A priori estimates for a class of degenerate elliptic equations

作者：de Miranda L H^*; Montenegro M

来源：Nodea-Nonlinear Differential Equations and Applications, 2013, 20(5): 1683-1699.

DOI：10.1007/s00030-013-0225-2

• Degenerate equations
• p-Laplacian
• Regularity theory
• Nikolskii spaces

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation %26lt;br%26gt;{-Delta pu + u = f in Omega, %26lt;br%26gt;partial derivative u/partial derivative v = 0 on partial derivative Omega, %26lt;br%26gt;when f is an element of L-q(Omega), p %26gt; 2 and q %26gt;= 2. If u is a weak solution in W-1,W-p (Omega), we obtain estimates for u in the Nikolskii space N-1+2/r,N-r (Omega), where r = q(p - 2) + 2, in terms of the L-q norm of f. In particular, due to embedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that parallel to u parallel to(r)(W1+2/r-epsilon,r(Omega)) %26lt;= C (parallel to f parallel to(q)(Lq(Omega)) + parallel to f parallel to(r)(Lq(Omega)) + parallel to f parallel to(2r/p)(Lq(Omega)) for every epsilon %26gt; 0 sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in W-1,W-r (Omega).

• 出版日期2013-10