Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems

We consider systems of partial differential equations of the first order int and of order 2s in thex variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid inR ⁿ×(0,T] fors≥2. For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle, to hold in the layerR ⁿ×(0,T] and in the cylinder μ×(0,T], where μ is a bounded subdomain ofR ⁿ. If the coefficients of the system do not depend ont, these conditions coincide. The necessary and sufficient condition in this case is that the principal part of the system is scalar and that the coefficients of the system satisfy a certain algebraic inequality. We show by an example that the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both onx andt. The research of the first author was supported by the Ministry of Absorption, State of Israel.     

Oscillations of higher order differential equations of neutral type

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of nth order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.     

Solving an optimal control problem for parabolic equations

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 90–98, 1986.     

On the evolution equation for magnetic geodesics

Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations (PDEs) whose solutions describe the orbits of closed p-branes under the effect of a “generalized physical force”. For the corresponding evolution equation, which is a system of parabolic nonlinear PDEs associated to the elliptic PDE, we can establish existence of short time solutions.     

A two-dimensional minimum-derivative spline

We consider the construction of a C ^(1,1) interpolation parabolic spline function of two variables on a uniform rectangular grid, i.e., a function continuous in a given region together with its first partial derivatives which on every partial grid rectangle is a polynomial of second degree in x and second degree in y. The spline function is constructed as a minimum-derivative one-dimensional quadratic spline in one of the variables, and the spline coefficients themselves are minimum-derivative quadratic spline functions of the other variable.     

Teoremi di esistenza e di unicitá per le soluzioni deboli dei problemi al contorno per una classe di sistemi parabolici di equazioni del secondo ordine

In this paper we give some existence and uniqueness theorems for weak solutions of boundary value problems for a particular class of parabolic systems of linear partial differential equations of the second order with real coefficients. In particular some uniqueness theorems for equations with complex coefficients are deduced from the previous results.     

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

In this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with L^p ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Orlicz regularity for higher order parabolic equations in divergence form with coefficients in weak BMO

We consider higher order parabolic equations in divergence form with measurable coefficients to find optimal regularity in Orlicz spaces of the maximum order derivatives of the weak solutions. The relevant minimal regularity requirement on the tensor matrix coefficients is of small BMO in the spatial variable and is measurable in the time variable. As a consequence we prove the classical W ^m,p regularity, m = 1, 2, . . . , 1 < p < ∞, for such higher order equations. In the same spirit the results easily extend to higher order parabolic systems as well as up to the boundary.     

Nonautonomous Parabolic Equations Involving Measures

In the first part of this paper, we study abstract parabolic evolution equations involving Banach space-valued measures. These results are applied in the second part to second-order parabolic systems under minimal regularity hypotheses on the coefficients. Bibliography: 16 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 16–52.     

Controllability of systems of Stokes equations with one control force: existence of insensitizing controls

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L² norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when the coupling terms concern first and second order derivatives, respectively), we prove this result for some general couplings.     

Parabolic Ito equations and second fundamental inequality

The paper studies stochastic partial differential equations of parabolic type with Dirichlet boundary conditions. Solvability, uniqueness, and a priori estimates similar to the second fundamental inequality are obtained for bounded and unbounded domains. For the case of discontinuous coefficients, some Cordes type conditions that ensure solvability are suggested.     

Homogenization of periodic semilinear parabolic degenerate PDEs

In this paper a second order semilinear parabolic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some part of Rd${\mathbf{R}}^d$.     

On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions

Summary In this paper we consider the question of existence of optimal controls for a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions and with controls appearing in the coefficients. In Theorems2.2 and2.3 we present existence and uniqueness of solutions of the first boundary problems. In Theorems3.1 and3.2 we prove that whenever the coefficients of the system converge in the w*-topology (L¹ topology on L^∞) the corresponding solutions converge weakly in an appropriate Sobolev space. Using these basic results we present two theorems (Theorems4.1 and4.2) on the existence of optimal controls. Entrata in Redazione il 21 gennaio 1978.     

Necessary conditions for optimality of cauchy problems for parabolic partial delay-differential equations

In this paper, we consider the question concerning the necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations indivergence form with Cauchy conditions. All the coefficients of the system are assumed measurable and contain controls and delays in their arguments. An integral maximum principle and its pointwise version for the corresponding controlled system are given.The authors wish to thank Dr. E. Noussair for his valuable discussions in the preparation of this paper.     

Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions

In this paper, we consider the question of necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain controls and delays in their arguments. The second-order parabolic partial delay-differential equation is in divergence form. In Theorem 4.1, we present results on the existence and uniqueness of weak solutions in the sense of Ladyzhenskaya-Solonnikov-Ural'ceva for this class of systems. An integral maximum principle and its point-wise version for the corresponding controlled system are established in Theorem 5.1 and Corollary 5.1, respectively.The authors wish to thank Dr. E. Noussair for his stimulating discussion and valuable comments in the preparation of this paper. Further, they also wish to acknowledge the referee of the paper for his valuable suggestions and comments. The discussion presented in Section 6 is in response to his suggestions.     

On the existence theorem for the barotronic motion of a compressible inviscid fluid in the half-space

Summary We give here an existence theorem, see Theorem 1.1,for the solution of the system of equations (1.1)that describes the motion of a compressible inviscid fluid in the half space. Moreover, we establish some sharp estimates, see Theorem 3.2,for the solution of the linear second order hyperbolic mixed problem (3.1)in terms of suitable norms of the coefficients. These estimates play a main rule here and in reference [BV3],where a first proof of Hadamard's classical well-posedness for the above nonlinear system of equations is given; see also [BV4].Here, we adapt and simplify the method followed in our previous paper [BV1].     

On doubling properties for non-negative weak solutions of elliptic and parabolic PDE

In this paper we study quantitative properties of non-negative (super) solutions for elliptic and parabolic partial differential equations of second order with strongly singular coefficients, in which we cannot expect Harnack's inequality in general. We show the doubling property ofu ^δ with some small exponent 1>δ>0 for non-negative weak supersolutionsu. Furthermore, we show the doubling property ofu ^q with large exponent 2(n+2)/n≥q>0 for non-negative weak solutionsu of parabolic equations.     

Local Solutions of Weakly Parabolic Quasilinear Differential Equations

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we lire able to prove L∞ - estimates.     

Second‐order elliptic and parabolic equations with BMO coefficients in the whole space

In this paper, we obtain the global regularity estimates in Orlicz spaces for second‐order divergence elliptic and parabolic equations with BMO coefficients in the whole space. In fact, the global result can follow from the local estimates. As a corollary we obtain L^p‐type regularity estimates for such equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Second order parabolic systems, optimal regularity, and singular sets of solutions

We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form

u_t−divA(x,t,u,Du)=0.