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

Hongya Gao; Shuang Liang; Yi Cui

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 101-110
  • ISSN: 0011-4642

Abstract

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The paper deals with very weak solutions u ∈ θ + W 0 1 , r ( Ω ) , max { 1 , p - 1 } < r < p < n , to boundary value problems of the p -harmonic equation - div ( | ∇ u ( x ) | p - 2 ∇ u ( x ) ) = 0 , x ∈ Ω , u ( x ) = θ ( x ) , x ∈ ∂ Ω . ( * ) We show that, under the assumption θ ∈ W 1 , q ( Ω ) , q > r , any very weak solution u to the boundary value problem ( * ) is integrable with u ∈ θ + L weak q * ( Ω ) for q < n , θ + L weak τ ( Ω ) for q = n and any τ < ∞ , θ + L ∞ ( Ω ) for q > n , provided that r is sufficiently close to p .

How to cite

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Gao, Hongya, Liang, Shuang, and Cui, Yi. "Integrability for very weak solutions to boundary value problems of $p$-harmonic equation." Czechoslovak Mathematical Journal 66.1 (2016): 101-110. <http://eudml.org/doc/276791>.

@article{Gao2016,
abstract = {The paper deals with very weak solutions $u\in \theta + W_0^\{1,r\}(\Omega )$, $\max \lbrace 1,p-1\rbrace <r<p<n$, to boundary value problems of the $p$-harmonic equation \[ \{\left\lbrace \begin\{array\}\{ll\} -\mbox\{div\}(|\nabla u(x)|^\{p-2\} \nabla u(x))=0, & x\in \Omega , \\ u(x)=\theta (x), & x\in \partial \Omega . \end\{array\}\right.\} \qquad \mathrm \{(*)\}\] 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 \[ u\in \{\left\lbrace \begin\{array\}\{ll\} \theta +L\_\{\rm weak\}^\{q^*\}(\Omega ) & \mbox\{for \} q<n, \\ \theta +L\_\{\rm weak\}^\tau (\Omega ) & \mbox\{for \} q=n \mbox\{ and any \} \tau <\infty , \\ \theta +L^\infty (\Omega ) & \mbox\{for \} q>n, \end\{array\}\right.\} \] provided that $r$ is sufficiently close to $p$.},
author = {Gao, Hongya, Liang, Shuang, Cui, Yi},
journal = {Czechoslovak Mathematical Journal},
keywords = {integrability; very weak solution; boundary value problem; $p$-harmonic equation},
language = {eng},
number = {1},
pages = {101-110},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Integrability for very weak solutions to boundary value problems of $p$-harmonic equation},
url = {http://eudml.org/doc/276791},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Gao, Hongya
AU - Liang, Shuang
AU - Cui, Yi
TI - Integrability for very weak solutions to boundary value problems of $p$-harmonic equation
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 101
EP - 110
AB - The paper deals with very weak solutions $u\in \theta + W_0^{1,r}(\Omega )$, $\max \lbrace 1,p-1\rbrace <r<p<n$, to boundary value problems of the $p$-harmonic equation \[ {\left\lbrace \begin{array}{ll} -\mbox{div}(|\nabla u(x)|^{p-2} \nabla u(x))=0, & x\in \Omega , \\ u(x)=\theta (x), & x\in \partial \Omega . \end{array}\right.} \qquad \mathrm {(*)}\] 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 \[ u\in {\left\lbrace \begin{array}{ll} \theta +L_{\rm weak}^{q^*}(\Omega ) & \mbox{for } q<n, \\ \theta +L_{\rm weak}^\tau (\Omega ) & \mbox{for } q=n \mbox{ and any } \tau <\infty , \\ \theta +L^\infty (\Omega ) & \mbox{for } q>n, \end{array}\right.} \] provided that $r$ is sufficiently close to $p$.
LA - eng
KW - integrability; very weak solution; boundary value problem; $p$-harmonic equation
UR - http://eudml.org/doc/276791
ER -

References

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