49 research outputs found
Existence, iteration procedures and directional differentiability for parabolic QVIs
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.Comment: 41 page
An abstract framework for parabolic PDEs on evolving spaces
We present an abstract framework for treating the theory of well-posedness of
solutions to abstract parabolic partial differential equations on evolving
Hilbert spaces. This theory is applicable to variational formulations of PDEs
on evolving spatial domains including moving hypersurfaces. We formulate an
appropriate time derivative on evolving spaces called the material derivative
and define a weak material derivative in analogy with the usual time derivative
in fixed domain problems; our setting is abstract and not restricted to
evolving domains or surfaces. Then we show well-posedness to a certain class of
parabolic PDEs under some assumptions on the parabolic operator and the data.Comment: 38 pages. Minor typos correcte
Stability of the solution set of quasi-variational inequalities and optimal control
For a class of quasi-variational 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 of
the solution set of the QVI with respect to {monotone} 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.Comment: 29 page
Well-posedness of a fractional porous medium equation on an evolving surface
We investigate the existence, uniqueness, and L1-contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface. To settle the existence, we reformulate the equation as a local problem on a semi-infinite cylinder, regularise the porous medium nonlinearity and truncate the cylinder. Then we pass to the limit first in the truncation parameter and then in the nonlinearity, and the identification of limits is done using the theory of subdifferentials of convex functionals. In order to facilitate all of this, we begin by studying (in the setting of closed Riemannian manifolds and Sobolev spaces) the fractional Laplace–Beltrami operator which can be seen as the Dirichlet-to-Neumann map of a harmonic extension problem. A truncated harmonic extension problem will also be examined and convergence results to the solution of the harmonic extension will be given. For a technical reason, we will also consider some related extension problems on evolving hypersurfaces which will provide us with the minimal time regularity required on the harmonic extensions in order to properly formulate the moving domain problem. This functional analytic theory is of course independent of the fractional porous medium equation and will be of use generally in the analysis of fractional elliptic and parabolic problems on manifolds
A Stefan problem on an evolving surface
We formulate a Stefan problem on an evolving hypersurface and study the
well-posedness of weak solutions given data. To do this, we first develop
function spaces and results to handle equations on evolving surfaces in order
to give a natural treatment of the problem. Then we consider the existence of
solutions for data; this is done by regularisation of the
nonlinearity. The regularised problem is solved by a fixed point theorem and
then uniform estimates are obtained in order to pass to the limit. By using a
duality method we show continuous dependence which allows us to extend the
results to data.Comment: 21 page
A coupled ligand-receptor bulk-surface system on a moving domain : well posedness, regularity and convergence to equilibrium
We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modeling 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 nonmoving 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
Recent trends and views on elliptic quasi-variational inequalities
We consider state-of-the-art methods, theoretical limitations, and open problems in elliptic Quasi-Variational Inequalities (QVIs). This involves the development of solution algorithms in function space, existence theory, and the study of optimization problems with QVI constraints. We address the range of applicability and theoretical limitations of fixed point and other popular solution algorithms, also based on the nature of the constraint, e.g., obstacle and gradient-type. For optimization problems with QVI constraints, we study novel formulations that capture the multivalued nature of the solution mapping to the QVI, and generalized differentiability concepts appropriate for such problems
On some linear parabolic PDEs on moving hypersurfaces
We consider existence and uniqueness for several examples of linear parabolic
equations formulated on moving hypersurfaces. Specifically, we study in turn a
surface heat equation, an equation posed on a bulk domain, a novel coupled
bulk-surface system and an equation with a dynamic boundary condition. In order
to prove the well-posedness, we make use of an abstract framework presented in
a recent work by the authors which dealt with the formulation and
well-posedness of linear parabolic equations on arbitrary evolving Hilbert
spaces. Here, after recalling all of the necessary concepts and theorems, we
show that the abstract framework can applied to the case of evolving (or
moving) hypersurfaces, and then we demonstrate the utility of the framework to
the aforementioned problems.Comment: 32 pages. Section 3 recalls the main results and notation of our
earlier work arXiv:1403.450
On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities
In this 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. Along 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. We conclude the paper with some examples and an
application to thermoforming.Comment: 13 page
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Existence, iteration procedures and directional differentiability for parabolic QVIs
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