Electrorheological Fluids: Modeling and Mathematical Theory

Initial-boundary-value problems for anisotropic elliptic-parabolicpseudoparabolic equations with variable exponents of nonlinearity

“…The functions p j are called exponents of nonlinearity. Nonlinear differential equations with variable exponents of nonlinearity describe many physical processes such us electromagnetic fields, electrorheological fluids, image reconstruction processes, current flow in variable temperature field [15]. Solutions of these problems belong to some generalized Lebesgue and Sobolev spaces.…”

Local boundedness of quasi-minimizers of integral functionals with variable exponent anisotropic growth and applications

“…We will also use the space [9,11,18,28,40,41] for basic results on W 1,p(·) (Ω). For the Banach spaces X and Y , the notation X → Y denotes that X is continuously imbedded into Y , and the notation X → → Y denotes that X is compactly imbedded into Y .…”

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

“…Then the singular operator S Γ is bounded in the space L p(·) (Γ, w) , if and only if 9) and also (1.10) in the case Γ is infinite.…”

“…This theory is known to develop rapidly last years in connection with various applications, see for instance [8][9][10], where other references may be also found. We give below the necessary definitions for the case of spaces on Carleson curves.…”