and Du = (Dαu)∣α∣≤m. If the coefficients Aα satisfy at most polynomial growth conditions in u and its space derivatives while g obeys no growth in u, but merely a sign condition, Landes and Mustonen [6] proved that the usual truncation can be utilized to obtain weak solutions of (1) when m = 1. In [1], Brézis and Browder considered (1) but under stronger hypotheses on g. Roughly speaking, they required g to be controlled from above and below by the derivative of some convex function. In [5], Landes proved that this assumption is not necessary provided a certain a priori bound for the time derivative of solutions were needed. In [2], Browder and Breézis established an existence and uniqueness result for a general class of variational inequalities for (1) when g obeys no growth condition while A is a regular elliptic operator. Their proof is based on a type of compactness result. In this note, we extend the result of [5] to the corresponding class of variational inequalities under weaker assumptions.
Assumptions and the main result
We start by assuming the following hypotheses.
Aα(x, t, ξ) : Ω × ]0, T[ × ℝs → ℝ is continuous in t and ξ for almost all x and measurable in x for all t and ξ. Moreover, there exist a constant c1 and a function λ1 ∈ Lp′(QT) with $p\in ]1,\infty[, p'=\frac{p}{p-1}$ such that
$$
|A_{\alpha}(x,t,\xi)|\leq c_1|\xi|^{p-1}+\lambda_1(x,t) \text{ for all } (x,t)\in Q_T \text{ and } \xi\in R^s.
$$
∑|α|≤m[Aα(x, t, ξ)−α(x, t, ξ)](ξα−ξ*α) ≥ 0 for all (x, t) ∈ QT and ξ ≠ ξ* in ℝs.
There exists a constant c2 > 0 and a function λ2 ∈ L2(QT) such that ∑∣α∣≤mAα(x, t, ξ)ξα ≥ c1∣ξ∣p−λ2(x, t) for all (x, t) ∈ QT and ξ ∈ ℝs.
There is a function F(x, t, ξ) continuous in ξ, measurable in x and differentiable in t such that $\frac{\partial F}{\partial \xi_{\alpha}}=A_{\alpha}$ for all (x, t) ∈ QT and all α with ∣α∣ ≤ m.
g(x, t, r)Ω × ]0, T[ × ℝ → ℝ is continuous in t and r for almost all x and measurable in x for all t and ξ. Moreover,
$$
|g(x,t,r)|\leq \lambda_4
(x,t)\psi(r)
$$
for some continuous function ψ: ℝ → ℝ and λ4 ∈ L1(QT).
g(x, t, r)r ≥ −λ5(x, t) for some function λ5 ∈ L1(QT).
There exists a function $\tilde{f}\in L^2(Q_T)$ such that $(f,v)=\int_{Q_T}\tilde{f}(x,t)v(x,t)dx\, dt$ .
The function spaces we shall deal with will be obtained by the completion of the space of smooth functions with respect to the appropriate norm. We denote by
Put W = X ∩ C (0, T;L2(Ω)). Finally, we choose a sequence $(\Phi_i)_{i=1}^{\infty}\subset \mathcal{C}_0^{\infty}(\Omega)$ such that $\cup_{n=1}^{\infty}V_n$ with Vn = span(Φ1, Φ2, …, Φn) is dense in Wj, p(Ω):jp > mp + N.
Denote by Yn = C (0, T;Vn). Since the closure of $\cup_{n=1}^{\infty}Y_n$ with respect to the Cm−topology contains $\mathcal{C}_0^{\infty}(Q_T)$ , then for f ∈ L2(QT) there exists $f_k\in \cup_{n=1}^{\infty}Y_n$ such that fk → f in L2(QT) [4]. For simplicity, we fix the constant c throughout this note. Now we are in a position to give our result.
Theorem
Let K be a closed convex subset of C (0, T;L2(Ω)) with 0 ∈ K. Let the hypothesesA1−A4, G, andDbe satisfied. Then for a givenf ∈ W* there exists a weak solutionu ∈ W ∩ Kwithu (0) = 0 such that
where c is a constant not depending on ε, k, and n.
For this aim, let Aα,ε, g(α, ε), and $\tilde{\tilde{f}}_{\epsilon}$ be the Friedrich’s mollification in the variables (x, t) ∈ ℝN + 1 of Aα, gk, and $\tilde{f}$ , respectively. There exists a Galerkin solution uϵ ∈ Yn∩ K for the mollified variational inequality
Let vk be the truncation at level k and the mollification with respect to the time and space variables, respectively, of the Galerkin’s solution un, i.e., $v_k=((u_n^k)_{\mu})_{\sigma}$ . Letting n → ∞ in (10), taking (8) into account, and the strong convergence of $((u_n^k)_{\mu})_{\sigma}$ into uk in X with respect to σ, μ, [6], we obtain (9) and hence, uk ∈ W ∩ K is a weak solution of (3), i.e.,