The Experts below are selected from a list of 138 Experts worldwide ranked by ideXlab platform
Alessandra Lunardi - One of the best experts on this subject based on the ideXlab platform.
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BV Functions on Convex Domains in Wiener Spaces
Potential Analysis, 2015Co-Authors: Alessandra Lunardi, Michele Miranda, Diego PallaraAbstract:We study functions of bounded variation defined in an abstract Wiener space X , relating the variation of a function u on a Convex Open Set Ω ⊂ X ${\Omega }\subSet X$ to the behavior near t =0 of T ( t ) u , T ( t ) being the Ornstein–Uhlenbeck semigroup in Ω.
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BV functions on Convex domains in Wiener spaces
arXiv: Functional Analysis, 2014Co-Authors: Alessandra Lunardi, Michele Miranda, Diego PallaraAbstract:We study functions of bounded variation defined in an abstract Wiener space X, relating the variation of a function u on a Convex Open Set O in X to the behavior near t=0 of T(t)u, T(t) being the Ornstein--Uhlenbeck semigroup in O.
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Ultraboundedness for parabolic equations in Convex domains without boundary conditions
Physica D: Nonlinear Phenomena, 2010Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:Abstract We consider the operator A u = 1 2 Δ u − 〈 D U , D u 〉 , where U is a Convex real function defined in a Convex Open Set O ⊂ R N and lim | x | → ∞ U ( x ) = lim x → ∂ O U ( x ) = + ∞ . We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure e − 2 U ( x ) d x .
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On a class of elliptic and parabolic equations in Convex domains without boundary conditions
Discrete & Continuous Dynamical Systems - A, 2008Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a Convex real function defined in a Convex Open Set $\O \subSet \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.
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On a class of self-adjoint elliptic operators in L 2 spaces with respect to invariant measures
Journal of Differential Equations, 2007Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:Abstract We consider the operator A u = Δ u / 2 − 〈 D U , D u 〉 , where U is a Convex real function defined in a Convex Open Set Ω ⊂ R N and lim | x | → ∞ U ( x ) = + ∞ . Setting μ ( d x ) = exp ( − 2 U ( x ) ) d x , we prove that the realization of A in L 2 ( Ω , μ ) with domain { u ∈ H 2 ( Ω , μ ) : 〈 D U , D u 〉 ∈ L 2 ( Ω , μ ) , ∂ u / ∂ n = 0 at Γ 1 } , is a self-adjoint dissipative operator. Here Γ 1 is the Set of points y in the boundary of Ω such that lim sup x → y U ( x ) + ∞ . Then we discuss several properties of A and of the measure μ , including Poincare and log-Sobolev inequalities in H 1 ( Ω , μ ) .
Stefano Gioffrè - One of the best experts on this subject based on the ideXlab platform.
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Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces
The Journal of Geometric Analysis, 2018Co-Authors: Antonio De Rosa, Stefano GioffrèAbstract:We prove qualitative and quantitative stability of the following rigidity theorem: the only anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $$n \ge 2$$ , $$p\in (1, \, +\infty )$$ and $$\Sigma $$ an n-dimensional, closed hypersurface in $$\mathbb {R}^{n+1}$$ , which is the boundary of a Convex, Open Set. We show that if the $$L^p$$ -norm of the trace-free part of the anisotropic second fundamental form is small, then $$\Sigma $$ must be $$W^{2, \, p}$$ -close to the Wulff shape, with a quantitative estimate.
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Quantitative stability for anisotropic nearly umbilical hypersurfaces
arXiv: Differential Geometry, 2017Co-Authors: Antonio De Rosa, Stefano GioffrèAbstract:We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $\Sigma$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a Convex, Open Set. We show that if the $L^p$ norm of the trace-free part of the anisotropic second fundamental form is small, then $\Sigma$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.
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A $W^{2, \, p}$-estimate for nearly umbilical hypersurfaces
arXiv: Differential Geometry, 2016Co-Authors: Stefano GioffrèAbstract:Let $n \ge 2$, $p \in (1, \, +\infty)$ be given and let $\Sigma$ be a $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A - \frac{1}{n} A^i_i g$ where $g = \delta |_{\Sigma}$.Assuming that $\Sigma$ is the boundary of a Convex, Open Set we prove that if the $L^p$-norm of $\mathring{A}$ is small, then $\Sigma$ must be $W^{2, \, p}$-close to a sphere, with a quantitative estimate.
Diego Pallara - One of the best experts on this subject based on the ideXlab platform.
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On functions of bounded variation on Convex domains in Hilbert spaces
Journal of Evolution Equations, 2021Co-Authors: Luciana Angiuli, Simone Ferrari, Diego PallaraAbstract: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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On functions of bounded variation on Convex domains in Hilbert spaces
arXiv: Functional Analysis, 2020Co-Authors: Luciana Angiuli, Simone Ferrari, Diego PallaraAbstract:We study functions of bounded variation (and Sets of finite perimeter) on a Convex Open Set $\Omega\subSeteq X$, $X$ being an infinite dimensional 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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BV Functions on Convex Domains in Wiener Spaces
Potential Analysis, 2015Co-Authors: Alessandra Lunardi, Michele Miranda, Diego PallaraAbstract:We study functions of bounded variation defined in an abstract Wiener space X , relating the variation of a function u on a Convex Open Set Ω ⊂ X ${\Omega }\subSet X$ to the behavior near t =0 of T ( t ) u , T ( t ) being the Ornstein–Uhlenbeck semigroup in Ω.
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BV functions on Convex domains in Wiener spaces
arXiv: Functional Analysis, 2014Co-Authors: Alessandra Lunardi, Michele Miranda, Diego PallaraAbstract:We study functions of bounded variation defined in an abstract Wiener space X, relating the variation of a function u on a Convex Open Set O in X to the behavior near t=0 of T(t)u, T(t) being the Ornstein--Uhlenbeck semigroup in O.
Jonathan M. Fraser - One of the best experts on this subject based on the ideXlab platform.
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The visible part of plane self-similar Sets
Proceedings of the American Mathematical Society, 2012Co-Authors: Kenneth Falconer, Jonathan M. FraserAbstract:Given a compact subSet F of R2, the visible part VθF of F from direction θ is the Set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1, then dimH VθF = 1 for almost all θ, where dimH denotes Hausdorff dimension. We confirm this when F is a self-similar Set satisfying the Convex Open Set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.
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The visible part of plane self-similar Sets
arXiv: Metric Geometry, 2010Co-Authors: Kenneth Falconer, Jonathan M. FraserAbstract:Given a compact subSet $F$ of $\mathbb{R}^2$, the visible part $V_\theta F$ of $F$ from direction $\theta$ is the Set of $x$ in $F$ such that the half-line from $x$ in direction $\theta$ intersects $F$ only at $x$. It is suggested that if $\dim_H F \geq 1$ then $\dim_H V_\theta F = 1$ for almost all $\theta$, where $\dim_H$ denotes Hausdorff dimension. We confirm this when $F$ is a self-similar Set satisfying the Convex Open Set condition and such that the orthogonal projection of $F$ onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and $F$ need not be connected.
Giuseppe Da Prato - One of the best experts on this subject based on the ideXlab platform.
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Ultraboundedness for parabolic equations in Convex domains without boundary conditions
Physica D: Nonlinear Phenomena, 2010Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:Abstract We consider the operator A u = 1 2 Δ u − 〈 D U , D u 〉 , where U is a Convex real function defined in a Convex Open Set O ⊂ R N and lim | x | → ∞ U ( x ) = lim x → ∂ O U ( x ) = + ∞ . We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure e − 2 U ( x ) d x .
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On a class of elliptic and parabolic equations in Convex domains without boundary conditions
Discrete & Continuous Dynamical Systems - A, 2008Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a Convex real function defined in a Convex Open Set $\O \subSet \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.
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On a class of self-adjoint elliptic operators in L 2 spaces with respect to invariant measures
Journal of Differential Equations, 2007Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:Abstract We consider the operator A u = Δ u / 2 − 〈 D U , D u 〉 , where U is a Convex real function defined in a Convex Open Set Ω ⊂ R N and lim | x | → ∞ U ( x ) = + ∞ . Setting μ ( d x ) = exp ( − 2 U ( x ) ) d x , we prove that the realization of A in L 2 ( Ω , μ ) with domain { u ∈ H 2 ( Ω , μ ) : 〈 D U , D u 〉 ∈ L 2 ( Ω , μ ) , ∂ u / ∂ n = 0 at Γ 1 } , is a self-adjoint dissipative operator. Here Γ 1 is the Set of points y in the boundary of Ω such that lim sup x → y U ( x ) + ∞ . Then we discuss several properties of A and of the measure μ , including Poincare and log-Sobolev inequalities in H 1 ( Ω , μ ) .
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elliptic operators with unbounded drift coefficients and neumann boundary condition
Journal of Differential Equations, 2004Co-Authors: Giuseppe Da Prato, Alessandra LunardiAbstract:Abstract We study the realization AN of the operator A = 1 2 Δ−〈DU,D·〉 in L 2 (Ω,μ) with Neumann boundary condition, where Ω is a possibly unbounded Convex Open Set in R N , U is a Convex unbounded function, DU(x) is the element with minimal norm in the subdifferential of U at x, and μ(dx)=c exp (−2U(x)) dx is a probability measure, infinitesimally invariant for A . We show that AN is a dissipative self-adjoint operator in L 2 (Ω,μ) . Log-Sobolev and Poincare inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by AN.