Browsing by Author "Elliott, Charles M."
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- ItemA 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, JoanaWe 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.
- ItemFunction 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.
- ItemGeometric Partial Differential Equations: Surface and Bulk Processes(Zürich : EMS Publ. House, 2015) Elliott, Charles M.; Kornhuber, Ralf; Sethian, James A.The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems.
- ItemGeometric Partial Differential Equations: Theory, Numerics and Applications(Zürich : EMS Publ. House, 2011) Elliott, Charles M.; Huisken, Gerhard; Kornhuber, RalfThis workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications.