On properness and related properties of quasilinear systems on unbounded domains

Abstract

Key words Proper nonlinear operators, quasilinear systems, compactness, decomposition lemma MSC (2010) 35G30, 35J55, 35A35The purpose of this paper is to provide tools for analyzing the compactness properties of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general class of nonlinear operators arising in quasilinear systems of partial differential equations of second order, in divergence form. Our approach, based on a suitable decomposition lemma, admits the discussion of problems with some inherent loss of compactness, for example due to a domain with infinite measure or a lower order term with critical growth. As an application, we obtain a characterization of properness which is considerably easier to verify than the definition.

On Compactness of Minimizing Sequences Subject to a Linear Differential Constraint

“…For a sequence of gradients on an unbounded domain, a corresponding result was obtained in [11]. In our present context, it would still be possible to give a proof relying on the abstract framework developed in [11], which provides a way to handle the numerous dierent properties of the component sequences w j n in a more systematic way.…”

“…For a sequence of gradients on an unbounded domain, a corresponding result was obtained in [11]. In our present context, it would still be possible to give a proof relying on the abstract framework developed in [11], which provides a way to handle the numerous dierent properties of the component sequences w j n in a more systematic way. However, the case of functionals is somewhat simpler than that of operators mapping into a Banach space which allows a reasonably-sized self-contained proof by hand, although our proof of (5.5) below only discusses the case j 0 = 5 in full detail, the other cases being more or less analogous.…”

“…This is essentially well known. For instance, the rst three lines of (4.1) immediately follow from Lemma 3.3Lemma 3.5 in [11], and the fourth line can be obtained analogously to the third. We omit the details.…”

“…We now derive a decomposition lemma in the tradition of [1], [8], [7] and [11], here for a sequence of A-free elds on the whole space. This result and suitable extensions to other unbounded domains will be our main tool for obtaining compactness of minimizing sequences.…”

On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals

“…A variant for unbounded Lipschitz domains can be found in [12]. For the convenience of the reader, a proof is given below.…”