aim is to estimate the L¹ norm of uεϕ_ε(u) in Q∗ independent of ε.

4. Estimating kuεϕε(u)k_1,Q∗.

From now on we will work with (3.5) over B4× (0, 1). We multiply it by the testing function

η^Λ+1(t)ϕε(u(x, t))ξ³(x) = wtξ³ and integrate over Q ≡ B1× (0, 1). Here and in what follows (4.1) ξ ∈ C_o^∞(B₁); 0 ≤ ξ ≤ 1; ξ ≡ 1 on |x| < 1

2; |Dξ| ≤ 4.

We also denote with z ≡ (x, t) points in Q and let dz = dxdt. Since (this is again the trick of [8]),

− Z

Q

∆wwtξ³dz ≥ 3 Z

Q

wtDwξ²Dξdz,

the indicated operations give (4.2)

Z

Q∗

uεϕε(u)dz ≤ Z

Q

vwtξ³dz− 3 Z

Q

wtDwξ²Dξdz.

By virtue of Proposition 3.1, Z

Q

vw_tξ³dz≤ C 1 + kϕ(u)k_1,K
m¹ Z

Q

w_tdz ≤ C 1 + kϕ(u)k_1,K
2

.

Moreover    Z

Q

wtDwξ²Dξdz    =

Z

Q

(wξ)tD(wξ)Dξdz − Z

Q

wtξw|Dξ|²dz    ≤ C

Z

Q

ft|Df |dz + Ckwk_∞,Q Z

Q

wtdz,

where we have set

(4.3) (x, t) 7→ f (x, t) ≡ w(x, t)ξ(x) ∈ C^∞(0, 1; C_o^∞(B₁)).

It follows again from Proposition 3.1 that

(4.4) 

Z

Q

w_tDwξ²Dξdz 

 ≤C 1 + kϕ(u)k_1,K
2

+ C Z

Q

f_t|Df |dz.

Therefore Lemma 2.1 will be proved if we can find an upper bound for the last integral independent of ε. Since f (·, t) ∈ C_o^∞(B₁), ∀t ∈ [0, 1], we write

f(x, t) = cN

Z

R^N

∆f (y, t)

|x − y|^N−2dy; |Df |(x, t) ≤ C Z

R^N

|∆f (y, t)|

|x − y|^N−1dy.

From (4.3) and (3.5)

(4.5) |∆f | ≤ |∆f − η^Λ+1uεξ| + η^Λ+1uεξ ≤ ∆f + 2|v|ξ + 4|DwDξ| + 2w|∆ξ|.

Putting this in (4.4) and using Proposition 3.1 we find Z

Q∗

u_εϕ_ε(u)dz ≤ C 1 + kϕ(u)k1,K
2

+ C Z

Q

f_t(x, t) Z

R^N

|DwDξ|(y, t)

|x − y|^N−1 dy (4.6) dxdt

+ C  

Z

Q

f_t(x, t) Z

R^N

∆f (y, t)

|x − y|^N−1dy dxdt

 .

Lemma 4.1. ∃C = C(N, Λ) such that ∀t ∈ (0, 1), ∀x ∈ B₁, ∀ε ∈ (0, 1)

J≡ Z

R^N

|Dw(y, t)Dξ(y)|

|x − y|^N−1 dy≤ C 1 + kϕ(u)k_1,K
.

Proof: By the structure of ξ, the domain of integration in J is B₁ ≡ {|y| < 1}. We view x7→ w(x, t) as the restriction to B₁ of

x 7→ w(x, t)l(x); where l ∈ C₀^∞(B₄); l ≡ 1 on B₃; 0 ≤ l ≤ 1; |Dl| ≤ 2.

Then we estimate

J≤ C Z

B2

|D(wl)|(y, t)

|x − y|^N−1 dy;

(wl)(y, t) = cN

Z

R^N

∆(wl)(λ, t)

|λ − y|^N−2dλ; |D(wl)|(y, t) ≤ C Z

R^N

|∆(wl)(λ, t)|

|λ − y|^N−1 dλ.

By an estimate similar to (4.5)

|∆(wl)| ≤ ∆(wl) + 2|v|l + 4|DwDl| + 2w|∆l| , and ∀y : |y| < 2

|D(wl)|(y, t) ≤ C Z

R^N

∆(wl)(λ, t)

|λ − y|^N−1dλ+ C 1 + kϕ(u)k1,K
+ C Z

{3<|λ|<4}

|DwDl|(λ, t)

|λ − y|^N−1 dλ.

If |y| < 2, the last integral is majorized by C

Z

B4

|Dw|(λ, t)dλ ≤ CkDw(·, t)k_2,B4 ≤ C 1 + kϕ(u)k_1,K
,

by (3.11) of Proposition 3.1. Putting these estimates in the expression of J we obtain J≤ C 1 + kϕ(u)k_1,K
+

Z

{|y|<2}

dy

|x − y|^N−1 Z

R^N

∆(wl)(λ, t)

|λ − y|^N−1dλ.

We deal with the last term by writing Z

{|y|<2}

dy

|x − y|^N−1 Z

R^N

∆(wl)(λ, t)

|λ − y|^N−1dλ= Z

R^N

Z

R^N

∆(wl)(λ, t)

|x − y|^N−1|y − λ|^N−1dλdy−

Z

{|y|>2}

dy

|x − y|^N−1 Z

R^N

∆(wl)(λ, t)

|λ − y|^N−1dλ= L₁+ L₂.

By M. Riesz composition formula (see [7])

|L₁| =  

Z

R^N

∆(wl)(λ, t)

|x − λ|^N−2dλ    ≤

Ckw(·, t)k_∞,B4 ≤ C 1 + kϕ(u)k_1,K
.

As for L2 notice that if |x| < 1 and N ≥ 3 by (1.3) and (1.4) we have

L₂ ≤ Z

{|y|>2}

dy

|x − y|^{2(N −1)}

¹₂

Z

R^N

∆(wl)(λ, t)

|λ − · |^N−1dλ 2,R^N

≤ CkDw(·, t)k_2,B4 ≤ C 1 + kϕ(u)k_1,K
.

by (3.11) of Proposition 3.1. The Lemma is proved.

We return to (4.6) which by virtue of Lemma 4.1 now becomes

(4.7) Z

Q∗

u_εϕ_ε(u)dz ≤ C 1 + kϕ(u)k1,K
2

+ C  

Z1

0

Z

B1

f_t(x, t) Z

R^N

∆f (y, t)

|x − y|^N−1dy dxdt

 . Therefore the proof of Lemma 2.1 reduces to bound the last integral uniformly in ε.

5. The harmonic extension of f .

For t ∈ [0, 1] let (x, s) 7→ F (x, s; t) be the unique classical solution of

∆xF + Fss= 0, in R^N × (t, ∞), (5.1)

F(x, t; t) = f (x, t), x∈ R^N.

Such an F can be represented by the Poisson kernel. Namely if H(x − y; s − t) = 2cN+1 |x − y|²+ (s − t)²
−^N−1₂

,

is the fundamental solution of Laplace’s equation in R^N+1 with pole (x, t), then F(x, s; t) =

Z

R^N

∂

∂sH(x − y; s − t)f (y, t)dy = (5.2)

2 σ_N+1

Z

R^N

(s − t)

[|x − y|²+ (s − t)²]^N+12 f(y, t)dy .

Remark 5.1. From the definitions (3.3) of w and (4.3) of f it follows that (x, s) 7→

F(x, s; 0) ≡ 0.

Using (5.1) and the representation (5.2), we gather a few facts about F . We let

∇F ≡ (DF, F_s) Lemma 5.1. For all t ∈ (0, 1), as s → t,

Fs(x, s; t) → Fs(x, t; t) = −2cN+1

Z

R^N

∆f (y, t)

|x − y|^N−1dy, in L²(R^N);

(5.3)

Ft(x, s; t) → Ft(x, t; t) = ft(x, t) + 2cN+1

Z

R^N

∆f (y, t)

|x − y|^N−1dy, in L²(R^N);

(5.4)

∃C = C(N, Λ) such that ∀t ∈ (0, 1)

∞

Z

t

Z

R^N

|∇F |²dxds≤ C 1 + kϕ(u)k_1,K
2

(5.5) .

Proof: In (5.2) take the s-derivative and observe that H_ss= −∆_yH. An integration by parts then gives

Fs(x, s; t) = −2cN+1

Z

R^N

∆f (y, t)

[|x − y|²+ (s − t)²]^N−12 dy ,

and (5.3) follows letting s → t by standard estimates, since f (·, t) ∈ C_o^∞(B1).

Taking the t-derivative in (5.2) and recalling that −Hts = Hss = −∆yH, gives F_t(x, s; t) =

Z

R^N

∂

∂sHf_t(y, t)dy + Z

R^N

H∆f (y, t)dy.

The second integral is −Fs(x, s; t) and therefore it converges in L²(R^N) to −Fs(x, t; t) as s → t. The first integral is the harmonic extension of ft(·, t) and as s → t it converges in L²(R^N) to ft(·, t) (see Stein [11], p. 62).

One checks that |∇F | ≤ C(t) |x| + s
−(N +1)

, as |x| + s → ∞ and that |F | ≤ kf (·, t)k_∞,B4 by the maximum principle. Multiply (5.1) by F and integrate by parts over {|x| < ρ} × (t + σ, t + ρ); σ ∈ (0, 1), ρ > 1. Letting σ → 0 and ρ → ∞ and using the

“initial” data (5.3) we get

∞

Z

t

Z

R^N

|∇F |²dxds = − Z

R^N

f(x, t)F_s(x, t; t)dx ≤ kf k_2,B4kF_s(·, t; t)k_2,R^N

≤ Ckf k_∞,B4

Z

R^N

∆f (y, t)

| · −y|^N−1dy

2,R^N ≤ by (1.3) and (1.4)

≤ Ckw(·, t)k_∞,B4kDf (·, t)k_2,R^N ≤ C

1 + kw(·, t)k²_∞,B4+ kDw(·, t)k²_2,B4 . Estimate (5.5) now follows from Proposition 3.1.

We may now conclude the proof of Lemma 2.1, by estimating the last integral on the right hand side of (4.7). Multiply (5.1) by Ft(x, s; t) and integrate over R^N × (t, ∞) by using the “initial” conditions (5.3) and (5.4). This gives

−2c_N+1 Z

R^N

f_t(x, t)Z

R^N

∆f (y, t)

|x − y|^N−1dy

dx= 4c²_N+1 Z

R^N

Z

R^N

∆f (y, t)

|x − y|^N−1dy2

(5.6) dx

− 1 2

Z∞

t

Z

R^N

∂

∂t|∇F (x, s; t)|²dxds.

By (1.3)–(1.4) and Proposition 3.1, the first integral on the right hand side of (5.6) is majorized by C 1 + kϕ(u)k_1,K
2

, ∀t ∈ (0, 1). Integrating now both sides of (5.6) in dt over (0, 1) we get

(5.7)   

Z1

0

Z

B1

ft(x, t)Z

R^N

∆f (y, t)

|x − y|^N−1dy dxdt

 ≤C 1 + kϕ(u)k_1,K
2

+ C  

Z1

0

Z(t)dt  ,

where

(5.8) Z(t) = −

Z∞

t

Z

R^N

∂

∂t|∇F (x, s; t)|²dxds.

Write

Z(t) = − ∂

∂t Z∞

t

Z

R^N

|∇F (x, s; t)|²dxds− Z

R^N

|∇F (x, t; t)|²dx.

Now ∀t ∈ (0, 1) Z

R^N

|∇F (x, t; t)|²dx= Z

R^N

|Df (x, t)|²dx+ Z

R^N

F_s²(x, t; t)dx

≤ C 1 + kϕ(u)k_1,K
2

+ C Z

R^N

 Z

R^N

∆f (y, t)

|x − y|^N−1dy2

dx

≤ C 1 + kϕ(u)k_1,K
2

.

Finally we integrate Z(·) over (0, 1) as indicated in (5.7). We use here (5.5) and Remark 5.1 to obtain

Z 1 0

Z(t)dt 

 ≤ 1 + kϕ(u)k1,K
2

+ Z ∞

1

Z

R^N

|∇F (x, s; 1)|²dxds+ Z∞

0

Z

R^N

|∇F (x, s; 0)|²dxds

≤ C 1 + kϕ(u)k_1,K
2

.

Putting this in (5.7) proves Lemma 2.1 and concludes the proof of the theorem.

References

[1] D. Andreucci, L^∞ estimates for local solutions of degenerate parabolic equations, SIAM Jour. on Math. Anal. (to appear).

[2] L.A. Caffarelli, L.C. Evans Continuity of the temperature in the two phase Stefan problem, Arch. Rat. Mech. Anal. 81 (1983), pp. 199-220.

[3] B.E. Dahlberg, C. Kenig, Weak solutions of the porous medium equations (preprint).

[4] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. Jour. 32 (1) (1983), pp. 83-118.

[5] E. DiBenedetto, A. Friedman, H¨older estimates for non-linear degenerate parabolic systems, J. f¨ur die Reine und Angew. Mathematik 357 (1985), pp. 1-22.

[6] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Ed. Springer Verlag (1983).

[7] N.S. Landkof, Foundations of Modern Potential Theory, Springer Verlag (1972).

[8] O.A. Oleinik, A.S. Kalashnikov, Yui-Lin Chzhou, The Cauchy problem and boundary value problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk USSR Ser. Mat. 22 (1958), pp. 667-704 (russian).

[9] M. Pierre, Uniqueness of the solutions of u_t = ∆ϕ(u) with initial datum a measure, Nonlinear Anal. TMA, 6 #2 (1982), pp. 175-187.

[10] P.E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal.

TMA, 7 (1983), pp. 387-409.

[11] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press (1970).