Krein-like extensions and the lower boundedness problem for elliptic operators

For selfadjoint extensions tilde-A of a symmetric densely defined positive operator A_min, the lower boundedness problem is the question of whether tilde-A is lower bounded {\it if and only if} an associated operator T in abstract boundary spaces is lower bounded. It holds when the Friedrichs extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets 1976); this applies to elliptic operators A on bounded domains. For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for general T. The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann extension A_0, appears here; its possible lower boundedness for all real a is decisive. We study this Krein-like extension, showing for bounded domains that the discrete eigenvalues satisfy N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .Comment: 35 pages, revised for misprints and accepted for publication in Journal of Differential Equation

Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains.

211 pagesThis is a preliminary version of the first part of a book project that will consist of four parts. We are making it available in electronic form now, because there is a demand for some of the technical tools it provides, in particular a detailed presentation of analytic elliptic regularity estimates in the neighborhood of smooth boundary points. In our proof of analytic a priori estimates, besides the classical Morrey-Nirenberg techniques of nested open sets and difference quotients, a new ingredient is a Cauchy-type estimate for coordinate transformations based on the Faà di Bruno formula for derivatives of composite functions. This first part can also serve as a general introduction into the subject of regularity for linear elliptic systems with smooth coefficients on smooth domains. We treat regularity in Sobolev spaces for a general class of second order elliptic systems and corresponding boundary operators that cover, in particular, many elliptic problems in variational form. Starting from the regularity of the variational solution, we follow the improvement of the regularity of the solution as the regularity of the data is raised, first for low regularity, and then going to ever higher regularity and finally to analytic regularity. Supported by the discussion of many examples, some of them new, such as the variational formulation of the electromagnetic impedance problem, we hope to provide new insight into this classical subject. We hope to be able to finish the whole project soon and to publish all four parts, but in the meantime this first part can be used as a starting point for proofs of elliptic regularity estimates in more complicated situations

On the Structure of Linear Dislocation Field Theory

Uniqueness of solutions in the linear theory of non-singular dislocations, studied as a special case of plasticity theory, is examined. The status of the classical, singular Volterra dislocation problem as a limit of plasticity problems is illustrated by a specific example that clarifies the use of the plasticity formulation in the study of classical dislocation theory. Stationary, quasi-static, and dynamical problems for continuous dislocation distributions are investigated subject not only to standard boundary and initial conditions, but also to prescribed dislocation density. In particular, the dislocation density field can represent a single dislocation line. It is only in the static and quasi-static traction boundary value problems that such data are sufficient for the unique determination of stress. In other quasi-static boundary value problems and problems involving moving dislocations, the plastic and elastic distortion tensors, total displacement, and stress are in general non-unique for specified dislocation density. The conclusions are confirmed by the example of a single screw dislocation.Comment: This is the published versio

Density-to-Potential Inversions in Density Functional Theory

Density functional theory and many of its extensions are formally exact quantum many-body theories. In practice, however, implementations of these theories use approximations for all but the most trivial systems. We present a set of inversion methods to numerically compute the exact potentials corresponding to given input densities. The results of these inversions may then be used to evaluate the quality of different density functional approximations and guide the design of new approximations. The inversion methods use classical gradient-based optimization routines that are constrained to satisfy the governing partial differential equations. Numerous examples are given to illustrate the strengths and weaknesses of the different inversion methods

Harnack Inequality and Fundamental Solution for Degenerate Hypoelliptic Operators

In this thesis we study subelliptic operators in divergence form on R^N, and we are interested in establishing Harnack inequalities related to these operators in various contexts. As a first result of the thesis, we prove a non-invariant Harnack inequality, passing through a Strong Maximum Principle; in doing so, we require the hypoellipticity of the operator to construct a Green function, that we have used (by means of techniques of Potential Theory) in order to obtain the Harnack inequality. In the second main result of this thesis, we prove a non-homogeneous invariant Harnack inequality for these subelliptic operators under low regularity assumption. Currently, it is known that the natural framework for Harnack-type theorems is the setting of doubling metric spaces; we suppose that the quadratic form of the operator can be naturally controlled by a family of locally-Lipschitz vector fields. Moreover, we assume that, with the associated Carnot-Carathéodory metric d, N-dimensional Euclidean space is endowed by d with the structure of a doubling space (globally) and a Poincaré inequality on any d-ball holds true. We use a Sobolev type inequality and the Moser iterative technique to prove a non-homegeneous invariant Harnack Inequality; as a consequence, we show the existence of the Green function using only the Harnack inequality

Navier-Stokes Equations with Navier Boundary Condition

164 p.En esta tesis se estudian diversos problemas relacionados con las ecuaciones de Stokes y Navier Stokes en dominios acotados y con condiciones de contorno de Navier.Por una parte se han obtenido resultados de existencia de soluciones de la ecuación estacionaria de Stokes con condiciones de contorno de Navier y se han obtenido estimaciones uniformes con respecto del coeficiente de fricción. Ello ha permitido demostrar la convergencia de estas soluciones a las soluciones de la ecuación estacionaria de Stokes con condición de contorno de Dirichlet cuando el coeficiente de fricción converge a infinito.Con estos resultados sobre la ecuación estacionaria de Stokes se ha estudiado el problema para las ecuaciones de evolución de Stokes y Navier Stokes con condiciones de Navier. Se ha obtenido una teoría de semigrupos en espacios Lp que extiende los resultados conocidos correspondientes a otra condiciones de contorno (como Dirichlet o de tipo Navier). Se ha demostrado aquí también la convergencia de las soluciones de estas ecuaciones con condiciones de Navier a las soluciones de la misma ecuación con condiciones Dirichlet cuando el coeficiente de fricción tiende a infinito.En un último capítulo se han obtenido estimaciones uniformes, con respecto del parámetro de fricción, de la regularidad de las soluciones de un operador elíptico en forma de divergencia con condiciones de tipo Robin en un dominio cuya frontera es de clase C1

Elementary Differential Equations with Boundary Value Problems

Written in a clear and accurate language that students can understand, Trench\u27s new book minimizes the number of explicitly stated theorems and definitions. Instead, he deals with concepts in a conversational style that engages students. He includes more than 250 illustrated, worked examples for easy reading and comprehension. One of the book\u27s many strengths is its problems, which are of consistently high quality. Trench includes a thorough treatment of boundary-value problems and partial differential equations and has organized the book to allow instructors to select the level of technology desired. This has been simplified by using symbols, C and L, to designate the level of technology. C problems call for computations and/or graphics, while L problems are laboratory exercises that require extensive use of technology. Informal advice on the use of technology is included in several sections and instructors who prefer not to emphasize technology can ignore these exercises without interrupting the flow of material. (From the 1st edition)https://digitalcommons.trinity.edu/mono/1008/thumbnail.jp