Maximal regularity for elliptic operators with second-order discontinuous coefficients
- PDF / 423,889 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 3 Downloads / 231 Views
Journal of Evolution Equations
Maximal regularity for elliptic operators with second-order discontinuous coefficients G. Metafune, L. Negro and C. Spina
Abstract. We prove maximal regularity for parabolic problems associated to the second-order elliptic operator L = + (a − 1)
N xi x j x Di j + c 2 · ∇ − b|x|−2 |x|2 |x|
i, j=1
with a > 0 and b, c real coefficients.
1. Introduction In this paper, we consider second-order elliptic operators of the form L = + (a − 1)
N xi x j x Di j + c 2 · ∇ − b|x|−2 2 |x| |x|
(1)
i, j=1
with a > 0 and b, c constant real coefficients. The leading coefficients are uniformly elliptic but discontinuous at 0, if a = 1, and singularities in the lower order terms appear when b or c is different from 0. The operator commutes with dilations, in the sense that Is−1 L Is = s 2 L, if Is u(x) = u(sx). When c = 0 and a = 1, L reduces to a Schrödinger operator with inverse square potential. Operators of this form have been widely investigated in previous works. In particular generation properties of analytic semigroups in L p spaces endowed with the Lebesgue measure, sharp kernel estimates and Rellich-type inequalities have been proved (see [3,10–16]). Here, we prove that the following parabolic problem associated with L ∂t u(t) − Lu(t) = f (t), t > 0, (2) u(0) = 0 has maximal L q regularity, that is for each f ∈ L q (0, ∞; X ) there exists u ∈ W 1,q (0, ∞; X ) ∩ L q (0, ∞; D(L)) satisfying (2). Here, X is the underlying function space where L acts and D(L) is the domain of L in X . Mathematics Subject Classification: 47D07, 35J70 Keywords: Elliptic operators, Discontinuous coefficients, Kernel estimates, Maximal regularity.
G. Metafune et al.
J. Evol. Equ.
The functional analytic approach we use for proving maximal regularity is widely described in [9] and in the new books [6,7]. The whole theory relies on a deep interplay between harmonic analysis and structure theory of Banach spaces but largely simplifies when the underlying Banach spaces are L p spaces, by using classical square function estimates. This last approach has been employed extensively in [4], showing that uniformly parabolic operators have maximal regularity, under very general boundary conditions. Here, we show that the same happens for a class of degenerate second-order operators. We deduce maximal regularity from the R-boundedness of the generated semigroup in closed sectors of the right half plane, see [4, Chapter 4]. This last is deduced through an extrapolation result in [1] which involves a family of Muckenhoupt weighted estimates. We show that L has maximal regularity in L p when it generates a semigroup in L p , that is when the necessary and sufficient conditions of Theorems 2.1 and 2.2 are satisfied. We consider L p spaces with respect to radial power weights |x|m not just for the sake of generality but because our proof relies on weighted estimates: we are unable to obtain the result just fixing the Lebesgue measure or the symmetrizing measure but we have to work simultaneously in different homog
Data Loading...
