Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

Cappa, Gianluca; Ferrari, S.

doi:10.1016/j.jde.2016.09.011

Cited by 22 publications

(38 citation statements)

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“…We simply realized that the continuity hypothesis was not needed. This implies that in the case Ω = X the results of this paper are also a generalization of the results in [12].…”

Section: Introductionsupporting

confidence: 71%

“…See [12, Section 3] for more details and [9] and [6,Section 12.4] for a treatment of the classical Moreau-Yosida approximations in Hilbert spaces, which are different from the one defined in (4.3). In the following proposition we recall some results contained in [12,Section 3].…”

Section: A Property Of the Moreau-yosida Approximations Along Hmentioning

confidence: 99%

“…Due to the lack of differentiability, at least in general, of the natural norm of a separable Banach space these methods cannot be applied in our case. The idea behind this paper is to replace "differentiability" by "differentiability along H" which is sufficient for our goals, because we use a modification of the Moreau-Yosida approximations, which already appeared in [12] (see Section 4), and of the distance function (see Section 3).…”

Section: Introductionmentioning

confidence: 99%

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Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

Cappa

Ferrari

2018

Journal of Mathematical Analysis and Applications

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Abstract. Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure µ. The associated Cameron-Martin space is denoted by H. Consider two sufficiently regular convex functions U : X → R and G : X → R. We let ν = e −U µ and Ω = G −1 (−∞, 0]. In this paper we are interested in the W 2,2 regularity of the weak solutions of elliptic equations of the typeand L ν,Ω is the self-adjoint operator associated with the quadratic form (ψ, ϕ) → Ω

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“…We simply realized that the continuity hypothesis was not needed. This implies that in the case Ω = X the results of this paper are also a generalization of the results in [12].…”

Section: Introductionsupporting

confidence: 71%

Section: A Property Of the Moreau-yosida Approximations Along Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

Cappa

Ferrari

2018

Journal of Mathematical Analysis and Applications

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“…Moreover the fact that U belongs to D 1,q (X, γ) for any q ∈ [1, ∞) allows us to conclude that the operator D H : FC 1 b (X) → L p (X, ν; H) is closable in L p (X, ν), p ∈ (1, ∞) and we may define the space D 1,p (X, ν), p > 1, as the domain of its closure (still denoted by D H ). In a similar way we can define D 2,p (X, ν), p ∈ (1, ∞) (for more details see [12] and [22]). The Gaussian integration by parts formula X D i f dγ = 1 √ λi X x, v i f dγ, which holds true for any f ∈ FC 1 b (X) and i ∈ N, yields that…”

Section: Assumptions and Preliminary Resultsmentioning

confidence: 99%

“…with ψ 1 ∈ D 1,2 (X, ν ε k ) and ψ 2 ∈ D(L ε k ) = D 2,2 (X, ν ε k ) (see [12,Theorem 6.2] for the characterisation of the domain of D(L ε k )). Theorem 2.8 guarantees that, for every t ≥ 0, the function u ε k ,n (t, ·) belongs to FC 3 b (X) and, as consequence, that G is differentiable in (0, t).…”

Section: Pointwise Gradient Estimatesmentioning

confidence: 99%

Gradient estimates for perturbed Ornstein–Uhlenbeck semigroups on infinite-dimensional convex domains

2019

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Let X be a separable Hilbert space endowed with a non-degenerate centred Gaussian measure γ and let λ 1 be the maximum eigenvalue of the covariance operator associated with γ. The associated Cameron-Martin space is denoted by H. For a sufficiently regular convex function U : X → R and a convex set Ω ⊆ X, we set ν := e −U γ and we consider the semigroup (T Ω (t)) t≥0 generated by the self-adjoint operator defined via the quadratic formwhere ϕ, ψ belong to D 1,2 (Ω, ν), the Sobolev space defined as the domain of the closure in L 2 (Ω, ν) of D H , the gradient operator along the directions of H.A suitable approximation procedure allows us to prove some pointwise gradient estimates for (T Ω (t)) t≥0 . In particular, we show thatfor any p ∈ [1, +∞) and f ∈ D 1,p (Ω, ν). We deduce some relevant consequences of the previous estimate, such as the logarithmic Sobolev inequality and the Poincaré inequality in Ω for the measure ν and some improving summability properties for (T Ω (t)) t≥0 . In addition we prove that if f belongs to L p (Ω, ν) for some p ∈ (1, ∞), thenwhere Kp is a positive constant depending only on p. Finally we investigate on the asymptotic behaviour of the semigroup (T Ω (t)) t≥0 as t goes to infinity. GRADIENT ESTIMATES ON INFINITE DIMENSIONAL CONVEX DOMAINS3 therein) and coupling methods (see for example [15,16,39]). On the other hand, in infinite dimensional Wiener spaces some partial results are also available. In the case of a Gaussian measure γ and Ω = X, the classical Mehler's representation formulawhere the equality has to be meant componentwise, (see [10, Proposition 1.5.6]). Again for the Gaussian measure γ on a convex subset Ω, in [11, Theorem 3.1] it is proved that |D H T (t)f | H ≤ e −t T (t)|D H f | H for any smooth function f . In this case, the idea consists in approximating the parabolic problem with a sequence of finite dimensional parabolic problems and using the factorisation of the Gaussian measure. Clearly, this approach does not work in our case since our measure in general does not decompose as a product of measures on orthogonal subspaces. Finally, the case of a weighted Gaussian measure is also considered in [20] where a version of (3) is proved when Ω = X and the H-derivative is replaced by the Fréchet one. We point out that, in this latter case, the proof of the gradient estimate is based on purely stochastic techniques. Hence, taking account of the existing literature, estimate (3) represents a generalisation of all the above results and the purely analytical proof we proposed, inspired by an idea due to Bakry andÉmery (see [4] and [38]), is a novelty in the proofs of gradient estimates.As announced, the pointwise gradient estimate (3) has several interesting consequences. First of all it yields that the semigroup T Ω (t) is smoothing, in the sense that it is bounded from L p (Ω, ν) into D 1,p (Ω, ν), for any p ∈ (1, ∞) and t > 0 as the estimatereveals. Due to the fact that the Sobolev embedding theorems fail to hold when we replace the Lebesgue measure with another general m...

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On functions of bounded variation on convex domains in Hilbert spaces

2021

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We study functions of bounded variation (and sets of finite perimeter) on a convex open set $${\varOmega }\subseteq X$$ Ω ⊆ X , X being an infinite-dimensional separable real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein–Uhlenbeck operator.

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Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

Cited by 22 publications

References 20 publications

Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: The convex subset case

Gradient estimates for perturbed Ornstein–Uhlenbeck semigroups on infinite-dimensional convex domains

On functions of bounded variation on convex domains in Hilbert spaces

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