The Experts below are selected from a list of 297 Experts worldwide ranked by ideXlab platform
Filip Rindler - One of the best experts on this subject based on the ideXlab platform.
-
Characterization of Generalized Gradient Young Measures Generated by Sequences in W^1,1 and BV
Archive for Rational Mechanics and Analysis, 2010Co-Authors: Jan Kristensen, Filip RindlerAbstract:Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L^1-Bounded sequences (or even by sequences of Bounded Radon measures), we turn to investigation of those Young measures that are generated by Bounded sequences of W^1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W^1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W^1,1- and BV-case and contains its version for weakly converging sequences in W^1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.
-
Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV
Archive for Rational Mechanics and Analysis, 2010Co-Authors: Jan Kristensen, Filip RindlerAbstract:Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L1-Bounded sequences (or even by sequences of Bounded Radon measures), we turn to investigation of those Young measures that are generated by Bounded sequences of W1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W1,1- and BV-case and contains its version for weakly converging sequences in W1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.
Jan Kristensen - One of the best experts on this subject based on the ideXlab platform.
-
Characterization of Generalized Gradient Young Measures Generated by Sequences in W^1,1 and BV
Archive for Rational Mechanics and Analysis, 2010Co-Authors: Jan Kristensen, Filip RindlerAbstract:Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L^1-Bounded sequences (or even by sequences of Bounded Radon measures), we turn to investigation of those Young measures that are generated by Bounded sequences of W^1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W^1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W^1,1- and BV-case and contains its version for weakly converging sequences in W^1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.
-
Characterization of Generalized Gradient Young Measures Generated by Sequences in W1,1 and BV
Archive for Rational Mechanics and Analysis, 2010Co-Authors: Jan Kristensen, Filip RindlerAbstract:Generalized Young measures as introduced by DiPerna and Majda (Commun Math Phys 108:667–689, 1987) provide a quantitative tool for studying the one-point statistics of oscillation and concentration in sequences of functions. In this work, after developing a functional-analytic framework for such measures, including a compactness theorem and results on the generation of such Young measures by L1-Bounded sequences (or even by sequences of Bounded Radon measures), we turn to investigation of those Young measures that are generated by Bounded sequences of W1,1-gradients or BV-derivatives. We provide several techniques to manipulate such measures (including shifting, averaging and approximation by piecewise-homogeneous Young measures) and then establish the main new result of this work, the duality characterization of the set of (BV- or W1,1-)gradient Young measures in terms of Jensen-type inequalities for quasiconvex functions with linear growth at infinity. This result is the natural generalization of the Kinderlehrer–Pedregal Theorem (Arch Ration Mech Anal 115:329–365, 1991; J Geom Anal 4:59–90, 1994) for classical Young measures to the W1,1- and BV-case and contains its version for weakly converging sequences in W1,1 as a special case. Finally, we give an application to a new lower semicontinuity theorem in BV.
Piotr Gwiazda - One of the best experts on this subject based on the ideXlab platform.
-
Existence and differentiability in parameter of the measure solution to a perturbed non-linear transport equation
arXiv: Analysis of PDEs, 2020Co-Authors: Piotr Gwiazda, Sander C Hille, Kamila ŁyczekAbstract:We consider a perturbation in the non-linear transport equation on measures i.e. both initial condition $\mu_0$ and the solution $\mu_t^h$ are Bounded Radon measures $\mathcal{M}(\mathbb{R}^d)$. The perturbations occur in the velocity field and also in the right-hand side scalar function. It is shown that the solution is differentiable with respect to the perturbation parameter $h$ i.e. that derivative is an element of a proper Banach space. This result extends our previous result which considered the linear transport equation. The proof exploits approximation of the non-linear problem which is based on the study of the linear equation.
-
differentiability in perturbation parameter of measure solutions to perturbed transport equation
Kinetic and Related Models, 2019Co-Authors: Piotr Gwiazda, Sander C Hille, Kamila łyczek, Agnieszka świerczewskagwiazdaAbstract:We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of Bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the Holder space \begin{document}$ \mathcal{C}^{1+\alpha}( {\mathbb{R}^d}) $\end{document} . This result on differentiability is necessary for application in optimal control theory, which we also discuss.
-
differentiability in perturbation parameter of measure solutions to perturbed transport equation
arXiv: Analysis of PDEs, 2018Co-Authors: Piotr Gwiazda, Sander C Hille, Kamila łyczek, Agnieszka świerczewskagwiazdaAbstract:We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of Bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the H\"older space $\mathcal{C}^{1+\alpha}(\mathbb{R}^d)$. This result on differentiability is necessary for application in~optimal control theory, which we also discuss.
-
Measure valued solutions to conservation laws motivated by traffic modelling
Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2006Co-Authors: Sylvie Benzoni-gavage, Rinaldo M. Colombo, Piotr GwiazdaAbstract:Motivated by a traffic model, we study a conservation law whose solutions naturally need to be Bounded Radon measures. We investigate the relations between this equation and finite systems of conservation laws that converge to it.
John J. Benedetto - One of the best experts on this subject based on the ideXlab platform.
-
Super-resolution by means of Beurling minimal extrapolation
Applied and Computational Harmonic Analysis, 2020Co-Authors: John J. BenedettoAbstract:Abstract We investigate the super-resolution capabilities of total variation minimization. Namely, given a finite set Λ ⊆ Z d and spectral data F = μ ˆ | Λ , where μ is an unknown Bounded Radon measure on the torus T d , the problem is to find the measures with smallest norm whose Fourier transforms agree with F on Λ. Our main theorem shows that solutions to the problem depend crucially on a set Γ ⊆ Λ , defined in terms of F and Λ. For example, when # Γ = 0 , the solutions are singular measures supported in the zero set of an analytic function, and when # Γ ≥ 2 , the solutions are singular measures supported in the intersection of ( # Γ 2 ) hyperplanes. By theory and example, we show that the case # Γ = 1 is different from other cases, and is deeply connected with the existence of positive solutions. This theorem has implications to the possibility and impossibility of uniquely recovering μ from F on Λ. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation [7] , [8] with Candes and Fernandez-Granda's work on super-resolution [12] . This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the total variation minimization problem.
-
Super-resolution by means of Beurling minimal extrapolation
arXiv: Functional Analysis, 2016Co-Authors: John J. BenedettoAbstract:Let $M(\mathbb{T}^d)$ be the space of complex Bounded Radon measures defined on the $d$-dimensional torus group $(\mathbb{R}/\mathbb{Z})^d=\mathbb{T}^d$, equipped with the total variation norm $\|\cdot\|$; and let $\hat\mu$ denote the Fourier transform of $\mu\in M(\mathbb{T}^d)$. We address the super-resolution problem: For given spectral (Fourier transform) data defined on a finite set $\Lambda\subset\mathbb{Z}^d$, determine if there is a unique $\mu\in M(\mathbb{T}^d)$ of minimal norm for which $\hat\mu$ equals this data on $\Lambda$. Without additional assumptions on $\mu$ and $\Lambda$, our main theorem shows that the solutions to the super-resolution problem, which we call minimal extrapolations, depend crucially on the set $\Gamma\subset\Lambda$, defined in terms of $\mu$ and $\Lambda$. For example, when $\Gamma=0$, the minimal extrapolations are singular measures supported in the zero set of an analytic function, and when $\Gamma\geq 2$, the minimal extrapolations are singular measures supported in the intersection of $\Gamma\choose 2$ hyperplanes. By theory and example, we show that the case $\Gamma=1$ is different from other cases and is deeply connected with the existence of positive minimal extrapolations. This theorem has implications to the possibility and impossibility of uniquely recovering $\mu$ from $\Lambda$. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation with Candes and Fernandez-Granda's work on super-resolution. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the super-resolution problem.
Hui Wang - One of the best experts on this subject based on the ideXlab platform.
-
A semilinear singular Sturm–Liouville equation involving measure data
Annales de l'Institut Henri Poincaré C Analyse non linéaire, 2016Co-Authors: Hui WangAbstract:Abstract Given α > 0 and p > 1 , let μ be a Bounded Radon measure on the interval ( − 1 , 1 ) . We are interested in the equation − ( | x | 2 α u ′ ) ′ + | u | p − 1 u = μ on ( − 1 , 1 ) with boundary condition u ( − 1 ) = u ( 1 ) = 0 . We establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated.
-
ON A HARDY TYPE INEQUALITY AND A SINGULAR STURM-LIOUVILLE EQUATION
2014Co-Authors: Hui WangAbstract:OF THE DISSERTATION On a Hardy type inequality and a singular Sturm-Liouville equation by Hui Wang Dissertation Director: Haim Brezis In this dissertation, we first prove a Hardy type inequality for u ∈ W 0 (Ω), where Ω is a Bounded smooth domain in RN and m ≥ 2. For all j ≥ 0, 1 ≤ k ≤ m − 1, such that 1 ≤ j + k ≤ m, it holds that ∂ ju(x) d(x)m−j−k ∈ W k,1 0 (Ω), where d is a smooth positive function which coincides with dist(x, ∂Ω) near ∂Ω, and ∂l denotes any partial differential operator of order l. We also study a singular Sturm-Liouville equation −(x2αu′)′ + u = f on (0, 1), with the boundary condition u(1) = 0. Here α > 0 and f ∈ L2(0, 1). We prescribe appropriate (weighted) homogeneous and non-homogeneous boundary conditions at 0 and prove the existence and uniqueness of H2 loc(0, 1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator Lu := −(x2αu′)′ + u under homogeneous boundary conditions. Finally, we are interested in the equation −(|x|2αu′)′ + |u|p−1u = μ on (−1, 1) with boundary condition u(−1) = u(1) = 0. Here α > 0, p ≥ 1 and μ is a Bounded Radon measure on the interval (−1, 1). We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated.
-
A SINGULAR STURM–LIOUVILLE EQUATION INVOLVING MEASURE DATA
Communications in Contemporary Mathematics, 2013Co-Authors: Hui WangAbstract:Let α > 0 and let μ be a Bounded Radon measure on the interval (-1, 1). We are interested in the equation -(|x|2αu′)′ + u = μ on (-1, 1) with boundary condition u(-1) = u(1) = 0. We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. The cases 0 < α < 1 and α ≥ 1 must be considered separately. We also study the limiting behavior of two different approximation schemes: one is the elliptic regularization and the other is to approximate a measure μ by a sequence of L∞-functions.