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Existence and regularity for a class of non-uniformly elliptic equations in two dimensions. (English) Zbl 0980.35054
The author considers the following elliptic problem: \[ -\text{div}(a(x,u)\nabla u)= f \quad\text{in}\quad \Omega \]
\[ u=0\quad\text{on}\quad \partial\Omega, \] where \(\Omega\) is bounded open set of \(\mathbb R^2, a(x,s):\Omega\times \mathbb R\to\mathbb R\) is a bounded Caratheodory function, satisfying the following assumption: \[ \frac{1}{(1+|s|)^\theta}\leq a(x,s). \] The datum \(f\) is assumed to belong to the space \(L^1(\Omega)\). It is proved existence of entropy solution \(u(x),\) such that \(u\in L^p(\Omega)\) for every \(p\geq 1\) and \(\nabla u\) belongs to the Marcinkievicz space \(M^p(\Omega)\). Additionally, if \(f\) belongs to the Lorentz space \(L(1,1)\), than \(u(x)\) is bounded. It is proved the sharpness of the mentioned regularity result in the scale of Lorentz spaces.

35J70 Degenerate elliptic equations
35B45 A priori estimates in context of PDEs