*Accepted Paper*

**Inserted:** 31 jul 2012

**Last Updated:** 20 jun 2013

**Journal:** Adv. Calc. Var.

**Pages:** 21

**Year:** 2013

**Abstract:**

Motivated by applications to congested optimal transport problems,
we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following:
\[
\partial_x \left[(

u_{x}

-\delta_1)_+^{q-1}\, \frac{u_{x}}{\vert u_{x}\vert}\right]+\partial_y \left[(

u_{y}

-\delta_2)_+^{q-1}\, \frac{u_{y}}{\vert u_{y}\vert}\right]=f,
\]
for $2\le q<\infty$ and some non negative parameters $\delta_1,\delta_2$. Here $(\,\cdot\,)_+$ stands for the positive part. We prove that if $f\in L^\infty_{loc}$, then $\nabla u\in L^r_{loc}$ for every $r\ge 1$.

**Keywords:**
Traffic congestion, Degenerate elliptic equations, Anisotropic problems

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