Fully anisotropic elliptic problems with minimally integrable data

Abstract

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic N -function, which is not necessarily of power type and need not satisfy the ∆2 nor the ∇2condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutionsin the approximable sense -is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of L 1 -data.

Gradient estimates for problems with Orlicz growth

“…Definition A.2), Mingione in [45] states that f ∈ L q,θ =⇒ |Du| p−1 ∈ L θq θ−q ,θ for 2 ≤ p < θ and 1 < q ≤ θp θp − θ + p . (6) Note that in comparison to (5) describing the range of parameters we change n to θ. See the classical paper by Stampacchia [50] for the first proof of the linear case (p = 2) of (6), [1] by Adams for the sharp linear version, and [45] by Mingione for the nonlinear one (p ≥ 2), its sharpness, and the corresponding result in the Lorentz-Morrey setting f ∈ L θ (q, s) =⇒ |Du| p−1 ∈ L θ nq n − q , ns n − q within the whole above range of parameters and including all bordeline cases.…”

“…We stress that the issue of gradient estimates for L 1 or measure data is deeply investigated in the Sobolev setting, but little is known in the Orlicz spaces, where we want to contribute. To our best knowledge in the Orlicz setting the Marcinkiewicz estimates are restricted to [6,18,22], while the Lorentz or the Morrey estimates for problems posed in the Orlicz spaces are not known yet at all.…”

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

“…There are also certain other notions also sharing fundamental property of uniqueness for L 1 -data. Recently in [60] in the Orlicz setting the notion of approximable solutions has been introduced, somehow merging the ideas of SOLA and entropy solutions, see also [10,46]. Some of the mentioned results are relevant in the context of measure data problems.…”

“…Note that [24,195] concern isotropic variable exponent spaces, while [22,23,141] the anisotropic ones. Isotropic and reflexive Orlicz spaces are employed in [7,26,60,61,80], isotropic and nonreflexive in [46], while anisotropic and nonreflexive ones in [10]. In [92,117,147] isotropic, separable and reflexive Musielak-Orlicz spaces are employed, [81] studies separable, but not reflexive Musielak-Orlicz spaces, while in anisotropic and non-reflexive Musielak-Orlicz spaces in [106,112,113].…”

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

“…Nonetheless, infering regularity estimates for special classes of very weak solutions and their gradients in Marcinkiewicz-type spaces has already become classical [12,14] and has been investigated futher in Lorentz and Morrey scale too [48]. For the corresponding recent results in the Orlicz setting we refer to [20,24,4]. On the other hand, it is known that the presence of the lower-order term satisfying the sign condition brings a regularizing effect on solutions to standard-growth problems [13,54].…”