Geodesic
Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 379-390
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We prove boundedness and continuity for solutions to the Dirichlet problem for the equation \[ -{\rm div}(a(x,\nabla u))=h(x,u)+\mu ,\quad \text {in} \ \Omega \subset \mathbb R^N, \] where the left-hand side is a Leray-Lions operator from $W_0^{1,p} (\Omega )$ into $W^{-1,p'}(\Omega )$ with $1
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation \[ -{\rm div}(a(x,\nabla u))=h(x,u)+\mu ,\quad \text {in} \ \Omega \subset \mathbb R^N, \] where the left-hand side is a Leray-Lions operator from $W_0^{1,p} (\Omega )$ into $W^{-1,p'}(\Omega )$ with $1$, $h(x,s)$ is a Carathéodory function which grows like $|s|^{p-1}$ and $\mu $ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu $.
DOI : 10.21136/CMJ.2018.0322-17
Classification : 35B45, 35B65, 35J15, 35J25, 35J60, 35J92
Keywords: bounded solution; $p$-Laplacian; renormalized solution; measure data 
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     journal = {Czechoslovak Mathematical Journal},
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     year = {2019},
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Dall'Aglio, Andrea; Segura de León, Sergio. Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 2, pp. 379-390. doi: 10.21136/CMJ.2018.0322-17

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