Differential Equations In Divergence Form Research Articles

Necessary optimality conditions for systems described by nonlinear elliptic equations. I (in Russian)

This paper considers optimal control problems for systems governed by second order elliptic differential equations in divergence form with nonlinear terms and control appear in all terms. The representation of the main part of increment for a solution of the state equation is given in the case of needle-like variations.

Existence of optimal controls for systems governed by parabolic partial differential equations with Cauchy boundary conditions

This paper considers the existence of optimal controls for systems governed by a second order parabolic partial differential equation in divergence form with Cauchy conditions. As preliminary results, theorems concerning the convergence of the sequence of weak solutions corresponding to a sequence of admissible controls are proved. Two general forms of criteria are considered. The first one is taken as a function of the weak solution of the system, and the other is taken as a function of the solution of the system and control. Several theorems and corollaries on the existence of optimal controls are then presented.

Regularit�tsuntersuchungen von L�sungen Elliptischer Systeme von Quasilinearen Differentialgleichungen Zweiter Ordnung

Solutions of general, uniformly elliptic systems of quasilinear second order partial differential equations in divergence form are studied under the assumption that the right hand side of the system grows at most quadratically with respect to the gradient of the solution. A partial regularity result is obtained, asserting that in the general case, any weak solution with sufficiently small modulus (an explicit bound is given) has holder continuous first order derivatives in a neighbourhood of almost every point of the domain of definition. For diagonal systems where the coefficients depend on the modulus, but in no other way, of the gradient of the unknown function, it is shown that regularity in fact holds throughout the domain.

Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form

Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.

Principio di massimo generalizzato e valutazione del primo autovalore per problemi ellittici del secondo ordine di tipo variazionale

I prove a generalized maximum principle and an evaluation of the first eingenvalue for linear second order elliptic partial differential equations in divergence form with discontinuous coefficients.