Problems and methods in partial differential equations

Summary Finite part and logarithmic part of some divergent integrals with applications to theCauchy problem. A Giovanni Sansone nel suo 70^mo compleanno. The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office under Contract AF 61 (056)-86.     

On a regular cauchy problem for the Euler-Poisson-Darboux equation

Summary The explicit solution of a particularCauchy problem for the n-dimensionalEuler-Poisson-Darboux equation is found. To obtain the solution the method ofM. Riesz is extended to include non self-adjoint equations. Existence and uniqueness are shown. This research was supported in part by the United States Air Force under Contract No. AF18(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Series expansions of solutions of the heat equation inn dimensions

Summary This paper is an extensions of a recent result byD. V. Widder andP. C. Rosenbloom concerning the expansion of a solution u(x, t) of the heat equation, u_x ^x=u_t, in series of polyonomial solutions. Whereas that result involved two variables only, the present one permits an arbitrary finite number of variables. It is found that the polynomial solutions of the heat equation in n dimensions are factorable into polynomials in two variables. The expansion problem may be considered as an investigation of the space spanned by linear combinations of these factorable solutions To Enrico Bompiani on his scientific Jubilee This research was supported (in part) by the U. S. Air Force, through the Air Force Office of Scientific Research, Air Research and Development Command, under Contract AF 49(638)-574. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

The Dirichlet problem for the wave equation

Summary We consider a vibrating string fixed at the ends for which the position is known at two different times. This corrisponds to a classical not well posed problem, theDirichlet problem for the wave equation, which we reconsider here in order to determine under what conditions it is possible to obtain useful information about the physical phenomenon. This problem is related to a functional equation from which the principal results can be deduced.

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Sunto Si riesamina un classico esempio di problema ? non ben posto ?, il problema diDirichlet corrispondente al modello fisico di una corda vibrante, fissa agli estremi e assumente posizioni note in due diversi istanti. Lo studio relativo all'esistenza ed unicità della soluzione è ricondotto a quello di una equazione funzionale ed è strettamente connesso a questioni di teoria dei numeri. Si esamina poi la dipendenza della soluzione dai dati e si discute il problema dal punto di vista delle sue applicazioni.

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This research was supported in part by the United States Air Force under contract No. AF 49 (638) 228 monitored by the Office of Scientific Research, Air Research and Development Command.     

Locking effects in the finite element approximation of elasticity problems

Summary We consider the finite element approximation of the 2D elasticity problem when the Poisson ratiov is close to 0.5. It is well-known that the performance of certain commonly used finite elements deteriorates asv0, a phenomenon calledlocking. We analyze this phenomenon and characterize the strength of the locking androbustness of varioush-version schemes using triangular and rectangular elements. We prove that thep-andh-p versions are free of locking with respect to the error in the energy norm. A generalization of our theory to the 3D problem is also discussed.The work of this author was supported in part by the Office of Naval Research under Naval Research Grant N00014-90-J-1030The work of this author was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, U.S. Air Force, under grant AFOSR 89-0252     

The ergodic theorem for Markov processes

Most of the material in Sections 4-5-6-8-11 has been published in [4]-[10]. We shall deal with the asymptotical behavior of the iterates of a Markov transition function. Our aim is to generalize the results about the ‘cyclic’ convergence of the iterates of a Markov matrix. Throughout the paper functional analytic methods are used and not probabilistic arguments. The report is self contained, modulo standart results from functional analysis, except for the decomposition into conservative and dissipative parts. Also we assume the existence of an invariant σ finite measure on the conservative part. This has been proved, under some restrictions, by several authors using probabilistic methods. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.     

On a cauchy problem with subharmonic initial values

Summary Investigation of the Euler-Poisson-Darboux equation for special initial values. Generalized transformation ofTricomi-Cibrario. Extension of classical theorems ofF. Riesz andP. Montel on mean values of subharmonic functions. This research was supported in part by the United States Air Force under Contract AF 18(600)573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

Lokalisation bei absoluter Cesàro-Summierbarkeit von Potenzreihen und trigonometrischen Reihen. I

Ohne ZusammenfassungThis research was supported in part (as far as the work ofW. Jurkat is concerned) by the United States Air Force, through the Office of Scientific Research of the Air Research and Development Command.     

On the carleman's model for the boltzmann equation and its generalizations

Summary T. Carleman [1] showed that a certain 2 x 2 hyperbolic, semilinear system of partial differential equations has many features in common with the integro-differential equation ofBoltzmann. We produce here a global existence theory for theCauchy problem for this system and its generalizations. The main result is derived by exploiting the fact that the solution of a certainRiccati equation depends isotonically (with respect to a suitable partial order) on the forcing term and the initial data. Research supported by the National Science Foundation, Grant NSF G-21053.     

Theorems in the additive theory of numbers

Summary This paper extends some earlier results on difference sets andB ₂ sequences bySinger, Bose, Erd?s andTuran, andChowla. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF 49 (638)-213. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Alcune limitazioni per le soluzioni di equazioni paraboliche

Sunto Si discute il problema diCauchy relativo all'equazione del calore, u_t=u_xx, ed a condizioni iniziali sull'asse t. Si provano inoltre alcune proprietà delle soluzioni positive di tale problema e di analoghi problemi diCauchy relativi a più generali equazioni paraboliche.

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Summary We consider theCauchy problem for the heat equation, u_t=u_xx, with initial conditions on the t-axis. We prove some property of positive solutions of this problem, and of similarCauchy problems for more general parabolic equations.

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A Giovanni Sansone nel suo 70^mo compleanno.     

Nörlund-Verfahren für Funktionen

Ohne ZusammenfassungDem Andenken anIssai Schur gewidmet.This research was supported by the United States Air Force through the Office of Scientific Research, of the Air Research and Development Command, under Contract Number AF 18 (600)–691.     

Su un problema esterno per la equazione delle onde

Sunto Si considera il problema diCauchy per la equazione u_tt = u_xx + u_yy con dati iniziali sui due piani caratteristici t = ±y. Si dà un teorema di esistenza ed unicità della soluzione nella ?regione esterna? |t| ≤ |y|, e si prova che la soluzione non dipende in modo continuo dai dati.

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Summary We consider theCauchy problem for the wave equation u_tt = u_xx + u_yy with initial data on the characteristic planes t = ±y. We discuss the problem related to the ?exterior region? |t| ≤ |y|; we prove uniqueness of the solution, its instability with respect to the data, and the existence of the solution if the data are analytic functions of x (this hypothesis is essential, as we prove with an example).

     

Spherical means in spaces of constant curvature

Summary Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations. To Enrico Bompiani on his scientific Jubiles This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under ontract No. AF 49 (638)–228.     

Über das Riemann-Privalovsche Randwertproblem

Ohne ZusammenfassungHerrn Professor Dr.H. Petersson zum 60. Geburtstag gewidmetThis research was supported by the Air Force Office of Scientific Research.     

On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes

We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.This research was supported in part by the Advanced Technology Program of the State of Texas, in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, and in part by the Air Force Office of Scientific Research (AFSC) under Contract F49620-89-C-0044.     

Some singular cauchy problems

Summary Existence and uniqueness theorems for some generalizedEuler-Poisson-Darboux equations are proved and growth and convexity properties of the solutions are studied for multiply subharmonic initial values. This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49 (638)-228, at the University of Maryland, and in part by the National Science Foundation, under a fellowship grant.     

Die geometrische Realisierung eines schiefen kartesischen Produktes

Ohne ZusammenfassungDiese Arbeit wurde zum Teil durch Unterstützung durch die United States Air Force ermöglicht (Contract No. AF 18(600)-1383 vom AF Office of Scientific Research, Air Research and Development Command).Herrn Professor Dr.Hellmuth Kneser zum 60. Geburtstag gewidmet