Elliptic and parabolic problems in unbounded domains

Abstract

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi-explicit representation formulas via operator-valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles.

Problems in Unbounded Cylindrical Domains

“…x ∈ R m A proof of this result which is based on the Dunford functional calculus for sectorial operators can be found in [7]. It is interesting to observe that the fundamental solution can be understood in terms of an analytic function of the pseudodifferential operator √ A.…”

“…Problems of this type naturally arise from boundary value problems in cylindrical domains (see [7]) and fit into the framework of generalized fundamental solutions.…”

“…The approach is developed in [7] and extended in [6]. Here we briefly review its main ingredients and then consider a series of examples to illustrate the wide range of applicability of the ideas.…”

Discrete Fourier multipliers and cylindrical boundary-value problems

“…The use of operator-valued multipliers to treat cylindrical-in-space boundaryvalue problems was first carried out in [15,16] in a Besov space setting. In these papers, Guidotti constructs semi-classical fundamental solutions for a class of elliptic operators on infinite cylindrical domains R n × V .…”

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

“…On the other hand, the usage of operator-valued Fourier multipliers to treat cylindrical in space boundary value problems was first carried out in [12] in a Besov space setting. In that paper the author constructs semiclassical fundamental solutions for a class of elliptic operators on infinite cylindrical domains R n × V .…”