Elliptic boundary value problems with large parameter for mixed order systems

Abstract

Abstract. In this paper boundary value problems are studied for systems with large parameter, elliptic in the sense of Douglis-Nirenberg. We restrict ourselves on model problems acting in the half-space. It is possible to define parameter-ellipticity for such problems, in particular we formulate ShapiroLopatinskii type conditions on the boundary operators. It can be shown that parameter-elliptic boundary value problems are uniquely solvable and that their solutions satisfy uniform two-sided a priori estimates in parameterdependent norms. We essentially use ideas from Newton's polygon method and of Vishik-Lyusternik boundary layer theory.

MaximalLp-regularity of parabolic problems with boundary dynamics of relaxation type

“…To analyze boundary value problems with inhomogeneous symbol, the Newton polygon approach was developed, see, e.g., [3] and the references therein. One way of describing the inhomogeneity uses the r-principal part of the symbol where r > 0 denotes the weight of λ with respect to ξ .…”

Fredholm theory connected with a Douglis–Nirenberg system of differential equations over an exterior region in

“…The aforementioned authors then turned their attention to a genuine boundary problem for a parameter‐elliptic Douglis–Nirenberg system of differential operators acting over a bounded region in and over , and various results pertaining to a priori estimates for solutions as well as to spectral theory were established (see , , ). Subsequent work in this area of investigation was also undertaken in , , , and .…”

On the resolvent arising in a parameter‐elliptic multi‐order boundary problem

“…Boundary problems related to (1.1), (1.2), but under C ∞ conditions have been the subject of investigation in both 20 and 10. In 20 various results pertaining to the resolvent operator are derived, while in 10 a priori estimates are established for solutions for the case where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega =\mathbb {R}^n_+$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma = \mathbb {R}^{n-1}$\end{document}.…”

“…Boundary problems related to (1.1), (1.2), but under C ∞ conditions have been the subject of investigation in both 20 and 10. In 20 various results pertaining to the resolvent operator are derived, while in 10 a priori estimates are established for solutions for the case where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Omega =\mathbb {R}^n_+$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma = \mathbb {R}^{n-1}$\end{document}. However, because of the restrictions imposed, both of these papers are not able to deal with the important case when \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$s_j = t_j = t^{\prime }_j$\end{document} for j = 1, …, N (here \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\big \lbrace t^{\prime }_j\big \rbrace _1^N$\end{document} denotes a monotonic decreasing sequance of positive integers) and the boundary conditions are of Dirichlet type (see 4, Section 2], 15, p. 448]).…”