Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − u v , is considered under the boundary conditions ∂ u ∂ ν − u ∂ v ∂ ν = 0 and v = v ⋆ on ∂Ω, where Ω ⊂ R n is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

Variational principles of nonlinear magnetoelastostatics and their correspondences

We derive the equations of nonlinear magnetoelastostatics using several variational formulations involving the mechanical deformation and an independent field representing the magnetic component. An equivalence is also discussed, modulo certain boundary integrals or constant integrals, between these formulations using the Legendre transform and properties of Maxwell’s equations. Bifurcation equations based on the second variation are stated for the incremental fields as well for all five variational principles. When the total potential energy is defined over the infinite space surrounding the body, we find that the inclusion of certain terms in the energy principle, associated with the externally applied magnetic field, leads to slight changes in the Maxwell stress tensor and associated boundary conditions. Conversely, when the energy contained in the magnetic field is restricted to finite volumes, we find that there is a correspondence between the discussed formulations and associated expressions of physical entities. In view of a diverse set of boundary data and the nature of externally applied controls in the problems studied in the literature, along with an equally diverse list of variational principles employed in modelling, our analysis emphasises care in the choice of variational principle and unknown fields so that consistency with other choices is also satisfied.

Inverse problem of shape identification from boundary measurement for Stokes equations: Shape differentiability of Lagrangian

For Stokes equations under divergence-free and mixed boundary conditions, the inverse problem of shape identification from boundary measurement is investigated. Taking the least-square misfit as an objective function, the state-constrained optimization is treated by using an adjoint state within the Lagrange approach. The directional differentiability of a Lagrangian function with respect to shape variations is proved within the velocity method, and a Hadamard representation of the shape derivative by boundary integrals is derived explicitly. The application to gradient descent methods of iterative optimization is discussed.

EFFICIENT CALCULATION OF THE BEM INTEGRALS ON ARBITRARY SHAPES WITH THE FFT

This paper builds upon the results of a recent study which illustrates how the Fast Fourier Transformation (FFT) can be used to accelerate the Boundary Element Method (BEM) for arbitrary shapes. In the present work, we further deepen this understanding and focus especially on implementation details in order to calculate the boundary integrals with the FFT. Different numerical techniques are compared for an exemplary shape. Also, additions to the concept are mentioned such as the introduction of a high-resolution grid close to the boundary and a low-resolution grid farther away.

A note on L2-boundary integrals of the Bergman kernel

In this paper, we obtain some estimates on the [Formula: see text]-boundary norm of the Bergman kernel for pseudoconvex domains admitting a plurisubharmonic defining function.