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Elliptic and parabolic secondorder PDEs with growing coefficients
"... Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this ca ..."
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Cited by 7 (4 self)
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Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L∞norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
Discreteness of the Spectrum for Some Differential Operators With Unbounded Coefficients in R^n
"... We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coeffic ..."
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Cited by 3 (1 self)
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We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrödinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients.
On the Generation of C_0 semigroups in L¹(I)
"... In this paper we characterize completely the existence of a C0semigroup in L 1 (I) (I real interval) generated by a secondorder differential operator when suitable boundary conditions at the endpoints are imposed. In spaces of continuous functions similar characterizations have been obtained by ..."
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In this paper we characterize completely the existence of a C0semigroup in L 1 (I) (I real interval) generated by a secondorder differential operator when suitable boundary conditions at the endpoints are imposed. In spaces of continuous functions similar characterizations have been obtained by Timmermans [6] on the maximal domain, by Cl'ement and Timmermans [2] in the case of Ventcel's boundary conditions and by Campiti, Metafune and Pallara [1] in the case of Neumann's boundary conditions. 1 Introduction and preliminaries Let I =]r 1 ; r 2 [ (\Gamma1 r 1 ! r 2 +1) be a real interval and consider the secondorder differential operator Bu(x) = ff(x)u 00 (x) + fi(x)u 0 (x) ; x 2 I ; (1.1) where ff; fi : I ! R are continuous functions and ff(x) ? 0 for every x 2 I . In [2] Cl'ement and Timmermans gave necessary and sufficient conditions in order for B to be the generator of a C 0 semigroup in C(I) on the domain DV (B) := ae u 2 C(I) " C 2 (I) j lim x!r1 ;r 2 Bu(x) = 0...
L^pRegularity for Elliptic Operators with Unbounded Coefficients
, 2002
"... Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1. ..."
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Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1.
The Domain of the OrnsteinUhlenbeck Operator on an L^pSpace with Invariant Measure
, 2001
"... We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems. ..."
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We show that the domain of the OrnsteinUhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative DoreVenni theorems.
Global L p estimates for degenerate OrnsteinUhlenbeck operators
, 2009
"... We consider a class of degenerate OrnsteinUhlenbeck operators in R N, of the kind p0X A aij @ 2 NX xix + j i;j=1 i;j=1 ..."
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We consider a class of degenerate OrnsteinUhlenbeck operators in R N, of the kind p0X A aij @ 2 NX xix + j i;j=1 i;j=1
NORM DISCONTINUITY AND SPECTRAL PROPERTIES OF ORNSTEINUHLENBECK SEMIGROUPS
, 2005
"... Let E be a real Banach space. We study the OrnsteinUhlenbeck semigroup P = {P(t)}t≥0 associated with the OrnsteinUhlenbeck operator Lf(x) = 1 2 Tr QD2 f(x) + 〈Ax, Df(x)〉, x ∈ E. Here Q ∈ L (E ∗ , E) is a positive symmetric operator and A is the generator of a C0semigroup S = {S(t)}t≥0 on E. Unde ..."
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Let E be a real Banach space. We study the OrnsteinUhlenbeck semigroup P = {P(t)}t≥0 associated with the OrnsteinUhlenbeck operator Lf(x) = 1 2 Tr QD2 f(x) + 〈Ax, Df(x)〉, x ∈ E. Here Q ∈ L (E ∗ , E) is a positive symmetric operator and A is the generator of a C0semigroup S = {S(t)}t≥0 on E. Under the assumption that P admits an invariant measure µ ∞ we prove that if S is eventually compact and the spectrum of its generator is nonempty, then ‖P(t) − P(s) ‖ L(L 1 (E,µ∞)) = 2 for all t, s ≥ 0 with t ̸ = s. This result is new even when E = R n. We also study the behaviour of P in the space BUC(E). We show that if A ̸ = 0 there exists t0> 0 such that ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all 0 ≤ t, s ≤ t0 with t ̸ = s. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all t, s ≥ 0, t ̸ = s, or S is the direct sum of a nilpotent semigroup and a finitedimensional periodic semigroup. Finally we investigate the spectrum of L in the spaces L 1 (E, µ∞) and BUC(E).
Global L^p estimates for degenerate . . .
, 2011
"... We present a new approach to prove global L p estimates for degenerate OrnsteinUhlenbeck operators in R N. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to CalderónZygmund’s singular integrals theory in Eucl ..."
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We present a new approach to prove global L p estimates for degenerate OrnsteinUhlenbeck operators in R N. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to CalderónZygmund’s singular integrals theory in Euclidean and in nonEuclidean settings are also provided.