$L^p$-Theory for Elliptic Operators on $\mathbb R^d$ with Singular Coefficients

Abstract

Abstract. We study the generation of an analytic semigroup in L p (R d ) and the determination of the domain for a class of second order elliptic operators with unbounded coefficients in R d . We also establish the maximal regularity of type L q -L p for the corresponding inhomogeneous parabolic equation. In contrast to the previous literature the coefficients of the second derivatives are not required to be strictly elliptic or bounded. Interior singularities of the lower order terms are also discussed.

Lp-uniqueness for elliptic operators with unbounded coefficients inRN

“…Moreover, whenever pointwise gradient estimates for the function T (t)f are available for any f ∈ C ∞ c (R N ), one can partially characterize D(A p ), showing that it is continuously embedded in the Sobolev space W 1,p (R N , μ) (the set of functions f ∈ L p (R N , μ) such that the distributional gradient of f is in L p (R N , μ) N ) for any p ∈ (1, +∞). Anyway, a complete characterization of D(A p ) is known, to the best of our knowledge only in very few cases (see, e.g., [13,22,24,25]). …”

“…Hence, assumptions on the growth at infinity of the coefficients of the operator A more restrictive than in the C b -case are to be prescribed. Typically, the diffusion coefficients are supposed to be bounded or to grow at most slightly more than quadratically at infinity, and some suitable compensation conditions on the coefficients are prescribed (see, e.g., the papers [9][10][11]16,17,23,25,26,28,30,31]). The suitable L p -spaces where to analyze elliptic operators with unbounded coefficients are L p -spaces related to particular measures, the so-called invariant measures and infinitesimally invariant measures.…”

Cores for parabolic operators with unbounded coefficients

“…Nowadays many results are known in the literature concerning the main properties of the semigroup associated with the operator L both in spaces of bounded and continuous functions (see e.g., [35,[40][41][42][43]) and in L p -spaces related to particular measures, the so-called invariant measures (see e.g., [22,[36][37][38]44,45,47]). It is well known that the L p -spaces related to invariant measures are the right L p -spaces where to study elliptic operators with unbounded coefficients.…”

A weighted least squares finite element method for elliptic problems with degenerate and singular coefficients

“…Systems with such behavior include applications in quantum mechanics (see [18,28]) and some reformulations of stochastic PDE (see e.g., [23] and references therein). Others have considered analysis of such problems in the context of a generalization of strongly elliptic equations (see e.g., [25,24,21]). Problems similar to (1.1) arise, for example, when an elliptic problem with cylindrical or spherical symmetry is reduced to a lower dimensional problem (see e.g., [15,22]).…”