25 research outputs found

    On parabolic final value problems and well-posedness

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    We prove that a large class of parabolic final value problems is well posed.This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is evolution equations for Lax--Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.Comment: 6 pages. Accepted version; a short announcement of results from our full paper on final value problems. Appeared in Comptes Rendu Mathematique

    A class of well-posed parabolic final value problems

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    This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax--Milgram operators in vector distribution spaces constitute the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.Comment: 16 pages. To appear in "Applied and numerical harmonic analysis"; a reference update. Conference contribution, based on arXiv:1707.02136, with some further development

    Final value problems for parabolic differential equations and their well-posedness

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    This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax--Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.Comment: 39 pages. Revised version, with minor improvements. Essentially identical to the accepted version, which appeared in Axioms on 9 May 201

    Glosarium Fisika

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    Cellular biomechanics in 2D and 3D epithelial model tissues : from keratin intermediate filaments to breast gland in vitro reconstructed basement membranes

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    The mechanical organization of biological tissue is crucial to the load-transmitting capacity of our bodies, and follows a hierarchical architecture that macroscopically results in organ’s formation and function. At the base of such a tightly regulated structure we find single cells, whose mechanical properties are decisive in shaping their interaction with the surrounding environment. Developing a fundamental understanding of this interplay over a wide range of length scales is essential to reach the deep working knowledge of the biomechanics of multicellular systems required for tissue engineering, surface design and nanomedicine. In this work, the mechanical properties of epithelial model systems were analyzed at different organizational scales (namely, from single cells to microtissue) by means of atomic force microscopy (AFM) micro- and nano-indentation experiments. Despite the intrinsic difficulty in characterizing soft and heterogeneous biological samples in terms of mechanical response, this technique still offers an exciting possibility to quantitatively probe their viscoelastic behavior. At first, a murine epidermal cell line completely devoid of keratin intermediate filaments (knock-out keratinocytes) was compared to its wild type counterpart in order to assess the role played by this cytoskeletal component in conferring mechanical stability to single cells. Then, cellular monolayers were analyzed, to validate the relevance of our findings also in a more physiological context. Despite its presence in organs such as skin and nails, which obviously serve a barrier function, the mechanical role of keratin in deeper tissue remained controversial for a long time. Reconstructed keratin polymer gels in fact display properties resemblant of viscoelastic solids; in vivo, the networks are formed of bundles that are relatively sparse and show lower connectivity than other cytoskeletal components. This fact, together with the low values of bending stiffness and extremely high extensibility reported for these filaments, would suggest that keratin networks confer resilience and elasticity to cells, rather than a scaffolding function against compressive stress. Our results though clearly pointed at a substantial softening of keratin-lacking cells, with elasticity moduli differences of 25% to 35% between wild type and knock-out according to the cellular region probed. The presented data represent the first proof of this effect on the single cell level. Validation of this result further came from the observation that the difference could be partially suppressed by reintroducing a single keratin protein in the mutant cells. In the second part of this work, a three-dimensional cell culture system mimicking the elementary unit of a human breast gland was analyzed in terms of its biomechanical and permeation properties. The cell line used for this purpose (MCF10A), when grown in an extracellular matrix-resembling environment, can develop into growth-arrested acinar structures which follow the same substantial maturation steps of a human breast gland; cells organize according to an apico-basal polarization scheme, secrete a dense matrix of cross-linked extracellular matrix proteins to surround them (the so called basement membrane) and finally develop a hollow lumen necessary, in vivo, for milk production and secretion. The centrality of breast gland tissue in a context of cancer research cannot be overstated: alveolar units are the hotspot for tumor formation, and as such have been the focus of much attention in the past years. Relatively little effort, though, has been dedicated to understanding the mechanical interplay of healthy breast gland microtissues with their surrounding environment, despite the fact that one of the hallmarks of cancer progression is a set of strong alterations in the mechanical phenotype of aberrant cells. Here, we offer an experimental analysis of the mechanical properties of healthy 3D acinar structures at different developmental stages, and briefly compare them with those of invasive microtissues. The application of different hyperelastic models to the interpretation of nanoindentation experiments is discussed, along with a tentative clarification of some of controversies arising during AFM data analysis. Additionally, a characterization of isolated basement membranes performed by means of atomic force microscopy imaging, scanning electron microscopy and superresolution light microscopy is reported; experimental evidence suggests that basement membranes act as fundamentally elastic materials whose thickness and structural stability change throughout the different developmental stages. To complement this biomechanical analysis, we investigated the acinar permeation properties; in short, data elucidate that the basement membrane acts as a passive diffusion barrier with a size-selectivity threshold for the retardation of macromolecular permeation of about 40 kDa and a pore size of at least 9 nm. At the same time, it offers a fundamental mechanical shielding function, reaching elastic modulus values of up to about 400 kPa in the fully matured state. Taken together, the presented data underline how intra- and extra-cellular polymer networks serve a crucial function in defining the mechanical properties of epithelial tissue.</p

    Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport

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    Unlike equilibrium statistical mechanics, with its well-established foundations, a similar widely-accepted framework for non-equilibrium statistical mechanics (NESM) remains elusive. Here, we review some of the many recent activities on NESM, focusing on some of the fundamental issues and general aspects. Using the language of stochastic Markov processes, we emphasize general properties of the evolution of configurational probabilities, as described by master equations. Of particular interest are systems in which the dynamics violate detailed balance, since such systems serve to model a wide variety of phenomena in nature. We next review two distinct approaches for investigating such problems. One approach focuses on models sufficiently simple to allow us to find exact, analytic, non-trivial results. We provide detailed mathematical analyses of a one-dimensional continuous-time lattice gas, the totally asymmetric exclusion process (TASEP). It is regarded as a paradigmatic model for NESM, much like the role the Ising model played for equilibrium statistical mechanics. It is also the starting point for the second approach, which attempts to include more realistic ingredients in order to be more applicable to systems in nature. Restricting ourselves to the area of biophysics and cellular biology, we review a number of models that are relevant for transport phenomena. Successes and limitations of these simple models are also highlighted.Comment: 72 pages, 18 figures, Accepted to: Reports on Progress in Physic

    References, Appendices & All Parts Merged

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    Includes: Appendix MA: Selected Mathematical Formulas; Appendix CA: Selected Physical Constants; References; EGP merged file (all parts, appendices, and references)https://commons.library.stonybrook.edu/egp/1007/thumbnail.jp
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