Quasilinear Degenerate Elliptic Equations in Divergence Form

Pengfei Guan^{∗} Department of Mathematics McMaster University Hamilton, Ontario L8S 4K1 Canada

Dedicate to Francois Treves on the occasion of his sixty fifth birthday.

1 Introduction

In [G], the investigation of degenerate Monge-Ampère equations was led to the study of degenerate elliptic quasilinear equations in the form

∑_{i,j} ∂_{j}(a_{ij}(x, u)∂_{i}u) = f,  (1)

where a_{ij}(x, u) ∈ C^{∞}(R^{n} × R), semi-positive. Unlike in elliptic case, Schauder theory is not available for the degenerate equations in the form (1). Under elliptic extendability assumption (introduced in [G]), C^{∞} regularity of the solutions of equation (1) in subelliptic case was proved by establishing a commutator formula with careful control of coefficients in the first order terms. The elliptic exdendibililty condition was used in crucial way in the proof (to ensure the uniform subellipticity after localizing the problem), since we used elliptic smoothing process. Though in two variables case, we were able to verify the elliptic extendability condition for most of subelliptic operators. Unfortunately, it is difficult to verify this condition for general subelliptic operators in higher dimensions.

In this note, we shall establish C^{∞} regularity of equation (1) in the subelliptic case WITHOUT elliptic extendability assumption. The proof we will produce here don’t use elliptic theory, therefore, our proof is also applicable to elliptic case (without using Schauder theory). Instead of using elliptic smoothing process, we smooth the solution using standard mollifier operators. The similar idea was also used by Trudinger in [Tru] to prove Schauder theory for elliptic equations.

We first introduce some notations. Let Ω ⊂ R^{n} be a domain. Let (a_{ij}(x)) be a nonnegatively definite matrix function in Ω.

∗Supported in part by NSERC grant OGP0046732 and a Sloan Fellowship.

Definition 1 Let T = ∑_{j=1}^{n} α_{j}∂_{j} be a vector field in Ω, we say T is a subunit with respect to (a_{ij}(x)) if

∑_{j=1}^{n}(α_{j}ξ_{j})^{2} ≤ ∑_{i,j}^{n} a_{ij}(x)ξ_{i}ξ_{j}  (2)

for all ξ ∈ R^{n}. If γ is an absolutely continuous curve in Ω, we say γ is a subunit curve (with respect to (a_{ij}(x))) if γ′ is a subunit vector field. If x, y ∈ Ω, we define a distance function d(x, y) (with respect to (a_{ij}(x)) ) as

d(x, y) = inf{t > 0| there is a subunit curve γ : [0, t] → Ω
such that γ(0) = x, γ(t) = y}.  (3)

d(x, y) = ∞ if the above set is empty. Let x ∈ Ω, r > 0 be fixed. Let

C_{j}(x, r) = {γ_{j}(t), 0 ≤ t ≤ r|γ(t) = (γ_{1}(t), …, γ_{n}(t)) is a subunit curve with γ(0) = x} (4)

for j = 1, 2, …, n.

And we say the matrix A(x) = (a_{ij}(x) satisfies subunit condition in Ω if there is constant C such that,

|A_{x_{k}}(x) · ξ|^{2} ≤ C^{t}ξAξ,  (5)

∀ ξ ∈ R^{n}, and∀ xinΩ.

Now, we define subellipticity for linear equations.

Definition 2 Let L be a linear operator of the form

Lu = ∑_{i,j}^{n} ∂_{i}(a_{ij}(x)∂_{j}u)  (6)

with a_{ij} ∈ L^{∞}(Ω), (a_{ij}) ≥ 0 and tr(a_{ij}(x)) > 0 for all x in Ω. We say L is α-subelliptic, if there is α > 0, such that, for any subunit vector fields T_{1}, …, T_{N}, and for any f_{0}, f_{1}, …, f_{N} ∈ L^{∞}(Ω), if v is a weak solution of the equation

Lv = f_{0} + T_{1}f_{1} + ⋯ + T_{N}f_{N}  (7)

then, v ∈ C^{α}(Ω), and for each K ⊂⊂ Ω,

∥v∥_{C^{α}(K)} ≤ C

where C depends only on K, ∥f_{j}∥_{L^{∞}(Ω)}, ∥v∥_{L^{2}(Ω)}, ∥T_{j}∥_{C^{1}}.

Following is our main result. Main Theorem Suppose u ∈ C^{0,1}(Ω), u is a weak solution of equation

∑_{i,j}^{n} ∂_{i}(a_{ij}(x, u)∂_{j}u) = f(x)  (8)

with a_{ij} ∈ C^{∞}(Ω × R), (a_{ij}) ≥ 0, f ∈ C^{∞}(Ω). Then u ∈ C^{∞}(Ω), if the linear operator

Lv = ∑_{i,j}^{n} ∂_{i}(ã_{ij}(x)∂_{j}v),  (9)

where ã_{ij}(x) = a(x, u(x)), is subunit and α-subelliptic for some α > 0.

Remark 3 By De Giorgi-Nash-Moser theory (e.g., see [DG], [N], [M] and [GT]), if (a_{ij}(x)) in (6) is positive definite, L is α-subelliptic for some α > 0. Subellipticity is much weaker notion than ellipticity. In smooth coefficients case, the subelliptic theory for second order operators in the form of (6) is quite complete due to fundamental works by Hörmander [H], Kohn [K1], [K2], Fefferman and Phong [FP] and, Oleinik and Radkevitch [OR] (see also [Tre]). For each second order degenerate operator L, one can associate a suitable metric in such a way that it is natural for the operator L as the Euclidean metric for the Laplace operator (see [FP] and [NSW]). The subellipticity of the operator L can be completely characterized in terms of the geometry of the associated metric, which is the Fefferman-Phong condition. When the coefficients of L in (6) are only assumed in L^{∞}, there is an extensive literature in studying of subellipticity for the degenerate operators, we refer [F] for the references. Here we only quote some results by Franchi in [F]. Sufficient conditions for subellipticity in [F], we shall call them “subellipticity conditions”, are in some way a version of Fefferman-Phong condition in the smooth case. They can be stated as follows:

(SE1):

(i) ∃ ν > 0, such that

ν ∑_{j=1}^{n} λ_{j}^{2}(x)ξ_{j}^{2} ≤ ∑_{i,j}^{n} a_{ij}(x)ξ_{i}ξ_{j} ≤ \frac{1}{ν} ∑_{j=1}^{n} λ_{j}^{2}(x)ξ_{j}^{2}  (10)

∀ (x, ξ) ∈ Ω × R^{n}, where

(ii) λ_{1}, …, λ_{n} are in C^{0,1}(Ω̄), and d associated with ∂_{λ} is finite in Ω

(iii) Let Λ_{i}(x, r) = max_{s_{j}∈C_{j}(x,r)} λ_{i}(s_{1}, …, s_{n}). ∀x_{0} ∈ K ⊂⊂ Ω there is a neighborhood U of x_{0}, such that, if 0 < ε_{j} ≤ |ξ_{j}| ≤ 1 for j = 1, …, n and if we denote by H(·, x, ξ) = (H_{1}, …, H_{n}) the integral curve of the vector field ξ_{1}λ_{1}∂_{1} + ⋯ + ξ_{n}λ_{n}∂_{n} starting from x, we assume

∫_{0}^{t} λ_{j}(H(s, x, ξ))dx ≥ tC_{ε_{1},...,ε_{n}}Λ_{j}(x, t)  (11)

for j = 1, …, n, where C_{ε_{1},...,ε_{n}} is independent of t ∈ (0, t_{0}), x ∈ U and ξ ∈ ∏_{j=1}^{n}[ε_{j}, 1].

When n = 2, we may assume λ_{1} = 1. The condition (SE1) can be replaced by

(SE2): with the condition (iii) in (SE1) replaced by: there is a constant c > 0, such that

∫_{0}^{t} λ_{2}(x + s, y)ds ≥ ct max_{x<z<x+t} λ_{2}(z, y).  (12)

We finally deduce following from our Main Theorem and results in [F].

Theorem 4 Suppose u ∈ C^{0,1}(Ω), u is a weak solution of equation

∑_{i,j}^{n} ∂_{i}(a_{ij}(x, u)∂_{j}u) = f(x)  (13)

with a_{ij} ∈ C^{∞}(Ω × R), (a_{ij}) ≥ 0, f ∈ C^{∞}(Ω). Let ã_{ij}(x) = a(x, u(x)). If the operator L = ∑_{i,j}^{n} ∂_{i}(ã_{ij}(x)∂_{j}) satisfies subelliptic condition (SE1), then u ∈ C^{∞}(Ω).

This work was completed while the author was visiting Centre for Mathematics and Its Applications, Australian National University. I am grateful to Professor N. Trudinger for his kind arrangements and hospitality in Australia.

2 Proof of the Main Theorem

The proof of the Main Theorem relies on a commutator lemma (?). Before we state the lemma, we introduce notations.

Definition 5 Let Λ^{s}, s ∈ R be Hölder-Zygmund spaces. We denote C_{∗}^{s} = Λ^{s} if s > 0, s ∉ Z, and C_{∗}^{ℓ} = C^{ℓ,0}, if ℓ ∈ Z^{+}. If B is a linear operator, B is bounded from Λs+m_{comp}(R^{n}) → Λs_{loc}(R^{n}) for some m ∈ R, and ∀ 0 < s ≤ t, then we say B ∈ O_{t}^{m}; if B is bounded from C_{∗}^{s} → C_{∗}^{s+m} for some m ≤ s ∈ R, and ∀ 0 < s + m ≤ t, then we say B ∈ O_{∗t}^{m}.

Let ρ(t) ∈ C_{0}^{∞}(R), ∀ ϵ > 0, let ρ_{δ}(x) = δ^{−n} ∏_{i} ρ(\frac{x_{i}}{δ}). We also denote the ρ_{δ} be the convolution operator:

ρ_{δ}(g)(x) = ∫ ρ_{δ}(x − y)g(y)dy,  (14)

Commutator Lemma. Let a(x, u) ∈ C^{∞}(Ω × R), suppose u ∈ C_{∗}^{t}(Ω) for some t ≥ 1. (i) Let ρ_{δ} as above, there are operators B_{δ,j} ∈ O_{∗t}^{−1}, i = 1, 2, …, n + 1, and B_{δ,0} ∈ O_{∗t}^{−2}, with bounds independent of δ, such that,

[ρ_{δ}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B_{δ,j} + a_{u}(x, u(x))B_{δ,n+1} + B_{δ,0}.  (15)

Moreover, for any first order differential operator with constant coefficients D, there are operators B̄_{δ,j} ∈ O_{∗t−1}^{0}, i = 1, 2, …, n + 1, and B̄_{δ,0} ∈ O_{∗t}^{−1}, with bounds independent of δ, such that,

D[ρ_{δ}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B̄_{δ,j} + a_{u}(x, u(x))B̄_{δ,n+1} + B̄_{δ,0}.  (16)

(ii) Let

|D|^{s} = (1 − Δ)^{s/2}  (17)

for x ∈ R^{n}, where Δ = ∂_{1}^{2}+⋯+∂_{n}^{2}. Then, we have ∀ ε > 0, there are operators B_{j} ∈ O_{t−s}^{−1+ε}, i = 1, 2, …, n + 1, and B_{0} ∈ O_{t−s}^{s−2+ε}, such that

[|D|^{s}, a(x, u(x))] = ∑_{j=1}^{n} a_{x_{j}}(x, u(x))B_{j} + a_{u}(x, u(x))B_{n+1} + B_{0}.  (18)

Proof. We prove part (i) of the lemma, part (ii) was proved in [G]. By definition,

[ρ_{δ}, a(x, u(x))]g(x) = ∫ ρ_{δ}(x − y)(a(y, u(y)) − a(x, u(x)))g(y)dy.  (19)

By Taylor expansion, since a(x, z) is smooth,

a(x, u(x)) − a(y, u(y)) = a_{x}(x, u(x))(x − y) + a_{u}(x, u(x))(u(x) − u(y))
+r(x, u(x), x − y, u(x) − u(y))  (20)

where r ∈ C^{∞}(R^{n} × R × R^{n} × R) and

|r(x, u(x), x − y, u(x) − u(y))| ≤ C(|x − y|^{2} + (u(x) − u(y))^{2}).  (21)

Since u ∈ C_{∗}^{t} for t ≥ 1, we have,

|r(x, u(x), x − y, u(x) − u(y))| ≤ C|x − y|^{2}.  (22)

Define,

B_{δ,j}g(x) = − ∫ ρ_{δ}(x − y)(x_{j} − y_{j})g(y)dy, ∀j = 1, 2, ..., n;  (23)

B_{δ,n+1}g(x) = − ∫ ρ_{δ}(x − y)(u(x) − u(y))g(y)dy,  (24)

B_{δ,0}g(x) = − ∫ ρ_{δ}(x − y)r(x, u(x), x − y, u(x) − u(y))g(y)dy.  (25)

It is easy to check that B_{δ,j} ∈ O_{∗t}^{−1} for j = 1, 2, ..., n + 1, and B_{δ,0} ∈ O_{∗t}^{−2}, with norms independent of δ. Now (15) follows from (19), (20), (23), (24) and (25).

Similarily, for example, if D = ∂_{k}, let ρ^{∗} = Dρ and ρ_{δ}^{∗}(x) = \frac{1}{δ^{n+1}} ρ^{∗}(\frac{x}{δ}).

D[ρ_{δ}, a(x, u(x))]g(x) = ∫ ρ_{δ}^{∗}(x − y)(a(y, u(y)) − a(x, u(x)))g(y)dy
−a_{x_{k}}(x, u(x))ρ_{δ}g(x) − a_{u}(x, u(x))u_{k}(x)ρ_{δ}g(x).  (26)

. We define,

B̄_{δ,j}g(x) = − ∫ ρ_{δ}^{∗}(x − y)(x_{j} − y_{j})g(y)dy − δ_{k}^{j}ρ_{δ}g(x), ∀j = 1, 2, ..., n;  (27)

B̄_{δ,n+1}g(x) = − ∫ ρ_{δ}^{∗}(x − y)(u(x) − u(y))g(y) − u_{k}(x)ρ_{δ}g(x),  (28)

B̄_{δ,0}g(x) = − ∫ ρ_{δ}^{∗}(x − y)r(x, u(x), x − y, u(x) − u(y))g(y)dy.  (29)

Again, we can easily check that B̄_{δ,j} ∈ O_{∗t−1}^{0} for j = 1, 2, ..., n + 1, and B̄_{δ,0} ∈ O_{∗t}^{−1}. Therefore, (16) follows from (26), (20) and (27)-(29). The proof of the lemma is complete.

Proof of the Proposition. Let A(x, u) = (a_{ij}(x, u)). Equation (1) can be rewritten as

∇ · A(x, u) · ∇u = f .  (30)

Let ρ(t) ∈ C_{0}^{∞}(R), ρ(t) ≥ 0, and ∫ ρ = 1, and let ρ_{δ} be the convolution operaor defined as in (14). For D = ∂_{k}, if we denote v_{δ}(x) = Dρ_{δ}u(x). By the Commutator Lemma (part (i)),

∇(A(x, u)∇v_{δ}) = Dρ_{δ}f + ∇(∑_{j} A_{x_{j}}(x, u(x))B̄_{δ,j}∇u)
+∇(A_{u}(x, u(x))B̄_{δ,n+1}∇u) + ∇(B̄_{δ,0}∇u).  (31)

Since A(x, u(x)) is subunit in Ω, we may write (31) as

∇ · A(x, u) · ∇v_{δ} = f̃+ ∑_{i=1} T_{i}f̃_{i}  (32)

with T_{i} subunit, f̃, f̃_{1}, …, f̃_{N} ∈ L^{∞}. By the subellipticity assumption, v_{δ} ∈ C^{α}(Ω) for some α > 0, with the bounds independent of δ. By passing δ to 0, we may conclude that u ∈ C^{1,α}.

We hope to differentiate (31) to get higher regularity. But, since we only have C^{α} bounds for v_{δ} for some 0 < α < 1, we can not apply usual differentiation. Here comes in fractional differentiation. Suppose u ∈ C_{∗}^{1+β} for some 0 < β ≤ 1, we want to show u ∈ Λ^{1+β+ \frac{α}{2}} . We already have β ≥ α. Now, for the convenience, we drop index δ, write ρ for ρ_{δ}, v for v_{δ}, and B̄_{j} for B̄_{δ,j}. Let w = |D|^{(β− \frac{α}{2} )}v. Apply |D|^{(β− \frac{α}{2} )} on equation (31), w satisfies equation

∇ · A(x, u) · ∇w = |D|^{(β− \frac{α}{2} )}ρDf + ∇ · ∑_{j} A_{x_{j}}(x, u) (|D|^{(β− \frac{α}{2} )}B̄_{j}∇u)

+∇ · A_{u}(x, u) (|D|^{(β− \frac{α}{2} )}B̄_{n+1}∇u)) + |D|^{(β− \frac{α}{2} )}∇(B̄_{0}∇u)

+∇ · [A(x, u), |D|^{(β− \frac{α}{2} )}] · ∇v + ∇ · ∑_{j} [A_{x_{j}}(x, u), |D|^{(β− \frac{α}{2} )}] · B̄_{j}∇u

+∇ · [A_{u}(x, u), |D|^{(β− \frac{α}{2} )}] B̄_{n+1}∇u.  (33)

By the Commutator Lemma (part (ii)),

[A(x, u), |D|^{(β− \frac{α}{2} )}] = ∑_{j=1}^{n} A_{x_{j}}(x, u)B_{j} + A_{u}(x, u)B_{n+1} + B_{0}  (34)

with B_{j} ∈ O_{1+ \frac{α}{2}}^{β−1+ε+ \frac{α}{2}} , j = 1, 2, …, n + 1,B_{0} ∈ O_{1+ \frac{α}{2}}^{β−2+ε+ \frac{α}{2}} and

[A_{x_{k}}(x, u), |D|^{(β− \frac{α}{2} )}] , [A_{u}(x, u), |D|^{(β− \frac{α}{2} )}] ∈ O_{1}^{β−1+ε+ \frac{α}{2}} .  (35)

If we note that B_{j}∇v ∈ Λ^{\frac{α}{2} −ε}, j = 1, 2, …, n + 1, ∇u, ∂_{k}u ∈ C_{∗}^{β}, β ≥ α, we have

∇ · A(x, u) · ∇w = f^{∗} + ∑_{i=1}^{N} T̃_{i}f_{i}^{∗}  (36)

with f^{∗}, f_{i}^{∗} ∈ C_{∗}^{\frac{α}{2} −ε}, i = 1, 2, …, N, and T̃_{i} subunit. If we pick 0 < ε < \frac{α}{2}, again,by the subellipticity assumption, w ∈ C^{α}. Passing δ to 0, we get u ∈ Λ^{1+β+ \frac{α}{2}} . From this we conclude u ∈ C_{∗}^{2+ \frac{α}{2}} .

Now, at each stage, when we reach u ∈ C^{m} for m ≥ 2, m ∈ Z, we apply D^{m−1} on equation (31), using the assumption for the linear equation, to get u ∈ C_{∗}^{m+α̃}. Once u ∈ C_{∗}^{m+β} for β ≥ α̃, by acting |D|^{β− \frac{α̃}{2}} on the resulted equation, repeat the previous

argument (using the Commutator Lemma), we can conclude u ∈ C_{∗}^{m+1+ \frac{α̃}{2}} (U). Since x_{0} is arbitrary, the proof of the proposition is complete.

Finally, Theorem 4 is a directly consequence of our Main Theorem and following result of Franchi [F].

Theorem 6 Suppose a_{ij}(x) are bounded, measurable, (a_{ij}(x)) ≥ 0 satisfies subelliptic condition (SE1) (or (SE3) if n=2). Let T_{1}, …, T_{N} be subunit vector fields in Ω with C^{0,1} coefficients. If u is a weak solution of the equation

∑_{i,j} ∂_{i}(a_{ij}(x)∂_{j}u) = f_{0} + T_{1}f_{1} + ⋯ + T_{N}f_{N}  (37)

with f_{0}, f_{1}, …, f_{N} ∈ L^{∞}(Ω). Then, there is α > 0 depending only on Ω and Condition (SE1), such that u ∈ C^{α}(Ω), and for each K ⊂⊂ Ω,

∥u∥_{C^{α}(K)} ≤ C

where C depends only on K, ∥f_{j}∥_{L^{∞}}, ∥T_{j}∥_{C^{1}} and (SE1) ( or (SE2) when n=2).

Remark 7 In [F], the right hand side of equation (37) is of the form ∇_{λ}f. As one easily adapts the proof in [F] for Theorem 9.

References

[DG] E. De Giorgi, Sulla differenziabilita’ e l’analiticita’ delle estremali degli integrali multipli, Mem. Acad. Sci. Torino Cl. Sci. Fis. Mat. Natur (3)3 (1957), 25-43.

[FP] C. Fefferman and D.H. Phong, Subelliptic eigenvalue problems, Conference on Harmonic Analysis (Chicago, 1981) (W. Beckner et al., eds.) Wadsworth, 1981, 590-606.

[F] B. Franchi, Weighted Sobolev-Poincare inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. AMS. 327 (1991), 125-158.

[GT] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag (New York) 1983.

[G] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations. To appear in Advances in Mathematics.

[H] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171.

[K1] J.J. Kohn, Boundaries of complex manifolds, Proc. Conf. on Complex Anal., Minneapolis, 1964, Springer-Verlag (1965), 81-94.

[K2] J.J. Kohn, Pseudodifferential operators and hypoellipticity, Proc. Symp. Pure Math. AMS (1973), 61-69.

[Mo] J. Moser, On Harnack’s theorem for elliptic differential equation, Comm. Pure Appl. Math. 14 (1961), 577-591.

[NSW] A. Nagel, E.M. Stein, and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147.

[N] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954.

[OR] O. A. Oleinik and E. V. Radkevitch, Second Order Equations with Nonnegative Characteristic Form, Plenum Press, 1973.

[Tre] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1-2. Plenum Press (New York), 1982.

[Tru] N. Trudinger, A new approach to the Schauder estimates for linear elliptic equations, Proc. CMA, ANU 14 (1986), 52-59.