Hopf Maximum Research Articles

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.

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.

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.