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Orlando Gianluca - One of the best experts on this subject based on the ideXlab platform.
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A Lower Semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,\gamma}$, $\gamma>1$
2019Co-Authors: Crismale Vito, Orlando GianlucaAbstract:We prove the Lower Semicontinuity of functionals of the form \[ \int \limits_\Omega \! V(\alpha) \, \mathrm{d} |\mathrm{E} u| \, , \] with respect to the weak converge of $\alpha$ in $W^{1,\gamma}(\Omega)$, $\gamma > 1$, and the weak* convergence of $u$ in $BD(\Omega)$, where $\Omega \subset \mathbb{R}^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their Lower Semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $\gamma < n$
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A RESHETNYAK-TYPE Lower Semicontinuity RESULT FOR LINEARISED ELASTO-PLASTICITY COUPLED WITH DAMAGE IN $W^1,n$
HAL CCSD, 2017Co-Authors: Crismale Vito, Orlando GianlucaAbstract:In this paper we prove a Lower Semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $\alpha_k \mathrm{E}u_k$ with respect to the weak convergence $\alpha_k \rightharpoonup \alpha$ in $W^1,n(\Omega)$ and the weak$^*$ convergence $u_k \rightharpoonup^* u$ in $BD(\Omega)$ , $\mathrm{E}$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $\alpha \mathrm{E}u + \eta$, with $\eta$ supported on an at most countable set
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A Reshetnyak-type Lower Semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,n}$
2017Co-Authors: Crismale Vito, Orlando GianlucaAbstract:In this paper we prove a Lower Semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $\alpha_k\,\mathrm{E}u_k$ with respect to the weak convergence $\alpha_k\rightharpoonup \alpha$ in $W^{1,n}(\Omega)$ and the weak$^*$ convergence $u_k\stackrel{*}\rightharpoonup u$ in $BD(\Omega)$, $\mathrm{E}$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $\alpha\,\mathrm{E}u+\eta$, with $\eta$ supported on an at most countable set.Comment: 12 pages, 2 figure
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Some results on cohesive energies: approximation, Lower Semicontinuity and quasistatic evolution
place:Trieste, 2016Co-Authors: Orlando GianlucaAbstract:In this thesis, cohesive fracture is investigated under three different perspectives. First we study the asymptotic behaviour of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter, which forces damage concentration on regions of codimension one. We determine the Gamma-limit as the small parameter tends to zero and show that it contains an energy term involving the crack opening. The second problem we consider is the Lower Semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrised gradient. We introduce a suitable class of cohesive surface terms, which make the functional Lower semicontinuous with respect to L^1 convergence. Finally, we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e., a complete fracture may be produced by oscillation of small jumps
Robert Plebaniak - One of the best experts on this subject based on the ideXlab platform.
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Maximality Principle and General Results of Ekeland and Caristi Types without Lower Semicontinuity Assumptions in Cone Uniform Spaces with Generalized Pseudodistances
Fixed Point Theory and Applications, 2010Co-Authors: Kazimierz Włodarczyk, Robert PlebaniakAbstract:Our aim is twofold: first, we want to introduce a partial quasiordering in cone uniform spaces with generalized pseudodistances for giving the general maximality principle in these spaces. Second, we want to show how this maximality principle can be used to obtain new and general results of Ekeland and Caristi types without Lower Semicontinuity assumptions, which was not done in the previous publications on this subject.
Monica Patriche - One of the best experts on this subject based on the ideXlab platform.
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random fixed point theorems under mild continuity assumptions
Fixed Point Theory and Applications, 2014Co-Authors: Monica PatricheAbstract:In this paper, we study the existence of the random fixed points under mild continuity assumptions. The main theorems consider the almost Lower semicontinuous operators defined on Banach spaces and also operators having properties weaker than Lower Semicontinuity. Our results either extend or improve corresponding ones present in literature.
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random fixed point theorems under mild continuity assumptions
arXiv: Probability, 2013Co-Authors: Monica PatricheAbstract:In this paper, we study the existence of the random fixed points under mild continuity assumptions. The main theorems consider the almost Lower semicontinuous operators defined on Frechet spaces and also operators having properties weaker than Lower Semicontinuity. Our results either extend or improve corresponding ones present in literature.
Rindler Filip - One of the best experts on this subject based on the ideXlab platform.
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Liftings, Young measures, and Lower Semicontinuity
2020Co-Authors: Shaw Giles, Rindler FilipAbstract:This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,Du_j)j$ for $(u_j)_j \in \mathrm{BV}(\Omega;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional \[ \mathcal{F}\colon u\to\int_\Omega f(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}({\Omega};\mathbb{R}^m),\quad {\Omega}\in\mathbb{R}^d\text{ open}, \] to the space $\mathrm{BV}(\Omega; \mathbb{R}^m)$. Lower Semicontinuity results of this type were first obtained by Fonseca and M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that $f$ be Carath\'eodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising $\mathcal{F}$ in the $x$ and $u$ variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.Comment: 75 pages. Updated to correct a series of minor typos/ inaccuracies. The statement and proof of Theorem have also been amended- subsequent steps relying upon the Theorem did not require updatin
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Liftings, young measures, and Lower Semicontinuity
'Springer Science and Business Media LLC', 2019Co-Authors: Rindler Filip, Shaw GilesAbstract:This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (uj, Duj)j for (uj)j⊂BV(Ω;Rm) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional F:u↦∫Ωf(x,u(x),∇u(x))dx,u∈W1,1(Ω;Rm),Ω⊂Rd open, to the space BV(Ω;Rm) . Lower Semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure
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Liftings of BV-maps and Lower Semicontinuity
Banff International Research Station for Mathematical Innovation and Discovery, 2018Co-Authors: Rindler FilipAbstract:Liftings and their associated Young measures are new tools to study the asymptotic behaviour of sequences of BV-maps under weak* convergence. Their main feature is that they allow to keep track of the precise shape of the jump path and as such are natural objects whenever different ways of approaching a jump need to be distinguished. While this tool has several promising applications, in this talk I will focus on its use to prove Lower Semicontinuity for linear-growth functionals that depend on the value of the argument function, u(x), besides its gradient. It is well known that in this situation the particular shape of jumps cannot be neglected. Using the theory of liftings, we can prove relaxation theorems under essentially optimal assumptions, generalizing a classical theorem by Fonseca & Müller (1993). The key idea is that liftings provide the right way of localizing the functional in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow up procedure. This is joint work with Giles Shaw.Non UBCUnreviewedAuthor affiliation: University of WarwickResearche
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Lower Semicontinuity and relaxation of linear-growth integral functionals under PDE constraints
2017Co-Authors: Arroyo-rabasa Adolfo, De Philippis Guido, Rindler FilipAbstract:We show general Lower Semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known Lower Semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.Comment: 43 page
Virginia De Cicco - One of the best experts on this subject based on the ideXlab platform.
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Lower Semicontinuity FOR NONAUTONOMOUS SURFACE INTEGRALS
2020Co-Authors: Virginia De CiccoAbstract:Abstract. Some Lower Semicontinuity results are established for nonautonomous surface integrals depending in a discontinuous way on the spatial variable. The proof of the Semicontinuity results is based on some suitable approximations from below with appropriate functionals
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Lower Semicontinuity and relaxation results in bv for integral functionals with bv integrands
ESAIM: Control Optimisation and Calculus of Variations, 2008Co-Authors: Micol Amar, Virginia De Cicco, Nicola FuscoAbstract:New L 1 -Lower Semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the Lower Semicontinuity in the sense of the 1-capacity is assumed in order to obtain the Lower Semicontinuity of the functional. This condition is satisfied, for instance, by the Lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained. Mathematics Subject Classification. 49J45, 49Q20, 49M20. (x) u−(x) f ∞ (x, s, νu(x))ds.
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a chain rule formula in bv and application to Lower Semicontinuity
Calculus of Variations and Partial Differential Equations, 2007Co-Authors: Virginia De Cicco, Nicola Fusco, Anna VerdeAbstract:Dv = B′(ũ)∇uLN +B′(ũ)Dcu+ (B(u)−B(u))νuH Ju , where ∇u is the absolutely continuous part of Du, Du is the Cantor part of Du and Ju is the jump set of u (for the definition of these and other relevant quantities, see Sect.2). A delicate issue about this formula concerns the meaning of the first two terms on the right hand side. In fact, in order to understand why they are well defined, one has to take into account that B′(t) exists for L-a.e. t and that, if E is an L-null set in IR, not only ∇u vanishes L -a.e. on ũ−1(E), but also |Dcu|(ũ−1(E)) = 0 (see [2, Theorem 3.92]). The difficulty of giving a correct meaning to the various parts in which the derivative of a BV function can be split is even greater when u is a vector field, a case where a chain rule formula has been proved by Ambrosio and Dal Maso in [1]. In particular, their result applies to the composition of a scalar BV function with a Lipschitz function B depending also on x, namely to the function B(x, u(x)), where B : Ω × IR → IR is Lipschitz. In many applications, however, B has the special form
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a chain rule in l 1 div ω and its applications to Lower Semicontinuity
2004Co-Authors: Virginia De Cicco, Giovanni LeoniAbstract:A chain rule in the space L 1 (div; Ω) is obtained under weak regularity con- ditions. This chain rule has important applications in the study of Lower Semicontinuity problems for general functionals of the form Ω f (x, u, ∇u) dx with respect to strong con- vergence in L 1 (Ω) . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.
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a chain rule in l 1 left operatorname div omega right and its applications to Lower Semicontinuity
Calculus of Variations and Partial Differential Equations, 2003Co-Authors: Virginia De Cicco, Giovanni LeoniAbstract:A chain rule in the space \(L^{1}\left(\operatorname*{div};\Omega\right) \) is obtained under weak regularity conditions. This chain rule has important applications in the study of Lower Semicontinuity problems for general functionals of the form \(\int_{\Omega}f(x,u,\nabla u) dx\) with respect to strong convergence in \(L^{1}\left(\Omega\right) \) . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.
