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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