Gradient estimates in parabolic problems with unbounded coefficients

Bertoldi, Marcello; Fornaro, Simona

doi:10.4064/sm165-3-3

Cited by 47 publications

(70 citation statements)

References 13 publications

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“…The problem of estimating the derivatives of T (t)f has already been studied in literature by both analytic ( [2,4,6,12]) and probabilistic methods ( [5,19]). …”

Section: Introductionmentioning

confidence: 99%

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Estimates of the derivatives for parabolic operators with unbounded coefficients

Bertoldi

Lorenzi

2005

Trans. Amer. Math. Soc.

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Abstract. We consider a class of second-order uniformly elliptic operators A with unbounded coefficients in R N . Using a Bernstein approach we provide several uniform estimates for the semigroup T (t) generated by the realization of the operator A in the space of all bounded and continuous or Hölder continuous functions in R N . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation λu − Au = f (λ > 0) and the nonhomogeneous Dirichlet Cauchy problem Dtu = Au + g. Then, we prove two different kinds of pointwise estimates of T (t) that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup T (t) in weighted L p -spaces related to the invariant measure associated with the semigroup.

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“…The problem of estimating the derivatives of T (t)f has already been studied in literature by both analytic ( [2,4,6,12]) and probabilistic methods ( [5,19]). …”

Section: Introductionmentioning

confidence: 99%

“…We assume dissipativity-type and growth conditions on the coefficients of A. We notice that some dissipativity condition is necessary, because in general estimate (1.2) fails; see [4].…”

Section: Introductionmentioning

confidence: 99%

Estimates of the derivatives for parabolic operators with unbounded coefficients

Bertoldi

Lorenzi

2005

Trans. Amer. Math. Soc.

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“…[2,3,8,1,9]. Since the very first studies it was apparent that operators of the type Au = Tr Q(x)D 2 u(x) + F (x)Du(x) , without potential terms, are not well settled in L p spaces with respect to the Lebesgue measure, unless the matrix Q and the vector F satisfy very severe restrictions, such as global Lipschitz continuity (see [9,7]).…”

Section: Introductionmentioning

confidence: 99%

“…This is what we do in this paper. We consider the operator A defined by (1) Au = ∆u − DΦ, Du = e Φ div (e −Φ Du), where Φ : R N → R is a C 2 semi-convex function, i.e., there is α ≥ 0 such that (2) Φ α (x) := Φ(x) + α|x| 2 /2 is convex, or, equivalently, the matrix D 2 Φ(x) + αI is nonnegative definite at each x. We emphasize that we do not assume any growth restriction on Φ or on its derivatives.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet boundary conditions for elliptic operators with unbounded drift

Lunardi

Metafune

Pallara

2005

Proc. Amer. Math. Soc.

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“…for solutions of parabolic equations have been obtained by many authors, see, e.g., [1], [2], [11], [15], and the references therein. Such estimates do not require (Hb) and one has C(t) → +∞ as t → 0 or t → +∞.…”

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confidence: 99%

Gradient Bounds for Solutions of Elliptic and Parabolic Equations

Богачев¹,

Prato²,

Röckner³

et al. 2005

Lecture Notes in Pure and Applied Mathematics

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Let L be a second order elliptic operator on R d with a constant diffusion matrix and a dissipative (in a weak sense) drift b ∈ L p loc with some p > d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure µ satisfying the equation L * µ = 0 and that the closure of L in L 1 (µ) generates a Markov semigroup {T t } t≥0 with the resolvent {G λ } λ>0 . We prove that, for any Lipschitzian function f ∈ L 1 (µ) and all t, λ > 0, the functions T t f and G λ f are Lipschitzian andAn analogous result is proved in the parabolic case.Suppose that for every t ∈ [0, 1], we are given a a strictly positive definite symmetric matrix A(t) = (a ij (t)) and a measurable vector field x → b(t, x) = (b 1 (t, x), . . . , b n (t, x)). Let L t be the elliptic operator on R d given bySuppose that A and b satisfy the following hypotheses:Hb) b is dissipative in the following sense: for every t ∈ [0, 1] and every h ∈ R d , there exists a measure zero set N t,h ⊂ R d such that(Hc) for every t ∈ [0, 1], there exists a Lyapunov function V t for L t , i.e., a nonnegative C 2 -function V t such that V t (x) → +∞ and L t V t (x) → −∞ as |x| → ∞. We consider the parabolic equation1

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Gradient estimates in parabolic problems with unbounded coefficients

Abstract: Abstract. We study, with purely analytic tools, existence, uniqueness and gradient estimates of the solutions to the Neumann problems associated with a second order elliptic operator with unbounded coefficients in spaces of continuous functions in an unbounded open set Ω in R N .

Cited by 47 publications

References 13 publications

Estimates of the derivatives for parabolic operators with unbounded coefficients

Estimates of the derivatives for parabolic operators with unbounded coefficients

Dirichlet boundary conditions for elliptic operators with unbounded drift

Gradient Bounds for Solutions of Elliptic and Parabolic Equations

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