Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Pr\"uss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results

Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems

On bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail

Well-Posedness, Regularity, and Convergence Analysis of the Finite Element Approximation of a Generalized Robin Boundary Value Problem

In this paper, we propose the mathematical and finite element analysis of a second-order partial differential equation endowed with a generalized Robin boundary condition which involves the Laplace--Beltrami operator by introducing a function space $H^1(\Omega; \Gamma)$ of $H^1(\Omega)$-functions with $H^1(\Gamma)$-traces, where $\Gamma \subseteq \partial \Omega$. Based on a variational method, we prove that the solution of the generalized Robin boundary value problem possesses a better regularity property on the boundary than in the case of the standard Robin problem. We numerically solve generalized Robin problems by means of the finite element method with the aim of validating the theoretical rates of convergence of the error in the norms associated to the space $H^1(\Omega; \Gamma)$

Membrane-based electrochemical nanobiosensor for escherichia coli detection and analysis of cells viability

A sensitive and selective membrane-based electrochemical nanobiosensor is developed for specific quantitative label-free detection of Escherichia coli (E. coli) cells and analysis of viable but nonculturable (VBNC) E. coli cells which remain mostly undetected using current methods. The sensing mechanism relies on the blocking of nanochannels of a nanoporous alumina-membrane modified electrode, upon the formation of immune complexes at the nanoporous membrane. The resulting obstacle to diffusive mass transfer of a redox probe in the analysis solution to the underlying platinum electrode reduces the Faradaic signal response of the biosensor, measured using cyclic voltammetry. Antibody loading under conditions of varying antibody concentrations and pHs are optimized. The biosensor gives a low detection limit of 22 cfu mL ̄¹ (R² = 0.999) over a wide linear working range of 10 to 10⁶cfu mL ̄¹. It is specific toward E. coli with minimal cross-reactivity to two other pathogenic bacteria (commonly found in waters). Relative standard deviation (RSD) for triplicate measurements of 2.5% indicates reasonably useful level of reproducibility. Differentiation of live, VBNC, and dead cells are carried out after the cell capture and quantitation step, by simple monitoring of the cells’ enzyme activity using the same redox probe in the analysis solution, in the presence of glucose.Accepted versio