An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

Dong, Ge; Shi, Zhongrui

doi:10.1016/j.na.2006.12.004

Cited by 15 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When it comes to the Calderón-Zygmund type estimates for elliptic and parabolic equations, they fail in the case L 1 . So it would be worthwhile to find a natural counterpart of the Calderón-Zygmund type estimates in the setting of Orlicz space and is not surprising that there have been recently many research activities on the regularity theory in Orlicz space, see [4,10,11,16,17,24,[34][35][36], for the purpose of finding a natural extension of the L p regularity for the related partial differential operators. We refer to monographs [2,22,29], for more details concerning Orlicz spaces and their applications.…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates for higher order elliptic equations on nonsmooth domains

Byun

Ryu

2011

Journal of Differential Equations

View full text Add to dashboard Cite

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical W m,p , m = 1, 2, . . . , 1 < p < ∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.

show abstract

Section: Introductionmentioning

confidence: 99%

Gradient estimates for higher order elliptic equations on nonsmooth domains

Byun

Ryu

2011

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…for k, m ∈ N. Having (36), for any fixed ε > 0, the number t can be chosen so large that ( 46) 27)-( 29) and a priori estimate (35), the sequence…”

Section: If (23)mentioning

confidence: 99%

“…The cornerstone of studies on quasilinear elliptic systems in divergence form has been laid in [57,58]. Existence of weak solutions to systems with Orlicz growth was proven in [35]. The studies on elliptic systems involving measure data started with [36,48,53] and were followed by preeminent works [32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Measure data systems with Orlicz growth

Chlebicka¹,

Youn²,

Zatorska–Goldstein³

2021

Preprint

View full text Add to dashboard Cite

We study the existence of very weak solutions to a systemu u u = 0 on ∂Ω with a datum µ µ µ being a vector-valued bounded Radon measure and A : Ω × R n×m → R n×m having measurable dependence on the spacial variable and Orlicz growth with respect to the second variable. We are not restricted to the superquadratic case.For the solutions and their gradients we provide regularity estimates in the generalized Marcinkiewicz scale. In addition, we show a precise sufficient condition for the solution to be a Sobolev function.

show abstract

“…We refer to some results in variable exponent Sobolev or Orlicz-Sobolev spaces [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

Dong

Fang

2021

Journal of Function Spaces

Self Cite

View full text Add to dashboard Cite

In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x , u , D u + a 0 x , u = f x , u , D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N , a 0 : Ω × ℝ ⟶ ℝ , and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering “≤,” which are called barrier solutions.

show abstract

An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

Cited by 15 publications

References 11 publications

Gradient estimates for higher order elliptic equations on nonsmooth domains

Gradient estimates for higher order elliptic equations on nonsmooth domains

Measure data systems with Orlicz growth

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

Contact Info

Product

Resources

About