Hopf Maximum Principle Research Articles

Diameter estimate for a class of compact generalized quasi-Einstein manifolds

In this paper, we discuss the lower diameter estimate for a class of compact generalized quasi-Einstein manifolds which are closely related to the conformal geometry. Using the Bochner formula and the Hopf maximum principle, we get a gradient estimate for the potential function of the manifold. Based on the gradient estimate, we get the lower diameter estimate for this class of generalized quasi-Einstein manifolds.

Application of the Hopf Maximum Principle to the Theory of Geodesic Mappings

In the present paper we consider some applications the Hopf maximum principle and its generalization to the classical theory of geodesic mappings. As a result, a series of classical theorems on geodesic mappings become consequences of our statements which we shall prove in the present paper.

Omori-Yau maximum principles and their applications

In the geometry and analysis on compact Riemannian manifolds, the Hopf maximum principle is a very useful tool. The Omori-Yau maximum principle is an important,basic and powerful tool on noncompact Riemannian manifolds corresponding to the Hopf maximum principle in the compact case.In this paper, we give a survey on the classical Omori-Yau maximum principle and its various generalizations, as well as their applications in the geometry and analysis on manifolds.

Time asymptotics of structured populations with diffusion and dynamic boundary conditions

This work revisits and extends in various directions a work by J.Z. Farkas and P. Hinow (Math. Biosc and Eng, 8 (2011) 503-513) on structured populations models (with bounded sizes) with diffusion and generalized Wentzell boundary conditions. In particular, we provide first a self-contained $L^{1}$ generation theory making explicit the domain of the generator. By using Hopf maximum principle, we show that the semigroup is always irreducible regardless of the reproduction function. By using weak compactness arguments, we show first a stability result of the essential type and then deduce that the semigroup has a spectral gap and consequently the asynchronous exponential growth property. Finally, we show how to extend this theory to models with arbitrary sizes and point out an open problem pertaining to this extension.

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p-Laplacian, p>1, on bounded cylindrical domains of RN+1,∂tu−Δpu−λ|u|p−2u≥0, with nonnegative initial–boundary conditions and λ≤0, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2, and give a counterexample to it for p>2. Also the Weak Maximum Principle for λ≤λ1 is established.

Maximum Principles for Fourth Order Nonlinear Elliptic Equations with Applications

The paper is devoted to prove maximum principles for a certain func- tionals defined for solution of the fourth order nonlinear elliptic equation. The max- imum principle so obtained is used to prove the nonexistence of nontrivial solutions of the fourth order nonlinear elliptic equation with some zero boundary conditions. Hopf's maximum principle is the main ingredient.

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.

Blow-up of Solutions for a Class of Nonlinear Parabolic Equations

In this paper, the blow up of solutions for a class of nonlinear parabolic equations u_t(x,t)=\nabla _{x}(a(u(x,t))b(x)c(t)\nabla _{x}u(x,t))+g(x,|\nabla _{x}u(x,t) |^2,t)f(u(x,t)) with mixed boundary conditions is studied. By constructing an auxiliary function and using Hopf's maximum principles, an existence theorem of blow-up solutions, upper bound of “blow-up time” and upper estimates of “blow-up rate” are given under suitable assumptions on a, b,c, f, g , initial data and suitable mixed boundary conditions. The obtained result is illustrated through an example in which a, b,c, f, g are power functions or exponential functions.

Subsonic solutions to compressible transonic potential problems for isothermal self-similar flows and steady flows

We establish boundary and interior gradient estimates, and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self-similar isothermal and steady problems. We establish an existence result for the self-similar isothermal problem, and improve the Hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem.

Maximum principles for a class of nonlinear second-order elliptic boundary value problems in divergence form

For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of and , where are the solutions of our problems. From these inequalities, we derive, using Hopf's maximum principles, some maximum principles for the appropriate combinations of and , and we list a few examples of problems to which these maximum principles may be applied.

Blow-up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions

The type of problem under consideration is u t = ∇ ⋅ ( a ( u ) b ( x ) ∇ u ) + g ( x , t ) f ( u ) , i n D × ( 0 , T ) , ∂ u ∂ n + σ ( x , t ) u = 0 , o n ∂ D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 , i n D _ where D is a smooth bounded domain of R N . By constructing an auxiliary function and using Hopf's maximum principles on it, existence theorems of blow-up solutions, upper bound of ‘blow-up time”, upper estimates of “blow-up rate”, existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a, b, f, g, σ, and initial data u 0( x). The obtained results are applied to some examples in which a, b, f, g, and σ are power functions or exponential functions.

Blow-up solutions for a class of nonlinear parabolic equations with Dirichlet boundary conditions

In the present paper, the blow up of smooth local solutions for a class of nonlinear parabolic equations u, t= ∇(a(u) ∇u)+f(x,u,q,t) (q=| ∇u| 2) with Dirichlet boundary conditions are studied. By constructing an auxiliary function and using Hopf's maximum principles on it, the sufficient conditions for blow-up solutions are obtained and the upper bound of “blow-up time” is given under some suitable assumptions on a, f and initial date. The obtained results are applied to some examples in which a and f are exponential functions.

Blow-up of solutions for a class of semilinear reaction diffusion equations with mixed boundary conditions

The Hopf's maximum principles are utilized to deal with the problem on blow-up of the solutions for a class of semilinear reaction diffusion equations u, t = Δ u + f( x, u, q, t) ( q = |▽ u| 2), subject to mixed boundary conditions. Some nonexistence theorems of global solutions and the bounds of “blow-up time” are obtained.

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R n . We establish such a maximum principle for parabolic surfaces in R3 with nonzero constant mean curvature and bounded Gaussian curvature.

Errata to "Hypersurfaces with Constant Mean Curvature in the Complex Hyperbolic Space"

Unfortunately, there is a mistake in the proof of Theorem 3.3 of our paper entitled Hypersurfaces with constant mean curvature in the complex space which appeared in the Transactions of the AMS (Vol. 339 (1993), 685-702). In the proof of this theorem we applied Hopf's maximum principle, which holds for a class of hypersurfaces satisfying an elliptic PDE, to a one-parameter family of hypersurfaces with constant mean curvature (cmc) of Q5 (using the notations of the paper) obtained by reflecting an initial cmc hypersurface of Q5 on a one-parameter family of totally geodesic hypersurfaces of Q5. However, while we know that the initial hypersurface satisfies an elliptic PDE since it is invariant by the S' group of isometries of Q5 and has cmc, the reflections of the hypersurface do not satisfy the equation since, although they have cmc, they are not S1 invariant any more. Therefore, Hopf's maximum principle cannot be used and the proof, as it stands in the paper, is not correct. This mistake was pointed out to us by Professor J. Eschenburg. So far, we have not found a way to correct it and, in fact, this seems to be a difficult question.

The maximum principle at infinity for minimal surfaces in flat three manifolds

Maximum principles are used as basic analytic tools for studying properties of functions defined on domains in R n and satisfying certain equations (e.g. elliptic). In general these maximum principles play a fundamental role in analysis on complete Riemannian manifolds, especially in the study of variational problems. For example, the well-known maximum principle for harmonic functions has had both a simplifying and unifying effect on the fields of harmonic and complex analysis. H. Hopf [18] gave an important general maximum principle for second order linear elliptic partial differential equations. The Hopf maximum principle easily yields a maximum principle for solutions of the minimal surface equation. In this context the principle states that if D c R 2 is a smooth connected domain and f~ ,f2 are two smooth functions on D that satisfy the minimal surface equation, then the difference fl -f2 cannot have an interior maximum or minimum unless the difference is constant. The maximum principle for minimal graphs gives rise to the following geometric result for minimal surfaces in Riemannian three-manifolds: IfM~ and ME are minimal surfaces in a Riemannian three-manifold that intersect at a common interior point p and M1 is on one side of M2 near p, then M~ intersects M2 in an open surface containing

A maximum principle for a class of generalized analytic functions

Let w be a generalized analytic function, that is. w satisfies the equation Hopf's maximum principle is used to derive sufficient conditions on the coefficients A and B such that any solution w of (∗) satisfies the maximum principle of the form for all = ∊Ğ, where G is a bounded domain of the complex plane C. Furthermore a new factorization theorem is proved which asserts that any solution w of (∗) can be factorized in the form w=W exp ω, where W is a solution of an equation of the same type (∗) with coefficients A, [Btilde] and ω is continuous in Ğ. This factorization theorem allows to improve the constant C in the estimate |w(z)|≤Csupζ∊∂G|w(ζ)| obtained by means of the factorization by analytic functions.

The strong maximum principle for the heat equation

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.

Gradient bounds for plasma confinement

AbstractThe problem of plasma confinement in a Tokomak machine has recently attracted considerable mathematical attention. In the present article, gradient bounds for the solution are obtained through the use of differential inequalities and the Hopf maximum principle. These estimates, which take slightly different forms in the plasma and vacuum regions, become exact in the one‐dimensional case. An estimate is also given for the distance from the boundary of the interior point where the maximum of the solution is attained.

Some maximum principles for nonlinear elliptic boundary-value problems

The Hopf maximum principles are utilized to obtain maximum principles for functions which are defined on solutions of nonlinear, second-order elliptic equations subject to Dirichlet, Robin, or mixed boundary conditions. The principles derived may be used to deduce bounds on important quantities in physical problems of interest.