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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.

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