Existence Theorem of Global Solutions for Monge-Ampere Equation

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The First Boundary Value Problem for General Parabolic Monge-Ampere Equation

In this note we consider the first boundary value problem for a general parabolic Monge-Ampere equation u_t - log det(D_{ij}u) = f(x, t, u,D_2u) in Q, \quad u = φ(x, t) on ∂, Q It is proved that there exists a unique convex in x solution to the problem from C^{1+β,2+β/2}(\overline{Q}) under certain structure aod smoothness conditions (H3) - (H7).     

Generalized Solution of the First Boundary Value Problem for Parabolic Monge-Ampere Equation

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).     

The Geometric Measure Theoretical Characterization of Viscosity Solutions to Parabolic Monge-Ampere Type Equation

By an involved approach a geometric measure theory is established for parabolic Monge-Ampère operator acting on convex-monotone functions, the theory bears complete analogy with Aleksandrov's classical ones for elliptic Monge-Ampère operator acting on convex functions. The identity of solutions in weak and viscosity sense to parabolic Monge-Ampère equation is proved. A general result on existence and uniqueness of weak solution to BVP for this equation is also obtained.     

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

We consider the non-homogeneous generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} -\nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.$$ Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to ${t \geq T_1}$ , where T ₁ depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for ${t \geq T_1}$ , and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.     

A Remark on Monge-Ampere Equations in Nonstrictly Convex Domains

In this paper, we discuss the regularity of the weak solutions and the existence of the classical solutions for Monge-Ampère equations on the bounded convex domains possessing uniform parabolic support. This paper improves the conclusion of [1].     

The Set of Measures Given by Bounded Solutions of the Complex Monge-Ampere Equation on Compact Kahler Manifolds

Consider the image of the Monge–Ampère operatoracting on bounded functions, defined on a compact Kählermanifold, whose sum with the local Kähler potential isplurisubharmonic. It is shown that a nonnegative Borel measurebelongs to this image if and only if it belongs to the imagelocally. In particular, those measures form a convex set.     

Two remarks on Monge-Ampere equations

Summary We consider real Monge-Ampère equations and we present two new properties of these equations. First, we show the existence of the «first eigenvalue of Monge-Ampère equation» i.e. we show the existence of a positive constant possessing all the properties of the first eigenvalue of a 2-nd order elliptic operator (positivity, uniqueness of the eigenfunction, maximum principle, bifurcation...).The second property concerns variational characterisations of solutions. Both properties are closely related to similar properties of the general class of Hamilton-Jacobi-Bellman equations.     

On Another Kind of Parabolic Monge－Ampere Equation：the Existence，Uniqueness and Regularity of the Vi

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A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

Variational problems connected with Monge-Ampere type operators

In this note we describe a geometric problem leading to Monge-Ampere type operators. Some variational problems and theorems on their solvability are formulated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 186–189, 1988.     

Sur les equations de Monge-Ampere. I

In this paper we study the real Monge-Arapère equations: det(D²u)= f(x) in 0, u convex in 0, u=0 on 0, and we introduce a new method for solving these equations which enables us to show the existence of regular solutions. This method uses only p.d.e. techniques and does not use any geometrical results. Furthermore, it enables us to solve quasilinear Monge-Ampère equations.     

Generalised elliptic functions

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ?-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study.The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given.The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.     

Generalised Fibonacci manifolds

Fibonacci manifolds have a hyperbolic structure which may be defined via Fibonacci numbers. Using related sequences of Lucas numbers, other 3-manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant trace-field examined.Supported by the Royal Society.     

A Classification of the Strongly Regular Generalised Johnson Graphs

The generalised Johnson graphs are the graphs J(n, k, m) whose vertices are the k subsets of {1, 2, . . . , n}, with two vertices J ₁ and J ₂ joined by an edge if and only if ${{|J_1 \cap J_2| = m}}$ . A graph is called d-regular if every vertex has exactly d edges incident to it. A d-regular graph on v vertices is called a (v, d, a, c)-strongly regular graph if every pair of adjacent vertices have exactly a common neighbours and every pair of non-adjacent vertices have exactly c common neighbours. The triangular graphs J(n, 2, 1), their complements J(n, 2, 0), the sporadic examples J(10, 3, 1) and J(7, 3, 1), as well as the trivially strongly regular graphs J(2k, k, 0) are examples of strongly regular generalised Johnson graphs. In this paper we prove that there are no other strongly regular generalised Johnson graphs.