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Author: Alphonse, Amal × DDC: 510 ×

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Now showing 1 - 9 of 9
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    Existence, iteration procedures and directional differentiability for parabolic QVIs
    (Berlin ; Heidelberg : Springer, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.
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    Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
    (Orlando, Fla. : Elsevier, 2023) Alphonse, Amal; Caetano, Diogo; Djurdjevac, Ana; Elliott, Charles M.
    We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev–Bochner spaces. An Aubin–Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev–Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.
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    Analysis of a quasi-variational contact problem arising in thermoelasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Rautenberg, Carlos N.; Rodrigues, José Francisco
    We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.
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    Stability of the solution set of quasi-variational inequalities and optimal control
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.
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    Optimal control and directional differentiability for elliptic quasi-variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
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    A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Alphonse, Amal; Elliott, Charles M.; Terra, Joana
    We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.
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    Directional differentiability for elliptic quasi-variational inequalities of obstacle type
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments.
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    On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.
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    Existence, iteration procedures and directional differentiability for parabolic QVIs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.
    We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.