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Integrability for solutions to quasilinear elliptic systems

Francesco Leonetti, Pier Vincenzo Petricca (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.

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

Hongya Gao, Shuang Liang, Yi Cui (2016)

Czechoslovak Mathematical Journal

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 .

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case q = 3 d d + 2

Jörg Wolf (2007)

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider weak solutions 𝐮 : Ω d to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain Ω d ( d = 2 or d = 3 ). For the critical case q = 3 d d + 2 we prove the higher integrability of 𝐮 which forms the basis for applying the method of differences in order to get fractional differentiability of 𝐮 . From this we show the existence of second order weak derivatives of u .

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