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