Class Of Fourth-order Equations Research Articles

Interior estimates for the Monge–Ampère type fourth order equations

In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial Legendre transform. As an application, we obtain a new proof of the Bernstein theorem without using Caffarelli-Guti\'errez's estimate, including the Chern conjecture on affine maximal surfaces. For the inhomogeneous equation, we also obtain a new proof in dimension two by an integral method relying on the Monge-Amp\`ere Sobolev inequality. This proof works even when the right hand side is singular. In higher dimensions, we obtain the interior regularity in terms of integral bounds on the second derivatives and the inverse of the determinant.

Second order splitting for a class of fourth order equations

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems. In this way we develop an approach to a class of fourth order elliptic partial differential equations using the idea of splitting into coupled second order equations. Our main motivation is to treat certain fourth order equations on closed surfaces arising in the modelling of biomembranes but the approach may be applied more generally. In particular we are interested in equations with non-smooth right-hand sides and operators which have non-trivial kernels. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g) be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.

An obstacle problem for a class of Monge–Ampère type functionals

In this paper we study an obstacle problem for Monge–Ampère type functionals, whose Euler–Lagrange equations are a class of fourth order equations, including the affine maximal surface equation and Abreuʼs equation.

Periodic and homoclinic solutions of a class of fourth order equations

Periodic and homoclinic solutions of a class of fourth order equations

The Bernstein theorem for a class of fourth order equations

In this paper, we prove the 2-dimensional Bernstein theorem for a class of fourth order equations including Abreu’s equation. The main ingredients of the paper are the a priori estimates and the proof of the strict convexity.

Global uniqueness of homoclinic orbits for a class of fourth order equations

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation $$\alpha x\prime \prime \prime \prime + \beta x\prime \prime + yx + k(x) = 0,x > 0,$$ whenα, γ>0>β,dk/dx 0 andK(0)=0. The existence problem is approached via the general theory of [1] and uniqueness follows from the Maximum Principle and some geometrical observations about the role of convexity. There are no small amplitude assumptions.

Homogenisation of a class of fourth order equations with application to incompressible elasticity

SynopsisUpon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.