Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds
From the introduction of the paper: ``We solve three basic potential theoretic problems: (I) Hodge decompositions for vector fields, (II) Poisson problems for the Hodge Laplacian, and (III) inhomogeneous Maxwell equations in Lipschitz subdomains of a smooth, compact, boundaryless, three-dimensional Riemannian manifold \({\mathcal M}\). They are all considered in the context of Sobolev-Besov spaces, that is, when the global smoothness of both the data and the solutions is measured on these scales. In hindsight, the problems (I)--(III) above turn out to be closely related. A manifestation of this is that they share a common, (asymptotically) sharp ``well-posedness region'', stemming from necessary limitations on the indices \(s\) (smoothness) and \(p\) (integrability), of the Sobolev-Besov spaces for which these partial differential equations (PDEs) have unique solutions, continuously dependent on the given data. In turn, this region, call it \({\mathcal R}_\Omega\), is entirely determined by the geometric characteristics of the underlying domain \(\Omega\subset {\mathcal M}\). More concretely, \[ \left(s,\frac{1}{p}\right)\in {\mathcal R}_\Omega \Longleftrightarrow\begin{cases} 0<\frac{ 1}{p}<1,\;-1+\frac{1}{p}<s<\frac{1}{p},\\ \frac{ 2}{3}(1-\frac{1}{p_\Omega})<\frac{1}{p}-\frac{s}{3} <\frac{1}{3}(\frac{2}{p_\Omega}+1).\end{cases} \] Here, \(p_\Omega\) is further defined in terms of the critical exponents intervening in the (regular) Dirichlet and Neumann problems for the Laplace-Beltrami operator in \(\Omega\) (as well as its complement) when optimal \(L^p\)-estimates for the associated nontangential maximal function are sought. A precise definition is given in \S 4. For the purpose of this introduction, we note that, generally speaking, \(1\leq p_\Omega <2\) (cf.\ \textit{B. E. J. Dahlberg} [Arch. Ration. Mech. Anal. 65, 272--288 (1977; Zbl 0406.28009) and Stud. Math. 66, 13--24 (1979; Zbl 0422.31008)], \textit{D. S. Jerison} and \textit{C. E. Kenig} [Bull. Am. Math. Soc. (N.S.) 4, 203--207 (1981; Zbl 0471.35026)], \textit{B. E. J. Dahlberg} and \textit{C. E. Kenig} [Ann. Math. (2) 125, 437--465 (1987; Zbl 0658.35027)], \textit{G. Verchota} [J. Funct. Anal. 59, 572--611 (1984; Zbl 0589.31005)] for the flat, Euclidean setting and \textit{M. Mitrea} and \textit{M. Taylor} [J. Funct. Anal. 163, 181--251 (1999; Zbl 0930.58014)], \textit{M. Mitrea} and \textit{M. Taylor} [Commun. Partial Differ. Equations 25, 1487--1536 (2000; Zbl 1157.35345)] for Lipschitz subdomains of Riemannian manifolds). One feature of \(\Omega\) which influences the size of \(p_\Omega\) is the local oscillations of the unit conormal \(\nu\) to \(\partial \Omega\), that is, \(\lim_{\varepsilon\rightarrow 0} \text{sup}\{\text{dist}(\nu(x),\nu(y));\text{dist} (x,y)<\varepsilon\}\) (cf. the comments in [\textit{A. P. Calderón}, North-Holland Math. Stud. 111, 33--48 (1985; Zbl 0608.31001)]'). In fact, \(p_\Omega\) tends to 1 as these local oscillations tend to zero. Moreover, \(p_\Omega =1\) when \(\partial\Omega\in C^1\) [see \textit{E. B. Fabes}, \textit{M. Jodeit jun.} and \textit{N. M. Rivière}, Acta Math. 141, 165--186 (1978; Zbl 0402.31009)], and this continues to be the case even when the unit conormal has only vanishing mean oscillations. Finally, for a Lipschitz polyhedron in the Euclidean setting, \(p_\Omega\) can be estimated in terms of the dihedral angles involved [cf. \textit{N. V. Grachev} and \textit{V. G. Maz'ya}, Solvability of a boundary integral equation on a polyhedron (Research report LiTH-MAT-R-91-50, Linköping University, Linköping, Sweden) (1991)].'' From the abstract of the paper: ``In each of the problems (I)--(III)'' we derive sharp estimates on Sobolev-Besov scales and establish integral representation formulas for the solution. The proofs rely on tools from harmonic analysis and algebraic topology, such as Calderón-Zygmund theory and de Rham theory.''
scientific article; zbMATH DE number 2141301
scientific article