Emmanuele DiBenedetto

Degenerate parabolic equations

This monograph evolved out of the 1990 Lipschitz Lectures presented by the author at the University of Bonn, Germany. It recounts recent developments in the attempt to understand the local structure of the solutions of degenerate and singular parabolic partial differential equations.

C1 + α local regularity of weak solutions of degenerate elliptic equations

Abstract : It is demonstrated the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

Harnack's inequality for degenerate and singular parabolic equations

Preface.- 1. Introduction.- 2. Preliminaries.- 3. Degenerate and Singular Parabolic Equations.- 4. Expansion of Positivity.- 5. The Harnack Inequality for Degenerate Equations.- 6. The Harnack Inequality for Singular Equations.- 7. Homogeneous Monotone Singular Equations.- Appendix A.- Appendix B.- Appendix C.- References.- Index.

Partial differential equations

Preliminaries.- Quasi-Linear Equations and the Cauchy#x2013 Kowalewski Theorem.- The Laplace Equation.- Boundary Value Problems by Double-Layer Potentials.- Integral Equations and Eigenvalue Problems.- The Heat Equation.- The Wave Equation.- Quasi-Linear Equations of First-Order.- Non-Linear Equations of First-Order.- Linear Elliptic Equations with Measurable Coefficients.- DeGiorgi Classes.

A Generalization of Theory for Two-Dimensional Fluorescence Recovery after Photobleaching Applicable to Confocal Laser Scanning Microscopes

Fluorescence recovery after photobleaching (FRAP) using confocal laser scanning microscopes (confocal FRAP) has become a valuable technique for studying the diffusion of biomolecules in cells. However, two-dimensional confocal FRAP sometimes yields results that vary with experimental setups, such as different bleaching protocols and bleaching spot sizes. In addition, when confocal FRAP is used to measure diffusion coefficients (D) for fast diffusing molecules, it often yields D-values that are one or two orders-of-magnitude smaller than that predicted theoretically or measured by alternative methods such as fluorescence correlation spectroscopy. Recently, it was demonstrated that this underestimation of D can be corrected by taking diffusion during photobleaching into consideration. However, there is currently no consensus on confocal FRAP theory, and no efforts have been made to unify theories on conventional and confocal FRAP. To this end, we generalized conventional FRAP theory to incorporate diffusion during photobleaching so that analysis by conventional FRAP theory for a circular region of interest is easily applicable to confocal FRAP. Finally, we demonstrate the accuracy of these new (to our knowledge) formulae by measuring D for soluble enhanced green fluorescent protein in aqueous glycerol solution and in the cytoplasm and nucleus of COS7 cells.

Simplified equation to extract diffusion coefficients from confocal FRAP data.

Quantitative measurements of diffusion can provide important information about how proteins and lipids interact with their environment within the cell and the effective size of the diffusing species. Confocal fluorescence recovery after photobleaching (FRAP) is one of the most widely accessible approaches to measure protein and lipid diffusion in living cells. However, straightforward approaches to quantify confocal FRAP measurements in terms of absolute diffusion coefficients are currently lacking. Here, we report a simplified equation that can be used to extract diffusion coefficients from confocal FRAP data using the half time of recovery and effective bleach radius for a circular bleach region, and validate this equation for a series of fluorescently labeled soluble and membrane‐bound proteins and lipids. We show that using this approach, diffusion coefficients ranging over three orders of magnitude can be obtained from confocal FRAP measurements performed under standard imaging conditions, highlighting its broad applicability.

Overexpression of Rhodopsin Alters the Structure and Photoresponse of Rod Photoreceptors

Rhodopsins are densely packed in rod outer-segment membranes to maximize photon absorption, but this arrangement interferes with transducin activation by restricting the mobility of both proteins. We attempted to explore this phenomenon in transgenic mice that overexpressed rhodopsin in their rods. Photon capture was improved, and, for a given number of photoisomerizations, bright-flash responses rose more gradually with a reduction in amplification--but not because rhodopsins were more tightly packed in the membrane. Instead, rods increased their outer-segment diameters, accommodating the extra rhodopsins without changing the rhodopsin packing density. Because the expression of other phototransduction proteins did not increase, transducin and its effector phosphodiesterase were distributed over a larger surface area. That feature, as well as an increase in cytosolic volume, was responsible for delaying the onset of the photoresponse and for attenuating its amplification.

A boundary modulus of continuity for a class of singular parabolic equations

Abstract Parabolic equations describing diffusion phenomena with change of phase are considered. It is demonstrated that weak solutions are continuous up to the parabolic boundary of the domain of definition. The continuity is quantitatively described by a modulus determined a priori only in terms of the data.

Diffusion of the Second Messengers in the Cytoplasm Acts as a Variability Suppressor of the Single Photon Response in Vertebrate Phototransduction

The single photon response in vertebrate phototransduction is highly reproducible despite a number of random components of the activation cascade, including the random activation site, the random walk of an activated receptor, and its quenching in a random number of steps. Here we use a previously generated and tested spatiotemporal mathematical and computational model to identify possible mechanisms of variability reduction. The model permits one to separate the process into modules, and to analyze their impact separately. We show that the activation cascade is responsible for generation of variability, whereas diffusion of the second messengers is responsible for its suppression. Randomness of the activation site contributes at early times to the coefficient of variation of the photoresponse, whereas the Brownian path of a photoisomerized rhodopsin (Rh*) has a negligible effect. The major driver of variability is the turnoff mechanism of Rh*, which occurs essentially within the first 2-4 phosphorylated states of Rh*. Theoretically increasing the number of steps to quenching does not significantly decrease the corresponding coefficient of variation of the effector, in agreement with the biochemical limitations on the phosphorylated states of the receptor. Diffusion of the second messengers in the cytosol acts as a suppressor of the variability generated by the activation cascade. Calcium feedback has a negligible regulatory effect on the photocurrent variability. A comparative variability analysis has been conducted for the phototransduction in mouse and salamander, including a study of the effects of their anatomical differences such as incisures and photoreceptors geometry on variability generation and suppression.

Periodic behaviour for the evolutionary dam problem and related free boundary problems Evolutionary dam problem

The time dependent dam problem describing the seepage of a compressible or incompressible fluid in a porous dam is studied. We prove existence of solutions in a suitable weak sense, and uniqueness for rectangular dams. Existence or periodic solutions is established and questions ofstability and periodic behavior for large time are studied.