Regularity results for solutions of linear elliptic degenerate boundary-value problems
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A. El Baraka
Arabian Journal of Mathematics
· M. Masrour
Regularity results for solutions of linear elliptic degenerate boundary-value problems
Received: 15 July 2019 / Accepted: 1 February 2020 © The Author(s) 2020
Abstract We give an a-priori estimate near the boundary for solutions of a class of higher order degenerate elliptic problems in the general Besov-type spaces B s,τ p,q . This paper extends the results found in Hölder spaces C s , Sobolev spaces H s and Besov spaces B sp,q , to the more general framework of Besov-type spaces. Mathematics Subject Classification
35J30 · 35J40 · 35B45 · 35B65
1 Introduction, definitions, and results 1.1 Introduction The aim of this article is to give an a-priori estimate for solutions of a class of linear degenerate elliptic boundary-value problems in Besov-type spaces involving the differential operator: L˜ =
min(k,m)
ϕ k−h P m−h (x, Dx ),
(1)
h=0
where k ∈ N, m ∈ N\{0}, the function ϕ is of class C ∞ from Rn+1 to R and associates with each element of Ω its distance from the boundary, with Ω = {x ∈ Rn+1 ; ϕ(x) > 0}, ∂Ω = {x ∈ Rn+1 ; ϕ(x) = 0} and dϕ = 0 on ∂Ω; P m−h (x, Dx ) is a differential operator with smooth coefficients on Ω and of order ≤ m − h. These operators were first introduced by Shimakura [16], who obtained a regularity result in Sobolev spaces ˜ Similar results were found by Bolley and H s for solutions of the boundary-value problems associated with L. Camus [3]. In addition, the same class was considered by C. Goulaouic and N. Shimakura [14] and also by Bolley et al. [1] in Hölder spaces C s . Later on, Rolland [15] gave an a-priori estimate of (1) in classical Besov spaces B sp,q with p = q. In this paper, we generalize the previous works to the more general frame of Besov-type spaces B s,τ p,q . They contain all the spaces cited previously: Hölder spaces C s , Sobolev spaces H s , and Besov spaces B sp,q , and include Goldberg spaces bmo and local Morrey–Campanato spaces l 2,λ , as a special case (see Remark 2). In the same spaces, the first author has established a regularity result for solutions of a class of regular elliptic boundary-value problems [12]. In [13], the authors investigated a particular case of operators (1) in B s,τ p,q spaces, while many researchers were interested in other degenerate operators; for example [4,7,10,20]. A. El Baraka (B)· M. Masrour FST Fes, Laboratory Modelling and Mathematical Structures, Department of Mathematics, University Sidi Mohamed Ben Abdellah, B.P. 2202, Route Immouzer, 30000 Fes, Morocco E-mail: [email protected]
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The results of this paper can be useful in several applications, namely, the study of the Lake equation. Indeed, a particular case of operators (1) models this phenomenon. For more details, we refer to [6] and [5, Section 7.2]. In addition, just as in [10, Section 2, Theorem 2.5], these estimates can be employed to prove the regularity of solutions of completely nonlinear boundary-value problems. The sections of this paper will be tackled in this o
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