Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

Abstract

Recent years have brought significant advances in the theory of higher order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary domains were established, followed by the higher order Wiener test. Certain boundary value problems for higher order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject in [25], this survey presents the current state of the art, emphasizing the most recent results and emerging open problems. Contents 1991 Mathematics Subject Classification. Primary 35-02; Secondary 35B60, 35B65, 35J40, 35J55.

Square function estimates on layer potentials for higher‐order elliptic equations

“…In the case where is the half-space R n+1 + , we will return to it in Section 2.4 below; in more general Lipschitz domains we refer the interested reader to [16], [17].…”

“…In this project we study elliptic differential operators of the form for , with general bounded measurable coefficients. As mentioned above, contrary to the second order case, most of the known well‐posedness results for higher order boundary value problems have been established only in the case of constant coefficients (see, for example, , , , , , , , or the survey paper ), or concern boundary‐value problems with data in fractional smoothness spaces, such as the Dirichlet problem where Ω is a Lipschitz domain and where lies in a boundary Besov space with smoothness parameter between zero and one. See , , .…”

“…Making sense of formulas and in the context of higher‐order operators is one of the key tasks in its own right. In the case where Ω is the half‐space , we will return to it in Section below; in more general Lipschitz domains we refer the interested reader to , .…”

“…for m ≥ 2, with general bounded measurable coefficients. As mentioned above, contrary to the second order case, most of the known well-posedness results for higher order boundary value problems have been established only in the case of constant coefficients (see, for example, [47], [51], [56], [62], [63], [69], [70], or the survey paper [16]), or concern boundary-value problems with data in fractional smoothness spaces, such as the Dirichlet problem…”

Bounds on layer potentials with rough inputs for higher order elliptic equations

“…The choice can be important; indeed, it was shown in that the Neumann problem for is well posed for some choices of and ill posed for others. These issues are discussed in and at length in .…”