We look at the mixed boundary value problem for elliptic operators in a bounded C1,1(ℝn) domain. The boundary is decomposed into disjoint parts, D and N, with Dirichlet and Neumann data, respectively. Expanding on work done by Ott and Brown, we find a larger range of values of p, 1 < p < n/(n-1), for which the Lp mixed problem has a unique solution with the non-tangential maximal function of the gradient in Lp(∂Ω).

We find an adequate interpretation of the Lamé operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lamé system.

In this paper results concerning a semilinear elliptic boundary value problem are proven. This problem has five solutions when the range of the derivative of the nonlinearity ƒ includes the first two eigenvalues. The existence and multiplicity or radially symmetric solutions under suitable conditions on the nonlinearity when Ω is a ball in R^N.

The thesis is devoted to the investigation of the initial-value problem for linearized Euler equations utilizing an idealized one-reaction detonation model in the case of three-dimensional perturbations in a circular pipe.The problem is solved using the Laplace transform in time, Fourier series in the azimuthal angle, and expansion into Bessel's functions of the radial variable.For each radial and azimuthal mode, the inverse Laplace transform can be presented as an expansion of the solution into the normal modes of discrete and continuous spectra. The dispersion relation for the discrete spectrum requires solving the homogeneous ordinary differential equations for the adjoint system and evaluation of an integral through the reaction zone.The solution of the initial-value problem gives a convenient tool for analysis of the flow receptivity to various types of perturbations in the reaction zone and in the quiescent gas.

The dissertation focuses on the initial boundary value problems (IBVPs) of a class of nonlinear Schrodinger equations posed on a half plane R x R+ and on a strip domain R x [0,L] with Dirichlet nonhomogeneous boundary data in a two-dimensional plane. Compared with pure initial value problems (IVPs), IBVPs over part of entire space with boundaries are more applicable to the reality and can provide more accurate data to physical experiments or practical problems. Although there is less research that has been made for IBVPs than that for IVPs, more attention has been paid for IBVPs recently. In particular, this thesis studies the local well-posedness of the equation for the appropriate initial and boundary data in Sobolev spaces H^s with non-negative s and investigates the global well-posedness in the H^1-space. The main strategy, especially for the local well-posedness, is to derive an equivalent integral equation (whose solution is called mild solution) from the original equation by semi-group theory and then perform the Banach fixed-point argument. However, along the process, it is essential to select proper auxiliary function spaces and prepare all the corresponding norm estimates to complete the argument. In fact, the IBVP posed on R x R+ and the one posed on R x [0,L] are two independent problems because the techniques adopted are different. The first problem is more related to the initial value problem (IVP) posed on the whole plane R^2 and the major ingredients are Strichartz's estimate and its generalized theory. On the other hand, the second problem can be studied as an IVP over a half-line and periodic domain, which is established on the analysis for series inspired by Bourgain's work. Moreover, the corresponding smoothing properties and regularity conditions of the solution are also considered. / Ph. D.

Two inverse problems in elasticity are addressed with motivation from cellular biomechanics. The first application is computation of holding forces on a cell during its manipulation and the second application is estimation of a cell’s interior elastic mapping (i.e., estimation of inhomogeneous distribution of stiffness) using only boundary forces and displacements. It is clear from recent works that mechanical forces can play an important role in developmental biology. In this regard, we have developed a vision-based force-sensing technique to estimate forces that are acting on a cell while it is manipulated. This problem is connected to one inverse problem in elasticity known as Cauchy’s problem in elasticity. Geometric nonlinearity under noisy displacement data is accounted while developing the solution procedures for Cauchy’s problem. We have presented solution procedures to the Cauchy’s problem under noisy displacement data. Geometric nonlinearity is also considered in order to account large deformations that the mechanisms (grippers) undergo during the manipulation. The second inverse problem is connected to elastic mapping of the cell. We note that recent works in biomechanics have shown that the disease state can alter the gross stiffness of a cell. Therefore, the pertinent question that one can ask is that which portion (for example Nucleus, cortex, ER) of the elastic property of the cell is majorly altered by the disease state. Mathematically, this question (estimation of inhomogeneous properties of cell) can be answered by solving an inverse elastic boundary value problem using sets of force-displacements boundary measurements. We address the theoretical question of number of boundary data sets required to solve the inverse boundary value problem.

Master of Science / Mathematics / Nathan Albin / Pietro Poggi-Corradini / This report explores singular integration, both real and complex, focusing on the the Cauchy type integral, culminating in the proof of generalized Sokhotski-Plemelj formulae and the applications of such to a Riemann-Hilbert problem.

We show that solutions to the mixed problem on a Lipschitz domain Ω can be approximated in the Sobolev space H1(Ω) by solutions to a family of related mixed Dirichlet-Robin boundary value problems which converge in H1(Ω), and we give a rate of convergence. Further, we propose a method of solving the related problem using layer potentials.

The purpose of this paper is to present a study of different methods of solving certain boundary value problems. In particular it will be concerned with solutions by classical methods and by operational methods. Of the various operational methods that may be considered, the Laplace transformation appears to be the best be used in this paper. In the 1951 Encyclopedia Americana Annual is this report on the activities in applied mathematics for the previous year: Progress was made on the general problem of finding the eigenvalues of matrices and systems of differential equations. Considerable effort was also expended in seeking methods of solution for the partial differential equations of physics. This gives an indication of the importance of this type of work at the present time. With the development of atomic energy, Jet air­planes, and guided missiles, have come many new and different boundary value problems. The solution of these problems is an important factor in the development. The paper will include a general description of what is meant by boundary value problem, followed by some examples. These examples will be solved by classical methods and then by operational method (Laplace transformation). Then where possible, comparisons between the two methods will be made. From the study of the solutions of these examples conclusions will be drawn.

The determination of the figure of the Earth and its gravity field has long relied on methodologies that approximate the Earth by a sphere, but this level of accuracy is no longer adequate for many applications, due to the advent of new and advanced measurement techniques. New, practical and highly accurate methodologies for gravity field modelling that describe the Earth as an oblate ellipsoid of revolution are therefore required. The foundation for these methodologies is formed by solutions to ellipsoidal geodetic boundary-value problems. In this thesis, new solutions to the ellipsoidal Dirichlet, Neumann and second-order boundary-value problems, as well as the fixed- and free-geodetic boundary-value problems, are derived. These solutions do not rely on any spherical approximation, but are nevertheless completely based on a simple spherical harmonic expansion of the function that is to be determined. They rely on new relations among spherical harmonic base functions. In the new solutions, solid spherical harmonic coefficients of the desired function are expressed as a weighted summation over surface spherical harmonic coefficients of the data on the ellipsoidal boundary, or alternatively as a weighted summation over coefficients that are computed under the approximation that the boundary is a sphere. / Specific applications of the new solutions are the computation of geopotential coefficients from terrestrial gravimetric data and local or regional gravimetric geoid determination. Numerical closed-loop simulations have shown that the accuracy of geopotential coefficients obtained with the new methods is significantly higher than the accuracy of existing methods that use the spherical harmonic framework. The ellipsoidal corrections to a Stokesian geoid determination computed from the new solutions show strong agreement with existing solutions. In addition, the importance of the choice of the reference sphere radius in Stokes's formula and its effect on the magnitude and spectral sensitivity of the ellipsoidal corrections are pointed out.