We give some diffusion models which can be used for restoration in image processing. The proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. The existence theorem for solutions and some numerical results are given.

New diffusion models in image processing

The Musielak–Orlicz setting unifies variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of developments of the theory of existence of PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev’s phenomenon.

A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.

/pdf/on-non-newtonian-fluids-with-a-property-of-rapid-thickening-4sadua7awk.pdf

On non-newtonian fluids with a property of rapid thickening under different stimulus

The paper concerns existence of weak solutions to the equations describing a motion of some non-Newtonian fluids with non-standard growth conditions of the Cauchy stress tensor. Motivated by the fluids of strongly inhomogeneous behavior and having the property of rapid shear thickening, we observe that the Lp framework is not suitable to capture the described situation. We describe the growth conditions with the help of general x-dependent convex function. This formulation yields the existence of solutions in generalized Orlicz spaces. As examples of motivation for considering non-Newtonian fluids in such spaces, we recall the electrorheological fluids, magnetorheological fluids, and shear thickening fluids. The existence of solutions is established by the generalization of the classical Minty method to non-reflexive spaces. The result holds under the assumption that the lowest growth of the Cauchy stress is greater than the critical exponent q=(3d+ 2)/(d+ 2), where d is for space dimension. The restriction on the exponent q is forced by the convective term. Copyright © 2009 John Wiley & Sons, Ltd.

Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids

We prove the Lewy–Stampacchia inequalities for the two obstacles problem in abstract form for T-monotone operators. As a consequence for a general class of quasi-linear elliptic operators of Ladyzhenskaya–Uraltseva type, including p(x)-Laplacian type operators, we derive new results of C 1,α regularity for the solution. We also apply those inequalities to obtain new results to the N-membranes problem and the regularity and monotonicity properties to obtain the existence of a solution to a quasi-variational problem in (generalized) Orlicz–Sobolev spaces.

On the two obstacles problem in Orlicz–Sobolev spaces and applications

An existence result for a strongly non-linear elliptic equation with natural growth condition on the non-linearity and L1-data is proved.

/pdf/non-linear-elliptic-problems-having-natural-growth-and-l1-3gahbehcd6.pdf

Non-linear elliptic problems having natural growth and L1 data in Orlicz spaces

Strongly nonlinear parabolic equations with natural growth terms and data in Orlicz spaces.

We prove existence and convergence theorems for nonlinear parabolic problems. We also prove some compactness results in inhomogeneous Orlicz-Sobolev spaces.

/pdf/parabolic-initial-boundary-value-problems-in-orlicz-spaces-2yok2va49w.pdf

Parabolic initial-boundary value problems in Orlicz spaces

Existence result for strongly nonlinear elliptic equation with a
natural growth condition on the nonlinearity is proved.

/pdf/existence-of-solutions-for-elliptic-equations-having-natural-24kdi775nd.pdf

Existence of solutions for elliptic equations having natural growth terms in Orlicz spaces

Elliptic inequalities with lower order terms and L1 data in Orlicz spaces

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