2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 121–131. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A note on W^{1,p} estimates for quasilinear parabolic equations ^{∗}

Ireneo Peral & Fernando Soria

Abstract

This work deals with the study of the W^{1,p} regularity for the solutions to parabolic equations in divergence form. An argument by perturbation based in real analysis is used.

1 Introduction

In this paper we study interior W^{1,p} estimates for solutions to quasilinear parabolic equations in divergence form, namely, solutions to the equation

u_{t} − div a(x, t, ∇u) = 0, x ∈ Ω, t > 0,  (1.1)

where a : Ω × (0, ∞) × ℝ^{N} → ℝ^{N}. We assume that a(x, t, ξ) is a Caratheodory function (in the sense that it is measurable in (x, t) and continuous with respect to ξ for each x) and that satisfies the following conditions:

(a1) a(x, t, 0) = 0

(a2) ⟨a(x, t, ξ) − a(x, t, η), (ξ − η)⟩ ≥ γ|ξ|^{2}.

(a3) |a(x, t, η)| ≤ Γ|η|,

where γ and Γ are positive constants.

We will work under the following hypotheses. (H1) (Reference operator.) For fixed a_{0} satisfying (a1)–(a3), we consider the corresponding parabolic equation

w_{t} − div(a_{0}(∇w)) = 0  (1.2)

If u is a weak solution to (1.2) (see Definition 3.2 below) then there exists γ > 0 such that

sup_{R′} |∇_{x}u(x, t)|^{2} ≤ γ \frac{1}{|R|} ∫_{R} |∇_{x}u(x, t)|^{2}dx dt,  (1.3)

∗Mathematics Subject Classifications: 35K10, 35K55, 42B25.

Key words: semilinear parabolic equations, gradient estimates, Calderon-Zygmund theory.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 22, 2002.

I. Peral was supported by grant BFM2001-0183 from M.C.Y.T. Spain.

F. Soria was supported by grant BFM2001-0189 from M.C.Y.T.

122 W^{1,p} estimates for quasilinear parabolic equations

for all parabolic rectangles

R = {(x, t) : |x_{i} − x_{i}^{0}| < ρ, i = 1, … N, t_{0} − ρ^{2} < t < t_{0}} ⊂ Ω,

where

R′ = \{(x, t) : |x_{i} − x_{i}^{0}| < \frac{ρ}{2}, i = 1, … N, t_{0} − (\frac{ρ}{2})^{2} < t < t_{0}\}.

(H2) (Approximation property.) The vector field a(x, t, ξ) is close to a_{0}(ξ) in the following sense:

|a(x, t, ξ) − a_{0}(ξ)| ≤ ϵ |ξ|,  (1.4)

uniformly in x and t.

The simplest example for a_{0}(ξ) is just a_{0}(ξ) = ξ, which corresponds to the heat equation. A classical example for a is a(x, t, ξ) = A(x, t)ξ with A(x, t) a bounded N × N matrix such that

∥A(x, t) − I_{N×N}∥_{∞} ≤ ϵ.

Even in the linear case, the results presented here seem to be new. We also point out that more general situations can be considered by the method that we develop. For simplicity, we restrict ourselves to the quadratic growth case. With these hypotheses we will be able to show some sort of parabolic Meyers type inequalities. More precisely we have the following main result.

Theorem 1.1 Let a be a vector field satisfying (a1)–(a3). Assume that (H1) holds. Given p > 2 there exists ϵ_{0} > 0 such that if for some 0 < ϵ < ϵ_{0} (H2) holds, then any weak solution to

u_{t} − div(a(x, t, ∇_{x}u)) = 0,

satisfies that |∇_{x}u| ∈ L_{loc}^{q}, 2 < q < p.

The idea is to estimate the level sets of ∇_{x}u and obtain the required growth of their measure to have the integrability property. To do that, we will use a form of the Calderón-Zygmund covering result. For the elliptic case see [1].

2 Preliminary results

In this section we present some tools that will be used along the paper. First of all we will prove the corresponding Calderón-Zygmund covering result and explain the properties of the maximal operator which naturally arises in the parabolic setting under study. To be systematic we will give a general Covering Lemma that includes the particular case that we will need.

The rectangle collection

For every k ∈ ℤ, let B_{k} denote a collection of rectangles satisfying the following properties:

i) All the rectangles of B_{k} have the same side lengths.

ii) Any two distinct rectangles of B_{k} have disjoints interiors.

iii) ⋃{R ∈ B_{k}} = ℝ^{N}

iv) If R ∈ B_{k} then R = ⋃{R′ ∈ B_{k−1}| R′ ⊂ R}

v) If δ_{k} denotes the length of the diagonal of any R ∈ B_{k} then δ_{k} ≥ 2δ_{k+1}

Observe that given R ∈ B_{k} there exists a unique rectangle in B_{k−1}, its predecessor which we denote by R^{∗} so that R ⊂ R^{∗}.

With these properties in hand we have the following observations.

Lemma 2.1 Fix R_{0} ∈ B_{k_{0}} and 0 < δ < 1. Assume that A ⊂ R_{0} and that

0 < |A| < δ|R_{0}|. Then, there exists a sequence {R_{d}} ⊂ ⋃_{k=k_{0}}^{∞} B_{k} such that:

1. |A \ ⋃_{d} R_{d}| = 0,

2. |A ⋂ R_{d}| ≥ δ|R_{d}|,

3. If R_{d} ⊂ R for some R ∈ ⋃_{k=k_{0}}^{∞} B_{k}, with R ⊂ R_{0}, then |A ⋂ R| ≤ δ|R| (in

particular |A ⋂ R^{∗}| ≤ δ|R^{∗}|).

Proof. The proof goes via the usual stopping-time argument. Observe that R_{0} does not satisfies 2) by hypothesis. We select all the rectangles in B_{k_{0}+1} for which 2) holds.

If R′ ⊂ R_{0}, R′ ∈ B_{k_{0}+1} has not been selected we consider the partition of R′ by rectangles of B_{k_{0}+2} and we take those satisfying 2).

In this way and by induction, we obtain a maximal sequence {R_{d}} satisfying 2) and 3). We call B_{A} = ⋃ R_{d}.

To see why {R_{d}} satisfies 1) too, we observe that if x ∈ R_{0}\⋃{∂R : R ∈ B_{A}} then there exists a unique sequence {R_{k}(x)} so that

x ∈ R_{k}(x) ∈ B_{k}, ∀k ≥ k_{0}.

Moreover, since diam R_{k}(x) = δ_{k} ↓ 0 as k ↑ ∞, we obtain that from the Lebesgue differentiation theorem,

lim_{k→∞} \frac{|A ⋂ R_{k}(x)|}{|R_{k}(x)|} = 1, a.e. x ∈ A.

In particular this says that A ⊂ ⋃ R_{d} a.e. and, therefore, 1) holds. □

Observe that the sequence {R_{d}} in Lemma 2.1, is given by maximal rectangles in R_{0} for which property 2) holds. We will call this sequence the Calderón-Zygmund covering of A. Note that given a rectangle R_{k} in a Calderón-Zygmund covering of the set A, there exists a nested finite family of parabolic rectangles

R̃_{k}^{1} ⊃ R̃_{k}^{2} ⊃ ⋯ ⊃ R̃_{k}^{r(k)} ⊃ R_{k},

for which

|R̃_{k}^{l} ∩ A| ≤ δ|R̃_{k}^{l}|.

We say that R̃_{k}^{r(k)} is the predecessor of R_{k} and for simplicity of notation we will write R̃_{k}^{r(k)} ≡ R̄_{k}.

We give some applications of Lemma 2.1 that are relevant for this work.

Lemma 2.2 The maximal operator

Mf(x) = sup ; x∈R∈⋃ B_{k} \frac{1}{|R|} ∫_{R} |f|

is of weak type (1, 1).

The proof uses the Vitali covering lemma and the observation that if R, R′ ∈ B = ⋃ B_{k} and with their interiors with nonempty intersection, then either R ⊂ R′ or R′ ⊂ R.

Lemma 2.3 Fix R_{0} ∈ B and 0 < δ < 1. Given A ⊂ B ⊂ R_{0} with the properties,

a) |A| < δ|R_{0}|,

b) If R_{d} is a rectangle in the Calderón-Zygmund decomposition of A (with respect to R_{0} and δ), then R_{d}^{∗} ⊂ B,

we conclude that |A| ≤ δ|B|

Proof. Consider the sequence {R_{d}^{∗}} of predecessors of the C-Z covering of A with respect to R_{0} and δ. Select a subsequence {R_{d_{l}}^{∗}} with disjoint interiors so that

⋃ R_{d_{l}}^{∗} = ⋃ R_{d}^{∗},

(a maximal subsequence). Then,

|A| = ∑_{l} |A ⋂ R_{d_{l}}^{∗}| ≤ δ ∑_{l} |R_{d_{l}}^{∗}| = δ| ⋃ R_{d_{l}}^{∗}| ≤ δ|B|.

3 Approximations Results

Denote the open unit cube in ℝ^{N+1} by Q_{0} = {(x, t) ∈ ℝ^{N+1} ||x_{i}| < 1, i = 1, 2, …, N, |t| < 1} and call Q_{0}^{+} = Q_{0} ∩ { t > 0 } and Q_{0}^{−} = Q_{0} ∩ { t < 0 }. A parabolic rectangle R is then the image of Q_{0} through any transformation in ℝ^{N+1} of the form

ϕ_{α}(x, t) = (x_{0} + αx, t_{0} + α^{2}t), α > 0, (x_{0}, t_{0}) ∈ ℝ^{N+1}.

Let us consider the following dyadic subdivision: Given a parabolic rectangle R

i) Divide into 2 equal parts each spatial side.

ii) Divide into 2^{2} equal parts the temporal side.

We call this procedure a parabolic subdivision. It is obvious that with this procedure we obtain 2^{N+2} new parabolic subrectangles. Associated to this subdivision we have the corresponding Calderón-Zygmund decomposition as in the previous section.

Before we continue, some notation is in order. Given a parabolic rectangle

R = {(x, t) : |x_{i} − x_{i}^{0}| < ρ, i = 1, … N, t_{0} − ρ^{2} < t < t_{0}}

we define the parabolic boundary of R as

∂_{p}R = {(x, t) : |x_{i}−x_{i}^{0}| ≤ ρ, t_{0}−ρ^{2} = t}∪{(x, t) : |x_{i}−x_{i}^{0}| = ρ, t_{0}−ρ^{2} < t < t_{0}}.

Example 3.1 In the case of the heat equation (the simplest model) we have the following result by Moser [4]. Let w be a measurable function in Q_{0} such that

(i) w(x, y) ≥ 0

(ii) w_{t}, ∇_{x}w, w ∈ L^{2}(Q_{0})

(iii) w_{t} − Δw ≤ 0 in D′(Q_{0}).

We call such w a positive subsolution of the heat equation.

According to the classical results by Moser in [4] we have that there exists a positive constant γ such that if w is a positive subsolution to the heat equation then

sup_{R′} w^{2}(x, t) ≤ γ \frac{1}{|R|} ∫_{R} w^{2}(x, t)dx dt,  (3.1)

where

R = {(x, t) : |x_{i} − x_{i}^{0}| < ρ, i = 1, … N, t_{0} − ρ^{2} < t < t_{0}}

and R′ = ϕ_{1/2}(R), that is,

R′ = \{(x, t) : |x_{i} − x_{i}^{0}| < \frac{ρ}{2}, i = 1, … N, t_{0} − (\frac{ρ}{2})^{2} < t < t_{0}\}.

We point out that if u is a solution to the heat equation then |∇_{x}u|^{2} is also a positive subsolution to the heat equation. Hence, by (2),

sup_{R′} |∇_{x}u(x, t)|^{2} ≤ γ \frac{1}{|R|} ∫_{R} |∇_{x}u(x, t)|^{2}dx dt,  (3.2)

where R and R′ are parabolic rectangles related as above.

It is clear that hypothesis (H1) extends the previous Example.

Definition 3.2 We say that u ∈ L^{2}(Q_{0}), with u_{t}, ∇_{x}u ∈ L^{2}(Q_{0}), is a weak solution to the equation

u_{t} − div_{x}(a(x, t, ∇_{x}u)) = 0

if we have

∫_{Q_{0}} uψ_{t}dxdt + ∫_{Q_{0}}⟨a(x, t, ∇_{x}u)∇ψ⟩dx dt = 0,

for all ψ in W_{0}^{1,2}, the completion of C_{0}^{∞}(Ω) with respect to the L^{2}-norm of the function and its gradient.

Lemma 3.3 Assume that (H1),(H2) hold. Let u be a weak solution to the equation

u_{t} − div_{x}(A(x, t, ∇_{x}u) = 0,  (3.3)

such that for R, a parabolic rectangle contained in Q_{0}, u satisfies

\frac{1}{|R|} ∫_{R} |∇_{x}u|^{2}dx dt ≤ μ.

If v is the solution to the problem

v_{t} − div_{x}(a_{0}(∇v)) = 0, (x, t) ∈ R
v|_{∂_{p}R} = u,

then

1. \frac{1}{|R|} ∫_{R} |∇_{x}(u − v)|^{2}dx dt ≤ μϵ^{2}

2. \frac{1}{|R|} ∫_{R} |∇_{x}v|^{2}dx dt ≤ μ(1 + ϵ)^{2} Proof. Write R = Q × (T_{1}, T_{2}). Observe that

γ \frac{1}{|R|} ∫_{R} |∇(u − v)|^{2}dx dt

≤ \frac{1}{|R|} ∫_{R}⟨a_{0}(∇u) − a_{0}(∇v), ∇(u − v)⟩dx dt
≤ \frac{1}{|R|} ∫_{Q} |u(x, T_{2}) − v(x, T_{2})|^{2}dx

+\frac{1}{|R|} ∫_{R}⟨a_{0}(∇u) − a_{0}(∇v), ∇(u − v)⟩dx dt

= \frac{1}{|R|} ∫_{R} [(u − v)_{t} − (div_{x}a_{0}(∇u) − div_{x}a_{0}(∇v))] (u − v)dx dt
= \frac{1}{|R|} ∫_{R}(u_{t} − div_{x}a_{0}(∇u))(u − v)dx dt

−\frac{1}{|R|} ∫_{R}(v_{t} − div_{x}a_{0}(∇v))(u − v)dx dt

= \frac{1}{|R|} ∫_{R}[u_{t} − div_{x}(a(x, t, ∇_{x}u)](u − v)dx dt

+\frac{1}{|R|} ∫_{R}⟨∇_{x}(u − v), (a_{0}(∇u) − a(x, t, ∇_{x}u))⟩dx dt

≤ ϵ\frac{1}{|R|}( ∫_{R} |∇_{x}(u − v)|^{2}dx dt)^{1/2}( ∫_{R} |∇_{x}u|^{2}dx dt)^{1/2},

where we have used for the last inequality that u is a solution to equation (1.1), condition (H2) and Cauchy-Schwarz inequality. We thus obtain (1). Now (2) is an easy consequence of (1) since, in fact,

(\frac{1}{|R|} ∫_{R} |∇_{x}v|^{2}dx dt)^{1/2}

= (\frac{1}{|R|} ∫_{R} |∇_{x}(u + (v − u))|^{2}dx dt)^{1/2}
≤ (\frac{1}{|R|} ∫_{R}[|∇_{x}u|^{2}dx dt)^{1/2} + (\frac{1}{|R|} ∫_{R} |∇_{x}(v − u))|^{2}dx dt)^{1/2}
≤ (1 + ϵ)μ^{1/2}.
□

We will now introduce the (dyadic) parabolic maximal operator, defined for f ∈ L_{loc}^{1} by

Mf(x) = sup_{x∈R} \frac{1}{|R|} ∫_{R} |f(y)|dy,

where the supremum is taken over all parabolic rectangles R containing x ∈ ℝ^{N+1}. Taking into account that ℝ^{N+1} is a homogeneous space with respect to the seminorm

d(x, t) = max{|x_{i}|, |t|^{1/2}, i = 1, 2 … N},

more precisely that

|ϕ_{2}(R)| ≤ 2^{N+2}|R|,

we have that M is of weak type (1, 1) as a consequence of the usual Besicovich covering lemma. The following result is the main ingredient in Theorem 1.1 below.

Lemma 3.4 Assume that u is a weak solution to

u_{t} − div_{x}(a(x, t, ∇_{x}u)) = 0, in Q_{0}.

Then, there exists a constant C > 1 so that for 0 < δ < 1 fixed, one can find ϵ_{0} = ϵ_{0}(δ) > 0 such that if (H2) holds with ϵ < ϵ_{0}, for all parabolic rectangle R_{k} in the Calderón-Zygmund δ-covering of

{(x, t) ∈ ℝ^{N+1} : M(|∇_{x}u|^{2})(x, t) > Cμ},

then its predecessor satisfies

R̄_{k} ⊂ {(x, t) : M(|∇_{x}u|^{2})(x, t) > μ}.

In particular, we have

|{(x, t) ∈ ℝ^{N+1} : M(|∇_{x}u|^{2})(x, t) > Cμ}| ≤ |{(x, t) : M(|∇_{x}u|^{2})(x, t) > μ}|.

Proof. The proof is similar to Lemma 3 in [1] and we sketch it here for the sake of completeness. Since we look for a local result, we can assume that a parabolic dilation of R_{k} is contained in Q_{0}, to be more precise, let us assume, say, that Q = ϕ_{4}(R_{k}) ⊂ Q_{0}. We argue by contradiction. If R_{k} satisfies the hypothesis, namely,

|R_{k} ∩ {x : M(|∇_{x}u|^{2}) > Cμ}| > δ|R_{k}|

and R̄_{k} does not satisfy the conclusion, there exists (x_{0}, t_{0}) ∈ R̄_{k} for which,

\frac{1}{|R|} ∫_{R} |∇_{x}u|^{2}dx dt ≤ μ, for all parabolic rectangles R with (x_{0}, t_{0}) ∈ R.

We solve the problem

v_{t} − div_{x}(a_{0}(∇v)) = 0, (x, t) ∈ Q_{0}
v|_{∂_{p}Q_{0}} = u.

Then, according to Lemma 3.3 we get:

1. \frac{1}{|Q|} ∫_{Q} |∇_{x}v|^{2}dx dt ≤ (1 + ϵ)^{2}μ

2. \frac{1}{|Q|} ∫_{Q} |∇_{x}(u − v)|^{2}dx dt ≤ ϵ^{2}μ Then the restricted maximal operator,

M^{∗}(|∇_{x}u|^{2})(x, t) = sup_{x∈R, R⊂ϕ_{2}(R_{k})} \frac{1}{|R|} ∫_{R} |∇_{x}u|^{2}dyds,

satisfies

M(|∇_{x}u|^{2})(x, t) ≤ max{M^{∗}(|∇_{x}u|^{2})(x, t), 4^{N+2}μ}.

Consider C = max{4^{N+2}, 4(1 + ϵ)^{2}}. Then

|{(x, t) ∈ R_{k} : M(|∇_{x}u|^{2}) > Cμ}|
≤ |\{(x, t) ∈ R_{k} : M^{∗}(|∇_{x}u|^{2}) > C \frac{μ}{2}\}|

+|\{(x, t) ∈ R_{k} : M^{∗}(|∇_{x}(u − v)|^{2}) > C \frac{μ}{2}\}|

≤ |\{(x, t) ∈ R_{k} : M^{∗}(|∇_{x}(u − v)|^{2}) > C \frac{μ}{2}\}|.

By the (1,1) weak type estimate for the maximal operator we conclude

|\{x ∈ R_{k} : M(|∇_{x}u|^{2}) > Cμ\}| ≤ |\{x ∈ R_{k} : M^{∗}(|∇_{x}(u − v)|^{2}) > C \frac{μ}{2}\}|
≤ A\frac{2}{Cμ} ∫_{R_{k}} |∇_{x}(u − v)|^{2}dx dt
≤ A\frac{2}{Cμ}ϵ^{2}|R_{k}|.

Taking ϵ > 0 so that A\frac{2}{Cμ}ϵ^{2} < δ we reach a contradiction. □

4 Proof of the main result

As a consequence of the approximation Lemma 3.3 and the behavior of the level sets of the maximal operator described in Lemma 3.4 we can formulate the following regularity result.

Theorem 4.1 Assume that (H1) holds. Given p > 2 there exists ϵ_{0} > 0 such that if for some 0 < ϵ < ϵ_{0}, (H2) holds, then any weak solution to

u_{t} − div(A(x, t)∇_{x}u) = 0,

satisfies that |∇_{x}u| ∈ L_{loc}^{q}, 2 < q < p.

Proof. For s > 0, call ω(s) = |{(x, t) : M(|∇_{x}u|^{2}(x, t) > s}|, the distribution function of the maximal operator. Take δ ∈ (0, 1) in such a way that C^{q/2}δ < 1, where C is as in Lemma 3.4. Now, there exists ϵ_{0}, such that if 0 < ϵ < ϵ_{0} and (H2) holds, then Lemmas 3.4 and 2.3 imply

ω(Cμ_{0}) ≤ δω(μ_{0}).

Hence by recurrence

ω(C^{k}μ_{0}) ≤ δ^{k}ω(μ_{0}).  (4.1)

Now |∇_{x}u|^{q} ∈ L^{1} if, in particular, M(|∇_{x}u|^{2}) ∈ L^{q/2}, and this is equivalent to the convergence of the series

∑_{k=1}^{∞} C^{k(q/2)}ω(C^{k}μ_{0}).

But, C^{q/2}δ < 1 and from estimate (7) we obtain

∑_{k=1}^{∞} C^{k(q/2)}ω(C^{k}μ_{0}) ≤ ∑_{k=1}^{∞}(C^{q/2}δ)^{k}ω(μ_{0}) < ∞

□

Corollary 4.2 Assume A(x, t) a N × N matrix which is continuous in Ω and such that

⟨A(x, t)ξ, ξ⟩ ≥ γ|ξ|^{2}.

Then if u is a weak solution to

u_{t} − div_{x}(A(x, t)∇_{x}u) = 0,

we have u ∈ W_{loc}^{1,p} for all 1 < p < ∞.

Proof. As reference equation we take the heat equation. The hypothesis (H2) is obtained easily by an orthogonal change of variables in ℝ^{N} and our assumptions on the continuity of A. □

By the same method we are able to get estimates for equations that are close to the p-Laplacian. For estimates to u_{t} − div(|∇u|^{p−2}∇u) see [2].

References

[1] L. A. Caffarelli and I. Peral, On W^{1,p} Estimates for Elliptic Equations in Divergence Form, Comm. Pure App. Math., 51, 1998, 1-21.

[2] E. Di Benedetto Degenerate Parabolic Equations, Springer-Verlag, 1993

[3] G. M. Lieberman, Second Order Parabolic Differential Equations. World Scientific, 1996.

[4] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17, 1964, 101–134.

[5] O. A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, ”Linear and Quasilinear Equations of Parabolic Type” Translations of Mathematics Monographs, A.M.S. 1968.

[6] P. Tolksdorff, Regularity for a more general class of quasilinear elliptic equations J. of Diff. Equations 51 (1984), 126-150

Ireneo Peral (e-mail: ireneo.peral@uam.es) Fernando Soria (e-mail: fernando.soria@uam.es) Departamento de Mátematicas U.A.M. 28049 Madrid, Spain