Regularity for solutions of nonlinear elliptic equations. (English) Zbl 0842.35014

Betta, M. F.; Ferone, V.; Mercaldo, A.

Bull. Sci. Math., II. Sér. 118, No. 6, 539-567 (1994).

The authors study Dirichlet problems for instance of type \[ -\text{div} \bigl( a(x,u, Du) \bigr) + d(x) Du |Du |^{p - 1} + H(x,u) = \text{div} f \text{ in } \Omega, \;u \in W^{1,p}_0 (\Omega) \] under structural assumptions on \(a,d,H\) and \(f\), which allow for a comparison of \(u\) via Schwarz-symmetrization with a solution \(v(r)\) of the associated problem \(-\Delta_p v = \text{div} F(r)\) in \(\Omega^\#\), \(u \in W_0^{1,p} (\Omega^\#)\). The construction of \(F\) from the data is quite interesting and nonstandard. Regularity of \(u\) in Orlicz spaces is derived when \(|f|\) and coefficients of the equation belong to suitable Lorentz spaces.

Reviewer: B.Kawohl (Köln)

MSC:

35B65	Smoothness and regularity of solutions to PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35B45	A priori estimates in context of PDEs
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

regularity in Orlicz spaces; Schwarz-symmetrization; Lorentz spaces