Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) and let \(\varphi:\Omega\times\mathbb{R}^+\rightarrow \mathbb{R}^+\) be a measurable function. The generalized Morrey space \(M_{p,\varphi}(\Omega)\), \(1\leq p<\infty\), consists of all \(f\in L^p(\Omega)\) such that \[ \|f\|_{M_{p,\varphi}(\Omega)}:=\sup_{x\in\Omega, r>0}\varphi(x,r)r^{-1}\Big(\int_{\Omega_r}|f(y)|^p\,dy\Big)^{1/p}<\infty. \] Here, \(\Omega_r=\Omega\cap\mathcal{B}_r(x)\), where \(\mathcal{B}_r\) is a ball in \(\mathbb{R}^n\) centered in some point \(x\in \Omega\). The generalized Sobolev-Morrey space \(W_{2,p,\varphi}(\Omega)\) is the set of all Sobolev functions \(f\in W_{2,p}(\Omega)\) with distributional derivatives \(D^sf\in M_{p,\varphi}(\Omega)\), endowed with the norm \[ \|f\|_{W_{2,p,\varphi}(\Omega)}=\sum_{0\leq |s|\leq2}\|D^sf\|_{M_{p,\varphi}(\Omega)}. \] Consider the Dirichlet problem for a linear uniformly elliptic operator \[ \begin{cases} \mathcal{L}u:=a^{ij}(x)D_{ij}u=f(x), \quad \text{ a.a. } x\in \Omega\\ u\in W_{2,p,\varphi}(\Omega)\cap W_{1,p}^0(\Omega), \quad p\in (1,\infty) \end{cases} \] where the coefficients \(a^{ij}\) belong to VMO\((\Omega)\), the class of functions of vanishing mean oscillation. This class contains also discontinuous functions. The authors prove that for any function \(f\in M_{p,\varphi}(\Omega)\), \(1<p<\infty\), there exists a unique solution of the Dirichlet problem above and it has second order derivatives in \(M_{p,\varphi}(\Omega)\). Besides \[ \|D^2u\|_{M_{p,\varphi}(\Omega)}\leq C(\|f\|_{M_{p,\varphi}(\Omega)}+\|u\|_{M_{p,\varphi}(\Omega)}). \] The core of the matter is the study of continuity in the spaces \(M_{p,\varphi}\) of certain sublinear integrals and their commutators with BMO functions which, in turn, allows them to obtain continuity of Calderón--Zygmund operators in \(M_{p,\varphi}(\mathbb{R}^n)\). With these results, by using representation formulas of \(D_{ij}u\) in terms of such operators, and standard techniques from the theory of PDE's, the authors get the regularity results abovementioned for the solution of the Dirichlet problem. Pioneer works concerning Dirichlet problem for linear uniformly elliptic operators with VMO coefficients are [\textit{F. Chiarenza, F. Frasca} and \textit{P. Longo}, Ric. Mat. 40, 149--168 (1991; Zbl 0818.35017); Trans. Am. Math. Soc. 336, No. 2, 841--853 (1993; Zbl 0772.35023)]; there, they study local and global Sobolev regularity of the strong solutions, and also prove that if \(\mathcal{L}u\in L^p(\Omega)\) then \(u\in W^{2,p}(\Omega)\), \(1<p<\infty\). These latter works were later extended in the Sobolev--Morrey spaces by \textit{G. Di Fazio, D. K. Palagachev} and \textit{M. A. Ragusa} [J. Funct. Anal. 166, No. 2, 179--196 (1999; Zbl 0942.35059)] and \textit{G. D. Fan, S. Lu} and \textit{D. Yang} [Georgian Math. J. 5, No. 5, 425--440 (1998; Zbl 0917.35017)]. The present paper generalizes some of these results. See also [\textit{N. V. Krylov}, Commun. Partial Differ. Equations 32, No. 3, 453--475 (2007; Zbl 1114.35079)] for a \(L^p\) theory of a larger class of operators with VMO coefficients.
scientific article; zbMATH DE number 6160854
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