Measure data elliptic problems with generalized Orlicz growth

2021

Preprint

Self Cite

We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.

“…See [3,17,20,25] for related existence and regularity results in the scalar case and [27,38,39,45,65] for vectorial existence results in various regimes.…”

Section: Remark 33 (Existence and Uniqueness Of Weak Solutions) Formentioning

confidence: 99%

Wolff potentials and measure data vectorial problems with Orlicz growth

Chlebicka¹,

Youn²,

2021

Preprint

Self Cite

“…The first a priori estimate. Let us recall (15) and structure assumption (6) imposed on A. Making use of ( 5), ϕ ϕ ϕ = T t (u u u j ) in (33) and by (32) for t > 0 and for any j ∈ N one obtains that…”

Section: Existence and Marcinkiewicz Regularity To Vectorial Measure ...mentioning

confidence: 99%

“…Recently the theory has been developed in Orlicz and generalized Orlicz setting e.g. by [3,15,18,23]. For pioneering work on the equations with Orlicz growth we refer to [37,49,56], while for regularity to their measure data counterparts see [5,10,14,16,19].…”

Section: Introductionmentioning

confidence: 99%

Measure data systems with Orlicz growth

Chlebicka¹,

Youn²,

2021

Preprint

Self 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.

“…In the general growth case Young's inequality reads as inequality (20) ts ≤ G(t) + G(s) for all s, t ≥ 0.…”

Section: 2mentioning

confidence: 99%

“…The definition coincides with SOLA (Solutions Obtained as a Limit of Approximation) introduced in [11] except for the use of truncations (19) as in [8], which broadens the class of admissible solutions. See Section 3.5 for the precise definition of this notion of very weak solutions, [29] for the existence and basic regularity, and [1,20,23] for related existence results. Such solutions can be unbounded if the measure concentrates (see Corollary 2.5 and Remark 2.7 for examples), though still one can provide pointwise estimates by a relevant potential from above and below.…”

Section: Introductionmentioning

confidence: 99%

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

Chlebicka¹,

Giannetti²,

2020

Preprint

Self Cite

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for A-superharmonic functions with nonlinear operator A : Ω × R n → R n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg-Wolff theorem on characterization of the dual of the Orlicz space.