Gradient estimates for Orlicz double phase problems with variable exponents

Baasandorj, Sumiya; Byun, Sun‐Sig; Lee, Hosik

doi:10.1016/j.na.2022.112891

Cited by 7 publications

(7 citation statements)

References 45 publications

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“…Then (3.5) holds for any θ > 0 in Cases ( 1)-( 2) and for θ = s * q−p in Case (3). The constant b 0 depends only on n, p, q, L, L ω and b.…”

Section: Model and Functionmentioning

confidence: 90%

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Regularity theory for non-autonomous problems with a priori assumptions

Hästö¹,

Ok²

2022

Preprint

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We study weak solutions and minimizers u of the non-autonomous problems divA(x, Du) = 0 and min v ´Ω F (x, Dv) dx with quasi-isotropic (p, q)-growth. We consider the case that u is bounded, Hölder continuous or lies in a Lebesgue space and establish a sharp connection between assumptions on A or F and the corresponding norm of u. We prove a Sobolev-Poincaré inequality, higher integrability and the Hölder continuity of u and Du. Our proofs are optimized and streamlined versions of earlier research that can more readily be further extended to other settings.Connections between assumptions on A or F and assumptions on u are known for the double phase energy F (x, ξ) = |ξ| p + a(x)|ξ| q . We obtain slightly better results even in this special case. Furthermore, we also cover perturbed variable exponent, Orlicz variable exponent, degenerate double phase, Orlicz double phase, triple phase, double variable exponent as well as variable exponent double phase energies and the results are new in most of these special cases.

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“…Then (3.5) holds for any θ > 0 in Cases ( 1)-( 2) and for θ = s * q−p in Case (3). The constant b 0 depends only on n, p, q, L, L ω and b.…”

Section: Model and Functionmentioning

confidence: 90%

“…See Corollary 1.1 for the corresponding expressions F and Table 1 for examples of our assumptions in some of these cases. We refer to [41] for references up to 2020 and [3,4,18,20,30,32,45,49] for some more recent advances on variants of the variable exponent and double phase models.…”

Section: Introductionmentioning

confidence: 99%

Regularity theory for non-autonomous problems with a priori assumptions

Hästö¹,

Ok²

2022

Preprint

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“…In recent years, there are many results of differential equations in Musielak-Orlicz-Sobolev spaces. For example, Fan [17] and we [18] proved the existence of weak solutions of a class of differential equations of divergence form by using a subsupersolution method in reflexive Musielak-Orlicz-Sobolev spaces and nonreflexive Musielak-Orlicz-Sobolev spaces, respectively; Li et al [19] proved the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces; we [20] proved the existence of barrier solutions of elliptic differential equations in Musielak-Orlicz-Sobolev spaces; and Baasandorj et al [21] established optimal regularity estimates for the gradient of solutions to nonuniformly elliptic equations of the Orlicz double phase with variable exponent types. Musielak-Orlicz functions (described in Section 2) have many applications such as non-Newtonian fluids (see, e.g., [22]), thermistor problem (see, e.g., [23]), and image restoration (see, e.g., [24]).…”

Section: Introductionmentioning

confidence: 95%

Existence of Solutions for Inclusion Problems in Musielak-Orlicz-Sobolev Space Setting

Dong

Fang

2023

Journal of Function Spaces

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In this paper, we mainly prove the existence of (weak) solutions of an inclusion problem with the Dirichlet boundary condition of the following form: L ∈ A x , u , D u + F x , u , D u , in Ω , and u = 0 , on ∂ Ω , in Musielak-Orlicz-Sobolev spaces W 0 1 L Φ Ω by using the surjective theorem, where Ω ⊂ ℝ N is a bounded Lipschitz domain, L belongs to the dual space W 0 1 L Φ Ω ∗ of W 0 1 L Φ Ω , A is a multivalued maximal monotone operator, and F is a multivalued convection term. Some examples for A and F are given in the paper. Then, we give some properties of the solution set of the inclusion problem. We also show the existence of (weak) solutions of the inclusion problem with an obstacle effect.

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“…On the other hand, since (1.1) features purely nonlocal behaviors on the set {a(x) = 0}, we could not compare (1.1) with a local problem as in [26]. We thus develop a different method motivated from the ones in [1,3,23], whose crucial tools include the expansion of positivity results described in Lemmas 4.2 and 4.3 below. For fractional p-Laplacian type problems, analogous results are proved in [23,Lemma 6.3], but their proofs are not directly applicable to our double phase setting.…”

Section: Introductionmentioning

confidence: 99%