On the continuity of the minima of variational integrals in orlicz-sobolev spaces

Abstract

In this paper I investigate the minima of the variational integral ∫_Ωf(x,u,▽u)dx, where f(x,u,p) : ω × R¹ × Rⁿ → R¹ satisfies the following condition: M(|p|) − K ≤ f(x,u,p) ≤ aM(|p|) + K with K ≥ 0, a ≥ 1, and M(t) − an N-function. I prove the continuity of such minima when M(t) satisfies the Δ₂ condition and the following further condition: there exists an m_ε with $\mathop {\lim }\limits_{\varepsilon \to 0^ + } \varepsilon m_\varepsilon = 0$for which M(βt) ≤ ≤ β^n-ε M(t) if 0 < β < 1 and t > 0.

The author is a member of the GNAFA of the CNR.