On the strong maximum principle for degenerate parabolic equations

Variants of the strong maximum principle are established for subsolutions to degenerate parabolic equations for which the standard version of the strong maximum principle does not hold. The results are formulated for viscosity solutions.     

On the maximum principle for ultraparabolic equations

We prove the maximum principle and various modifications of it for one class of degeneration of parabolic equations.     

Concavity maximum principle for viscosity solutions of singular equations

We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.     

Discrete maximum principle for the finite element solution of linear non-stationary diffusion–reaction problems

The maximum principle is one of the basic characteristic properties of solutions of second order partial differential equations of parabolic (and elliptic) types. The preservation of this property for solutions of corresponding discretized problems is a very natural requirement in reliable and meaningful numerical modelling of various real-life phenomena (heat conduction, air pollution, etc.). In the present paper we analyse a full discretization of a quite general class of linear parabolic equations and present sufficient conditions for the validity of a discrete analogue of the maximum principle in the case when bilinear finite elements are used for discretization in space.     

On the refined maximum principle for degenerate elliptic and parabolic problems

We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems.     

On the maximum principle for doubly nonlinear parabolic equations

For a general class of doubly nonlinear parabolic equations, some versions of the maximum principle are established. They play an important role in studying the regularity of generalized solutions of such equations. In particular, the results obtained can be used in studying equations of the form $$\frac{{\partial u}}{{\partial t}} - div\{ |u|^l |\nabla u\} = 0, m > 1, l > 1 - m.$$ which have numerous applications in the mechanics of continuous media. Bibliography: 17 tiles.     

On the maximum principle for linear parabolic equations

We prove extensions of our previous estimates for linear elliptic equations with inhomogeneous terms in L ^p spaces, p ≤ n to linear parabolic equations with inhomogeneous terms in L ^p , p ≤ n + 1. As with the elliptic case, our results depend on restrictions on parabolicity determined by certain subcones of the positive cone . They also extend the maximum principle of Krylov for the case p = n + 1, corresponding to the usual parabolicity.     

A maximum principle for combinatorial Yamabe flow

This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operator is introduced as an operator which satisfies the maximum principle, but may not be parabolic in the usual sense of operators on graphs. A maximum principle is derived for the curvature of combinatorial Yamabe flow under certain assumptions on the triangulation, and hence the heat operator is shown to be parabolic-like. The maximum principle then allows a characterization of the curvature as well was a proof of long term existence of the flow.     

Interior schauder estimates for parabolic differential- (or difference-) equations via the maximum principle

A-priori pointwise estimates to difference-quotients of solutions to elliptic or parabolic equations can be obtained by using the maximum property of appropriate higher-dimensional operators. This method, introduced by Brandt, is here used for a simple derivation of the interior Schauder estimates for second-order parabolic differential equations. The same derivation is applicable also for the analogous finite-difference equations.     

On the maximum principle and its application to diffusion equations

In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi‐discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h²) and O(h⁴) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h²) uniformly convergent discrete scheme or by O(h⁴) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Maximum principle for quasi-linear backward stochastic partial differential equations

In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDEs with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.     

A maximum principle for fourth order ordinary differential equations

We present a maximum principle for fourth order ordinary differential equations, based on a new approach involving counting of inflection points. We use our results to compute solutions of nonlinear equations describing static displacements of a uniform beam     

A maximum principle for fourth order ordinary differential equations

A maximum principle is obtained for solutions of fourth order ordinary differential equations. As an application, the existence of a nonnegative (positive) solution of a nonlinear boundary problem is established. Extension of the maximum principle for solutions of higher order equations is also indicated.     

The strong maximum principle revisited

In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.

(i)

two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);

(ii)

the exterior Dirichlet boundary value problem (Section 5);

(iii)

the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);

(iv)

Euler-Lagrange inequalities on a Riemannian manifold (Section 9);

(v)

comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

The case of p-regular elliptic inequalities is briefly considered in Section 11.     

Lower inequalities of heat semigroups by using parabolic maximum principle

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly ρ-local Dirichle form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.     

An extension of the Kaskosz maximum principle

We strengthen the conventional maximum principle for the optimal control of nonsmooth differential equations with nonsmooth unilateral constraints. This strengthened principle applies, in particular, to any admissible relaxed trajectory whose endpoint lies on the boundary of the attainable set generated by unrelaxed admissible trajectories. In this new principle the generalized Jacobian of the right-hand side can be replaced by the generalized Jacobian of any compatible selectionh(t, x) of the convexified right-hand side that is Lipschitzian inx. This extends a recent result of Barbara Kaskosz that applies to problems without unilateral constraints and with the functionh restricted to a certain form. We also show how our arguments extend to unilateral problems defined by functional-integral equations (and, in particular, delay-differential equations).This work was partially supported by the National Science Foundation under Grant DMS 8619002.     

A new maximum principle of elliptic differential equations in divergence form

In this paper will be presented a new maximum principle of elliptic differential equations in divergence form which can be regarded as the counterpart of the Alexandroff-Bakelman-Pucci maximum principle of elliptic differential equations in nondivergence form.

     

A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations

We formulate and prove a non-local “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integro-partial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain compare/uniqueness results for integro-partial differential equations, but proofs have so far been lacking.     

A maximum principle for an optimal control problem with integral constraints

In this paper, necessary corditions are obtained for an optimal control problem whose state variables are given in terms of integral equations. The conditions are obtained separately for Volterra equations and Fredholm equations. The main result for each case is the maximum principle and multiplier rule. For the Volterra equations, transversality conditions are obtained.     

An extension to Banach space of Pontryagin's maximum principle

An extension of Pontryagin's maximum principle to the case where the state space is infinite dimensional is given. The control process is governed by ordinary nonlinear differential equations. A property of control processes, which is analogous to well-known, nonlinear interior mapping theorems, makes up the basis for the proofs.