J. B. Diaz

Dirichlet, Neumann, and mixed boundary value problems for the wave equation uxx —uyy =0 for a rectangle

We reconsider here a classical “not well posed” problem, the Dirichlet problem for the wave equation for a rectangle with sides parallel to the coordinate axes. For the same domain, we also consider the Neumann, and many mixed Dirichlet–Neumann boundary value problems, including a “general mixed problem” where, at each point which is not a corner, the boundary condition is either of Dirichlet or of Neumann type.

Sturmian theorems for a class of elliptic equations of order 4m

Sturm fundamental comparison theorems deal with solutions of two ordinary differential equations . The purpose of this paper is to prove comparison theorems for the two partial differential equations: where m and n are positive integers, is the Laplacian, and δ m is the mth iterated Laplacian, Separation theorems are also proved.

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplaces equation is obtained, involves Greens function, and Kelvins method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplaces equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplaces equation has already solved...

A Generalization, to Higher Dimensions, of a Theorem of Lucas Concerning the Zeros of the Derivative of a Polvnomial of one Complex Variable

Let m and n be positive integers, and z1,z2,...,zm be m points in real n dimensional Euclidean space En, For eachj=1,2,...,m,let fj be a real valued continuously differentiate function of the positive real variable r, with fj(r)gt;0 for r>0, Consider the real valued function F, which is defined, for w in En, with w=zj forj = 1,2,..., m, by the equation

Sturm comparison and separation theorems for linear, second order, self-adjoint, ordinary, differential equations and for first order systems

Sturm comparison and separation theorems, for first order linear combinations of solutions of linear, second order, self-adjoint, ordinary, differential equations, are obtained as consequences of Sturm comparison and separation theorems for linear combinations of components of solutions of linear, first order systems of ordinary, differential equations.