Hölder continuity for non variational parabolic systems

Maria Stella Fanciullo

Let Ω be an open bounded subset of R^{n}, n ≥ 2, and Q the cylinder Ω × (−T, 0), T > 0, we consider the following second order non linear non variational system

a(X, u, Du, H(u)) − \frac{∂u}{∂t} = b(X, u, Du),  (1)

where a(X, u, p, ξ) : Q × R^{N} × R^{nN} × R^{n^{2}N} → R^{N}, N an integer ≥ 1, is measurable in X and continuous in (u, p, ξ). The system (1) is called parabolic if the vector a(X, u, p, ξ) satisfies the following condition: (A) there exist three positive constants α, γ and δ, with γ +δ < 1, such that, ∀u ∈ R^{N}, ∀p ∈ R^{nN}, ∀τ, ξ ∈ R^{n^{2}N} and for a.e. X ∈ Q, it results:

∥∑_{i=1}^{n} τ_{ii} − α[a(X, u, p, τ + ξ) − a(X, u, p, ξ)]∥^{2} ≤ γ∥τ∥^{2} + δ ∥∑_{i=1}^{n} τ_{ii}∥^{2}.

We suppose that the vector b(X, u, p) : Q × R^{N} × R^{nN} → R^{N} is measurable in X, continuous in (u, p) and satisfies for a.e. X ∈ Q, ∀u ∈ R^{N} and ∀p ∈ R^{nN},

∥b(X, u, p)∥ ≤ c(f(X) + ∥u∥^{β} + ∥p∥^{β′})

with f ∈ L^{2}(Q), 1 ≤ β < \frac{n+2}{n−2} and 1 ≤ β′ < \frac{n+2}{n}.

Set W^{2}(Q, R^{N}) = \{u : u ∈ L^{2}(−T, 0, H^{2}(Ω, R^{N})), \frac{∂u}{∂t} ∈ L^{2}(Q, R^{N})\} , first we prove that if u is solution in Q of the system (1) (that is u ∈ W^{2}(Q, R^{N}) and satisfies (1) for a.e. X ∈ Q) then u is Hölder continuous in Q if n = 2.

Thereafter, for quasibasic systems of the following kind

a(X, H(u)) − \frac{∂u}{∂t} = b(X, u, Du),

where a(X, ξ) satisfies the condition (A) and b(X, u, p) has linear growth (β = β′ = 1) we obtain that a solution u and the spatial gradient Du are Hölder continuous in Q for n ≤ 4 and n = 2 respectively.

Moreover if u ∈ W^{2}(Q, R^{N}) is solution in Q of the parabolic basic system

a(H(u)) − \frac{∂u}{∂t} = b(X, u, Du),

then u is Hölder continuous in Q for n ≤ 6 and the gradient Du for n ≤ 4.