Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

Abstract

Abstract. We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

Microlocal Analysis of Generalized Functions: Pseudodifferential Techniques and Propagation of Singularities

“…As a first application of Theorem 3.9 we prove an extension of the classical result on non-characteristic regularity for distributional solutions of arbitrary pseudodifferential equations (with smooth symbols). A generalization of this result for partial differential operators with Colombeau coefficients was achieved in [16]; here we present a version for pseudodifferential operators with slow-scale generalized symbols. = p(x, D) is a properly supported pseudodifferential operator with slow-scale symbol and u ∈ G(Ω), then…”

“…Note that here the use of the attribute 'slow scale' refers to the appearance of the slowscale lower bound in (2.6). This is a crucial difference from more general definitions of ellipticity given in [6,7,16], whereas a similar condition has already been used in [15, § 6] in a special case. In fact, due to the overall slow-scale conditions in Definition 2.1, any symbol which is slow-scale micro-elliptic at (x 0 , ξ 0 ) fulfils the stronger hypoellipticity estimates [7, Definition 6.1]; furthermore, (2.6) is stable under lower-order (slow-scale) perturbations.…”

“…Secondly, as soon as one puts oneself into the much wider setting of Colombeau spaces-with the possibility of allowing for highly singular symbols of (pseudo)differential operators-the question of good choices for appropriate generalized notions of the characteristic set or (micro-)ellipticity turns into a considerable and crucial part of the research issue (cf. [7,13,15,16]).…”

Propagation of singularities for generalized solutions to nonlinear wave equations

“…Together with assumption (B), this implies that g V,W is in G([0, T ] × R). Hence, by Lemma 3.2, the two integrals in (18) and (19) are in G([0, T ] × R). Furthermore:…”

“…Intrinsic regularity theory in Colombeau algebras (i.e., without recourse to distributional limits) started with the introduction of G ∞ (Ω) in [22]. It turned out that in linear partial differential equations, most of the classical regularity theory could be replicated with G ∞ (Ω) in place of C ∞ (Ω): elliptic regularity, hypoellipticity, microlocal elliptic regularity, wave front sets, and propagation along bicharacteristics, including the techniques of pseudodifferential operators and Fourier integral operators with Colombeau amplitudes and phase functions [11][12][13][14][15]18,20,26]. In addition, the G ∞ -singular support of solutions to linear wave equations with discontinuous coefficients could be precisely located in one space dimension and for radially symmetric solutions in higher space dimensions [6][7][8]17].…”