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November/December 2011 A few natural extensions of the regularity of a very weak solution
J.M. Rakotoson
Differential Integral Equations 24(11/12): 1125-1146 (November/December 2011). DOI: 10.57262/die/1356012880


We consider a linear operator $L_m$ with variable coefficients of order $2m$ and we study the regularity of the very weak solution $u$ integrable in a bounded open smooth set $\Omega$, \[ \int_\Omega uL^*_m {\varphi} \,{\rm dx} = \int_\Omega {\varphi} d\mu\quad\forall\,{\varphi}\in C^{2m}( \overline \Omega) \] with $ \frac{{{\partial}}^j{\varphi}}{{{\partial}}\nu^j}=0$ on the boundary ${{\partial}}\Omega$ for $j{\leqslant} m-1$, where $L^*_m$ is the adjoint operator of $L_m$ and $\mu$ is in the space of weighted bounded Radon measures $M^1(\Omega,dist(x,{{\partial}}\Omega)^m)$. In particular, we show that the solution $u$ and all its derivatives of order $|{\gamma}|,\ |{\gamma}|{\leqslant} m-1,$ are in Lorentz spaces. If the measure on the right-hand side belongs to a smaller space such as $$ M^1(\Omega, dist(x,{{\partial}}\Omega)^{m-1+a}), \quad 0{\leqslant} a<1, $$ then all its derivatives o f order $|{\gamma}|,\ | {\gamma}|{\leqslant} m,$ are in Lorentz spaces.


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J.M. Rakotoson. "A few natural extensions of the regularity of a very weak solution." Differential Integral Equations 24 (11/12) 1125 - 1146, November/December 2011.


Published: November/December 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35082
MathSciNet: MR2866015
Digital Object Identifier: 10.57262/die/1356012880

Primary: 35G05 , 35J250 , 35J60 , 35J67 , 35P30

Rights: Copyright © 2011 Khayyam Publishing, Inc.


Vol.24 • No. 11/12 • November/December 2011
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