Integrability for very weak solutions to boundary value problems of $p$-harmonic equation

Gao, Hongya ; Liang, Shuang ; Cui, Yi

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 101-110.

Résumé

The paper deals with very weak solutions $u\in \theta +W_0^{1,r}(\mathrm{\Omega })$, $max\{1,p-1\}$, 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 $\theta \in W^{1,q}(\mathrm{\Omega })$, $q>r$, any very weak solution $u$ to the boundary value problem ($\ast$) 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 $p$.

Détail

DOI : 10.1007/s10587-016-0242-5

Classification : 35D30, 35J25
Mots-clés : integrability; very weak solution; boundary value problem; $p$-harmonic equation