Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk

In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order 2m with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases m=1, 2, 3 are studied separately. For the case m=2, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 498–508.Original Russian Text Copyright © 2005 by V. P. Burskii, E. A. Buryachenko.This revised version was published online in April 2005 with a corrected issue number.     

On periodic solutions for even order differential equations

A class of sufficient conditions are obtained for the existence of a unique 2π$2\pi$-periodic solution of even order differential equations, employing an initial value problem.     

On a class of partial differential equations of even order

Summary The general solution for a class of equations of even order is expressed as a sum of solutions of equations of second order. To Mauro Picone on his 70^th birthday. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

On the Dirichlet problem for first order partial differential equations. A Baire category approach

((without abstract.)) Received September 22, 1997     

Multipoint problem for typeless systems of differential equations with constant coefficients

For typeless systems of differential equations with constant coefficients, we investigate the well-posedness of the problem with multipoint conditions for a selected variable and 2π-periodic conditions for the other coordinates. The conditions of univalent solvability are established and the metric theorems are proved for lower bounds of small denominators that appear in the construction of solutions of the problems.     

Oscillation criteria for even order differential equations

Summary Three theorems concerning oscillatory properties of solutions of even order ordinary differential equations are given. One theorem gives sufficient conditions for all solutions of (*) y⁽²ⁿ⁾(x) + f(y(x))p(x) = 0 to possess an infinite number of zeroes on the positive x axis. Another theorem gives sufficient conditions for all solutions of the general homogeneous linear equation of order 2n to have the same oscillatory property. Finally a necessary condition is given for (*) to have a solution with a last zero. This research was carried out under the support of Grant NSF GP-2067 with the National Science Foundation.     

A uniqueness theorem for solution of one Dirichlet problem

We consider the Dirichlet problem in a four-dimensional domain formed by characteristic surfaces of an equation of the 8th order with the double major partial derivative. We state sufficient conditions for the unique solvability of this problem in terms of control coefficients, based on the method of a priori estimates.     

Comparison theorems for even order half-linear differential equations

A generalization of Picone’s formula to the case of half-linear differential operators of the even order is given and comparison results concerning the associated differential equations are established with the help of this formula.     

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

Violation of unique solvability of the Dirichlet problem on the disk for systems of second-order differential equations

We study the Dirichlet problem for the system of elliptic equations with matrix complex-valued coefficients

On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

We investigate the structure of solutions of an equation y″(t) = By(t), where B is a weakly positive operator in a Banach space , on the interval (0, ∞) and establish the existence of their limit values as t → 0 in a broader locally convex space containing as a dense set. The analyticity of these solutions on (0, ∞) is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1462–1476, November, 2006.     

Dirichlet problem for complex model partial differential equations

ABSTRACT

Complex model partial differential equations of arbitrary order are considered. The uniqueness of the Dirichlet problem is studied. It is proved that the Dirichlet problem for higher order complex partial differential equations with one complex variable has infinitely many solutions.