The Hodge-Laplacian. Boundary value problems on Riemannian manifolds
From the authors' introduction: In their influential paper [Acta Math. 141, 165--186 (1978; Zbl 0402.31009)] on layer potential techniques for boundary value problems, \textit{E. Fabes}, \textit{M. Jodeit} and \textit{N. Rivière} have showed that for a bounded domain \(\Omega\subset \mathbb{R}^n\) of class \(\mathcal{C}^1,\) with outward unit normal \(\nu\) and boundary surface measure \(\sigma,\) the harmonic double layer \[ Kf(x):=\lim_{\varepsilon\to0^+} \frac{1}{\omega_{n-1}}\int\limits_{\underset{|x-y|>\varepsilon}{y\in\partial\Omega}} \frac{\langle \nu(y),y-x\rangle}{|x-y|^n}f(y)\text{d}\sigma(y),\quad x\in \partial\Omega,\eqno(1) \] is a compact operator on \(L^p(\partial\Omega)\) for each \(p\in(1,\infty),\) where \(\langle\cdot,\cdot\rangle\) is the standard inner product, and \(\omega_{n-1}\) denotes the area of the unit sphere in \(\mathbb{R}^n\). This continued a long line of work, originating with Erik Ivar Fredholm in the late 1800's and early 1900's whose motivation for the development of Fredholm theory was the use of such compactness results in order to treat boundary value problems via integral equations methods. In particular, this development made it possible to solve in [loc. cit.] boundary value problems for the Laplacian in a bounded \(\mathcal{C}^1\) domain \(\Omega\subset\mathbb{R}^n\) equipped with Dirichlet or Neumann boundary conditions. In the formulation of these problems, the boundary data is selected from Lebesgue spaces, the boundary traces are taken in a pointwise nontangential sense, and the size of the solution was measured through the membership of the nontangential maximal function to \(L^p(\partial\Omega)\). Concerning the compactness of \(K\) on \(L^p(\partial\Omega),\) the demand that \(\Omega\subset\mathbb{R}^n\) is of class \(\mathcal{C}^1\) may be significantly relaxed and, in [\textit{S. Hofmann} et al., Int. Math. Res. Not. 2010, No. 14, 2567--2865 (2010; Zbl 1221.31010)], the authors have identified the natural class of domains, dubbed \textit{regular SKT (Semmes-Kenig-Toro) domains,} in which principal value singular integral operators with an algebraic structure similar to that of the harmonic double layer \(K\) in \((1)\) induce compact mappings on each \(L^p(\partial\Omega),\) \(p\in(1,\infty).\) The membership of \(\Omega\) in the class of regular SKT domains is characterized by the following geometric measure theoretic conditions: \[ \begin{aligned} &\Omega\;\text{is an open set with compact Ahlfors regular boundary,}\\ &\Omega\;\text{satisfies a two-sided local John condition, and}\\ &\text{the outward unit conormal}\;\nu\;\text{of}\;\Omega\;\text{belongs to}\;\text{VMO}(\partial\Omega). \end{aligned} \eqno{(2)} \] Here VMO\((\partial\Omega)\) stands for the Sarason class of functions of vanishing mean oscillation on \(\partial\Omega\), relative to the ``surface measure'' \(\sigma:=\mathcal{H}^{n-1}\lfloor\partial\Omega\), where \(\mathcal{H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure in the ambient space (viewed as a metric space). Domains satisfying the conditions listed in \((2)\) include bounded domains of class \(\mathcal{C}^1\) or, more generally, bounded domains locally given as the upper-graph of continuous functions with gradients in VMO. This being said, regular SKT domains need not be locally graph domains since, in the plane, this class contains the category of chord-arc domains with vanishing constant. An alternative way of asserting that \(\Omega\) is a regular SKT domain is via demanding that \(\Omega\) is a two-sided NTA domain (in the sense of \textit{D. Jerison} and \textit{C. Kenig} [Adv. Math. 46, 80--147 (1982; Zbl 0514.31003)]), with a compact Ahlfors regular boundary, and whose unit conormal belongs to VMO\((\partial\Omega).\) \(\ldots\)'' Based on previous works of the authors, devoted to boundary integral methods for Lipschitz subdomains of Riemannian manifolds, the general goal of the monograph under review is to develop potential/geometric measure theoretic techniques in the study of the natural boundary value problems for the Hodge-Laplacian \[ \Delta_{\text{HL}}:= -(d+\delta)^2=-(d\delta+\delta d), \] acting on differential forms of arbitrary degree in the class of regular SKT subdomains of a compact, orientable, \(n\)-dimensional Riemannian manifold without boundary and with a \(\mathcal{C}^2\)-smooth metric tensor. Here \(d\) and \(\delta\) denote, respectively, the exterior derivative operator and its formal adjoint, the codifferential. Detailed study and solvability results are given for a natural class of boundary value problems for \(\Delta_{\text{HL}},\) that includes the absolute and relative boundary conditions used in the Hodge-type representation of absolute and relative cohomology classes of the underlying domain by harmonic forms. In addition, the \(L^p\)-boundary problems with arbitrary \(p\in(1,\infty)\) are studied for the Hodge-Laplacian equipped with classical Dirichlet, Neumann, transmission, nontangential Poincaré and Robin boundary conditions in regular SKT domains.
boundary value problems on Riemannian manifolds
scientific article; zbMATH DE number 6608462