THE REGULARITY OF SOLUTIONS TO SOME VARIATIONAL PROBLEMS, INCLUDING THE p-LAPLACE EQUATION FOR 3 ≤ p < 4

摘要

We consider the higher differentiability of solutions to the problem of minimisingintegral(Omega) [L(del v(x)) + g(x, v(x))]dx on u(0) + W-0(1,p) (Omega)where Omega subset of R-N, L(xi) = l(vertical bar xi vertical bar) = 1/p vertical bar xi vertical bar(p) and u(0) is an element of W-1,W-p(Omega) and hence, in particular, the higher differentiability of weak solution to the equationdiv(vertical bar del u vertical bar(p-2)del u) = f.We show that, for 3 <= p < 4, under suitable assumptions on g, there exists a solution u* to the Euler-Lagrange equation associated to the minimisation problem, such thatdel u* is an element of W-loc(s,2)(Omega)for 0 < s < 4-p. In particular, for p = 3, we show that the solution u* is such that del u* is an element of W-loc(s,2)(Omega) for every s < 1. This result is independent of N. We present an example for N = 1 and p = 3 whose solution u is such that del u* is not in W-loc(s,2)(Omega), thus showing that our result is sharp.