Classical Solutions of the Divergence Equation with Dini Continuous Data
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Journal of Mathematical Fluid Mechanics
Classical Solutions of the Divergence Equation with Dini Continuous Data Luigi C. Berselli
and Placido Longo
Communicated by G. P. Galdi
Abstract. We consider the boundary value problem associated with the divergence operator on a bounded regular subset of Rn , with homogeneous Dirichlet boundary condition. We prove the existence of a classical solution under slight assumptions on the datum. Mathematics Subject Classification. Primary 26B12; Secondary 35C05, 35F15. Keywords. Divergence equation, Classical solutions, Boundary value problem, Dini continuity.
1. Introduction In this paper we deal with the existence of classical solutions for the first order boundary value problem div u = F in Ω, (1) u=0 on ∂Ω. We look for solutions u : Ω → Rn , belonging at least to C 1 (Ω) ∩ C 0 (Ω) but, actually, we will prove a sharper result of regularity at the boundary (see Theorem 1). Here, Ω is a smooth, bounded, open subset of Rn , n ≥ 2, while F is a given continuous function (as expected, F will be required to fulfill a condition slightly stronger than the bare continuity), satisfying the compatibility condition Ω F (x) dx = 0. This is a classical problem in mathematical fluid mechanics, strictly connected with the Helmholtz decomposition and the div–curl lemma (see Kozono and Yanagisawa [20]). We recall that, if the boundary condition is dropped, a solution of the divergence equation can be readily obtained by taking the gradient of the Newtonian potential of F , provided it is in C 2 (Ω). These aspects are extensively covered in Galdi [15, Ch. III], with special attention to the work of Bogovski˘ı [7], where the problem (1) is solved in the setting of the Sobolev spaces H01,p (Ω). Further developments may also be found in Borchers and Sohr [8]. For different approaches and results, the reader should consider the books by Ladyzhenskaya [21] and Tartar [26], which especially cover the Hilbert case, while Amrouche and Girault [1] devised an approach based on the negative norm theory developed in Neˇcas [23]. Our approach follows closely the Bogovski˘ı’s one, where the representation formula (2) below, in analogy with the Sobolev’s “cubature” formulae, provides explicitly a special solution of the problem (1). We recall that, per se, problem (1) has infinitely many solutions. The representation formula (2) turns out to be extremely flexible in the applications to many different settings as, for instance, in the recent older spaces have been shown results for weighted and Lp(x) -spaces (see Huber [17]). Classical results in H¨ in Kapitansk˘ı and Piletskas [18], as corollaries of a more general result, which seems to be obtained in a way different from ours. We point out that our methods, which can be considered as classical, can be also easily modified to obtain the corresponding results in H¨ older spaces, for which we also mention the recent review in Csat´o et al. [11]. In addition, we also note that the non-uniqueness feature of the first order system (1) allows some existence results
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