Geodesic
Integrability for very weak solutions to boundary value problems of $p$-harmonic equation
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 101-110
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The paper deals with very weak solutions $u\in \theta + W_0^{1,r}(\Omega )$, $\max \{1,p-1\}r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$ u\in \begin {cases} \theta +L_{\rm weak}^{q^*}(\Omega ) \mbox {for } qn, \end {cases} $$ provided that $r$ is sufficiently close to $p$.
The paper deals with very weak solutions $u\in \theta + W_0^{1,r}(\Omega )$, $\max \{1,p-1\}$, to boundary value problems of the \mbox {$p$-harmonic} equation $$ \begin {cases} -\mbox {div}(|\nabla u(x)|^{p-2} \nabla u(x))=0, x\in \Omega , \\ u(x)=\theta (x), x\in \partial \Omega . \end {cases} \eqno (*) $$ We show that, under the assumption $\theta \in W^{1,q}(\Omega )$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$ u\in \begin {cases} \theta +L_{\rm weak}^{q^*}(\Omega ) \mbox {for } q, \\ \theta +L_{\rm weak}^\tau (\Omega ) \mbox {for } q=n \mbox { and any } \tau \infty , \\ \theta +L^\infty (\Omega ) \mbox {for } q>n, \end {cases} $$ provided that $r$ is sufficiently close to $p$.
DOI : 10.1007/s10587-016-0242-5
Classification : 35D30, 35J25
Keywords: integrability; very weak solution; boundary value problem; $p$-harmonic equation 
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Gao, Hongya; Liang, Shuang; Cui, Yi. Integrability for very weak solutions to boundary value problems of $p$-harmonic equation. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 101-110. doi: 10.1007/s10587-016-0242-5

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