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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