FRECHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT

Let Omega be a domain in R-N, let p(.) is an element of C((Omega) over bar) be such that p(x) > 1 for all x is an element of (Omega) over bar, let W-1,W-p(.)(Omega) be the Sobolev space with variable exponent p(.), let Gamma(0) be a d Gamma-measurable subset of Gamma = partial derivative Omega that satisfies d Gamma-meas Gamma(0) > 0, and let U-Gamma 0 = {u is an element of W-1,W-p(.)(Omega); tr u = 0 on Gamma(0)}. It is shown that the map u is an element of U-Gamma 0 bar right arrow parallel to u parallel to(0),(p(.)),(del) = parallel to vertical bar del u vertical bar parallel to(0),(p(.)) is a Frechet-differentiable norm on U-Gamma 0, and a formula expressing the Frechet derivative of this norm at any nonzero u is an element of U-Gamma 0 is given. We also show that, if p(x) >= 2 for all x is an element of (Omega) over bar, (U-Gamma 0, parallel to u parallel to(0),(p(.)),(del)) is uniformly convex. Using properties of duality mappings defined on Banach spaces having a Frechet-differentiable norm, we give the explicit form of continuous linear functionals on (U-Gamma 0, parallel to u parallel to(0),(p(.)),(del)). It is also shown that the space U-Gamma 0 and its dual have the same Krein-Krasnoselski-Milman dimension.