The higher order regularity Dirichlet problem for elliptic systems in the upper-half space

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is well-posed in the class of functions whose nontangential maximal operator of their derivatives up to, and including, order $\ell$ is $L^p$-integrable. This class includes all scalar, complex coefficient elliptic operators of second order, as well as the Lam\'e system of elasticity, among others

Mean value formulas for differential operators

Abstract only availableThe main goal of our research is to discuss Mean Value Formulas for solutions of partial differential equations of a certain form (see 1 below) where A is a symmetric, positive definite, real-valued matrix. Such equations arise when modeling various steady state (time independent) phenomena. A mean value formula is a precise way to relate the value of the solution at a point c to the values taken in a neighborhood of that point. A neighborhood being simply a region around the point c. Mean Value Formulas have a rich history stretching back to the early 1800 when the famous mathematician C.F. Gauss initiated their study by considering the case when the matrix A is the identity. This corresponds to the Laplace equation (see 2 below). In this case, the neighborhood around a point c is a sphere. The more general settings we discuss correspond to: integer powers of the Laplacian, the anisotropic (direction dependent) model and the Lame system of isotropic elasticity. We were able to establish mean value formulas in each of these cases. There are many aspects of the project that lend themselves to future study. This includes considering other types of partial differential equations, such as the heat equation and Helmholtz equation, and proving converses to the Mean Value Theorems already established, obtaining the most general version of the Mean Value Formula, valid for all solutions to the Lame system.MU Undergraduate Research Scholars Progra

The Poisson Problem for the Exterior Derivative Operator with Dirichlet Boundary Condition on Nonsmooth Domains

International audienceWe formulate and solve the Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Lipschitz domains, of arbitrary topology, for data in Besov and Triebel-Lizorkin spaces

Distributions, partial differential equations, and harmonic analysis

The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity. The main additions to the current, second edition, pertain to fundamental solutions (through the inclusion of the Helmholtz operator, the perturbed Dirac operator, and their iterations) and the theory of Sobolev spaces (built systematically from the ground up, exploiting natural connections with the Fourier Analysis developed earlier in the monograph).