On the role of energy convexity in the web function approximation

For a given p > 1 and an open bounded convex set Ω ⊂ℝ2,we consider the minimization problem for the functional Jp (u) = ∫Ω(1/p|∇ μ|p - u)over W01,pp Ω.Since the energy of the unique minimizer u p may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary δΩ. In this case, a representation formula for the unique minimizer v p is available. Hence the problem of estimating the error one makes when approximating J p (u p ) by J p (v p ) arises. When Ω varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p → ∞, we also prove that u p-v p converges to zero in W01,m(Ω) for all m<∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J p increases. © Birkhäuser Verlag, Basel 2005.