Dirichlet integrals and Gaffney-Friedrichs inequalities in convex domains (Q2716431)

Let \(\Omega\) be a Lipschitz domain on a Riemannian manifold \(M\). Geometric conditions of \(\Omega\) guaranteeing the validity of the classical Gaffney-Friedrichs estimates for differential forms are studied. Two types of conditions are regarded: \(\ell\)-convexity, \(0\leq\ell\leq\dim M\), and a uniform exterior ball condition. In the simplest case of smooth domains of \(\mathbb{R}^n\) the \(\ell\)-convexity means that any collection of \(n-\ell\) principal curvatures of the boundary \(\partial\Omega\) has a nonnegative sum. The Riemannian case is the same in spirit but more technical. The condition can be extended to Lipschitz domains by approximation.NEWLINENEWLINENEWLINEThe main theorem asserts that if \(\Omega\subset M\) is \(\ell\)-convex, \(u\in L^2(\Omega,\Lambda^\ell TM)\), \(du\in L^2(\Omega, \Lambda^{\ell+1}TM)\), \(\delta u\in L^2(\Omega, \Lambda^{\ell-1} TM)\) and \(\nu\wedge u= 0\) on \(\partial\Omega\), then \(u\) belongs to the Sobolev space \(H^{1,2}(\Omega, \Lambda^\ell TM)\) and NEWLINE\[NEWLINE\|u\|_{H^{1,2}(\Omega)}\leq C(\|du\|_{L^2(\Omega)}+ \|\delta u\|_{L^2(\Omega)}+\|u\|_{L^2(\Omega)}).NEWLINE\]NEWLINE Here \(d\), \(\delta\), \(\nu\) stand for the exterior differentiation operator, its formal adjoint and outward unit conormal on the boundary \(\partial\Omega\), respectively.NEWLINENEWLINENEWLINEThe similar estimate is proved for bounded Lipschitz domains of \(\mathbb{R}^n\) satisfying the uniform exterior ball condition. Roughly speaking, the last conditions mean that boundary singularities are directed outward. Applications to the regularity of PDEs in Lipschitz domains are discussed.