Geometry in Partial Differential Equations

The following sections are included:

• PREFACE

• CONTENTS

In this article we give two applications of the coarea formula of Federer and Fleming to problems in P.D.E's. The first question we address concerns a functional inequality involving a Jacobian (know as Wente's estimate) and the use of the coarea formula allows us to show that the constant appearing in this inequality can be chosen independently of the domain. In the second part we address a totaly different question which is related to the stationary Euler equations for inviscid incompressible fluids. We propose a new proof of the fact that, in some circonstances, weak limits of approximate solutions are still weak solutions. This result is then applied to an existence result for weak solutions to a damped Euler equation.

In this paper, we present recent developments concerning the multiplicity of solutions to the equation of surfaces of prescribed mean curvature.

Optical Hamiltonian functions can be defined (at least locally) on a symplectic manifold endowed with a Lagrangian distribution. They can be considered as a far going generalization of fiberwise convex Hamiltonians on a cotangent bundle. It turns out that critical points of these functions have very specific properties which lead to global restrictions on the topology of underlying symplectic manifolds. We apply these results to a geometrical study of the group of Hamiltonian diffeomorphisms.

The following sections are included:

• Introduction

• The Hodge–De Rham Laplace Operators

• Explicit Examples of p-Spectra

• Asymptotic Expansions

• p-Isospectral Closed Riemannian Manifolds

• p-Spectra and Geometry

• References

A minimal surface is reflected in a constant elasticity of substitution (CES) function used in economic theory and in specifying extremum conditions. The paper gives an account of some of the main developments connected with minimal surfaces in economic theory.

This work is devoted to solving the problem of finding particular solutions of one non-linear partial differential equation of second order. This problem was posed by T. Morimoto. First, the symmetry algebra of this equation is found. The work also gives the classification (up to the group of automorphisms) of its one-dimensional subalgebras and their normalizers, which are then used for finding particular solutions of the equation. The methods employed in this work can also be applied to quadrature of other partial differential equations.

Let M be a compact Riemannian manifold with a smooth boundary. We examine various asymptotic expansions arising in the heat equation trace and the heat content in spectral geometry. MOS classification number 58G25.

In the differential geometric framework an exposition of Richardson-Nijenhuis and Frőlicher-Nijenhuis brackets is given. The theory of deformations in exposed and deformations of a number of classical integrable equations are constructed leading to recursion operators for these equations.

We first construct, from the point of view of the modified Kadomtsev–Petviashvili (mKP) hierarchy, the mKP equation together with its map into the KP one, and a deformed KP equation together with its contraction onto the KP equation itself. Similar formulae are deduced for the Zabolotskaya–Khokhlov (ZK) equation. The rest of the paper addresses the problem of Hamiltonian forms for the constructed equations. The algebraic approach fails here, and we have to take up the geometric point of view by treating our equations as submanifolds in 4-jet bundles. In this new language, purely local Hamiltonian forms are found for the KP and ZK equations; first of these reduces to a novel Hamiltonian structure for the Boussinesq equation.

We present the classification of mixed type Monge-Ampere equations on a plane.

We present various algebraic and geometric applications of a spectral sequence approach to non-integrability.

We describe properties of the Spencer complexes and resolve the problem of duality for Spencer cohomologies. We propose construction of spectral sequence converging to Spencer cohomologies. By using symmetry groups of differential equations systems we manage to unify computations by reduction to the invariant systems over a homogeneous space. The conditions of coincidence of Spencer cohomologies with the cohomologies of invariant Spencer complex we obtain from arithmetic of Œ-characteristic manifold with respect to fundamental weights of the homogeneous space.

This is a new geometric study in the theory of partial differential equations: an application of Fourier integral operators to equations with distributional coefficients. As an example, a derivation of the existence and spectrum of Hawking's radiation from a collapsing star is given.

Superspaces and supermanifolds are introduced by using the concept of weak differentiability as usually given for locally convex spaces. This allows us to consider in algebraic way superdual spaces and superderivative spaces. In this way we obtain a good generalization of just known superstructures general enough to develop a formal theory for super PDE's that directly extends previous ones for standard manifolds of finite dimension. In particular, we give a Goldschmidt-type criterion of formal superintegrability for super PDE's, and show that a geometric theory of singular supersolutions, with singularities of Thom-Bordman type, can be formulated in the framework of super PDE's too. Conservation superlaws associated to super PDE's are considered and related with some spectral sequences and wholly cohomological character of these equations.

By using a geometric framework for PDE's we prove that the set of solutions of the J. D'Alembert equation

The geometric prequantization of the Einstein's vacuum field equations is described and its connection with the reduction under the symmetry group of all space time diffeomorphisms is pointed out.

We describe a method for constructing a normal projective connection associated with a second order ordinary differential equation. Our construction shows that differential equations generate Cartan's connections in the same way as Riemannian metrics generate Levi-Civita's connections. We use this consruction to give a classification, in terms of projective curvature, of differential equations admitting Lie groups of point symmetries.

The following sections are included:

• Condition of Geodesic Convexity

• Regular Finite-Dimensional Reductions

• Equivalence of Regular Reductions

• Intermediate Reduction

• Special One-dimensional Reduction for Even Functionals

• Reduction in the Problem of Equilibrium of Elastic Plate

• Elastically Strengthened Kirchhoff's Rod

• Bifurcation of Stationary Rotations of Multi-Dimensional Top

• References

A curious method of finding an approximate solution to the Schrödinger equation for an N-electron atom is described. The approximate solution turns out to be the sum of a thousand exact solutions of similar but different second order ordinary differential equations. The geometric concept of orthogonality plays a crucial role in this theory.

Generating functions are found for all local symmetries and conservation laws of Euler-Darboux equation 𝒴_n with the parameter n ∉ ℤ. It is proved that in the symmetries algebra the basis of subalgebra, which is the direct complement to the infinite-dimensional Abelian ideal consisting of smooth solutions of the equation 𝒴_n, is generated by three recursion operators from zero-order trivial symmetry. Explicit differential form is given for conservation laws corresponding to the generating functions found.

We review the relationship between string theory and theories of higher-dimensional extended objects. Considered in particular are the coupling to Yang-Mills background fields in the presence of gauge anomalies and the relationship between string and membrane k fermionic gauge symmetries.

The two-dimensional ideal magnetohydrodynamic (MHD) equations derived in the limit of small Alfven-Mach numbers are considered.

The magnetic field in this situation is generally described by complex analytical functions having several branch points. We discuss the methods for finding such functions from some boundary conditions.

If the evolution of magnetic field is found, the MHD flows may be described at small and large Mach numbers (hyposonic and hypersonic cases, respectively) by a system of ordinary differential equations.

In hypersonic case there exist the flows with frozen-in potential of magnetic field. They are determined by a linear diffusion-like equation. Two classes of solutions to this equation are found analytically.

The following sections are included:

• AUTHOR INDEX