Connections with prescribed first Pontrjagin form

Let be a principal bundle over a manifold of dimension . If , then we prove that every differential 4-form representing the first Pontrjagin class of is the Pontrjagin form of some connection on .     

Graphs with prescribed Levi form trace

Abstract We prove existence and uniqueness of a viscosity solution of the Dirichlet problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uniform limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers. Keywords: Levi mean curvature, Quasilinear degenerate elliptic PDE’s, Viscosity solutions, Comparison principle, Global Lipschitz estimates     

Inverse problem for the diffusion equation with overdetermination in the form of an external volume potential

An initial-boundary value problem for the diffusion equation with an unknown initial condition is considered. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplacian calculated for the solution of the initial-boundary value problem. Uniqueness theorems for the inverse problem are proved in the case when the spatial domain of the initial-boundary value problem is a spherical layer or a parallelepiped.     

An extremal case of the equation of prescribed Weingarten curvature

In this paper we investigate the existence and regularity of solutions to a Dirichlet problem for a Hessian quotient equation on the sphere. The equation in question arises as the determining equation for the support function of a convex surface which is required to meet a given enclosing cylinder tangentially and whose k-th Weingarten curvature is a given function. This is a generalization of a Gaussian curvature problem treated in [13]. Essentially given ${\Omega \subset \mathbb{R}^n}$ we seek a convex function u such that graph(u) has a prescribed k-th curvature ψ and |Du(x)| → ∞ as x → ?Ω. Under certain regularity assumptions on ψ and Ω we are able to demonstrate the existence of a solution whose graph is C ^3,α provided that ${\psi^{-\frac{1}{k}} = \psi^{-\frac{1}{k}}(x, \nu)}$ is convex in x and a certain compatibility condition between ψ| _?Ω and Ω is satisfied.     

Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem

     

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W^2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W^2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.     

Fundamental solutions to Pell equation with prescribed size

We prove that the number of parameters D up to a fixed x ≥ 2 such that the fundamental solution ? _D to the Pell equation T ² ? DU ² = 1 lies between \(D^{\tfrac{1}{2} + \alpha _1 }\) and \(D^{\tfrac{1}{2} + \alpha _2 }\) is greater than \(\sqrt x \log ^2 x\) up to a constant as long as α ₁ < α ₂ and α ₁ < 3/2. The starting point of the proof is a reduction step already used by the authors in earlier works. This approach is amenable to analytic methods. Along the same lines, and inspired by the work of Dirichlet, we show that the set of parameters D ≤ x for which log ? _D is larger than D ^¼ has a cardinality essentially larger than x ^¼ log² x.     

Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation

We propose a tensor structured preconditioner for the tensor train GMRES algorithm (or TT-GMRES for short) to approximate the solution of the all-at-once formulation of time-dependent fractional partial differential equations discretized in time by linear multistep formulas used in boundary value form and in space by finite volumes.Numerical experiments show that the proposed preconditioner is efficient for very large problems and is competitive, in particular with respect to the AMEn algorithm.     

Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem

depending on the behaviour at the origin and at infinity of the potential . Besides solutions in W^2,1(0,1), we also consider solutions in which are possibly discontinuous at the endpoints of [0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.     

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem

     

Symmetric form of the Hudson-Parthasarathy stochastic equation

We prove that the Hudson-Parthasarathy equation corresponds, up to unitary equivalence, to the strong resolvent limit of Schrödinger Hamiltonians in Fock space and that the symmetric form of this equation corresponds to the weak limit of the Schrödinger Hamiltonians.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 726–750, November, 1996.     

Remark on the anisotropic prescribed mean curvature equation on arbitrary domains

In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on ${x \in \varOmega \subset \mathbb {R}^n}In this article we consider the Dirichlet problem for hypersurfaces of aniso- tropic prescribed mean curvature H = H(x, u, N) depending on x ? \varOmega ì \mathbb Rⁿ{x \in \varOmega \subset \mathbb {R}^n}, the height u of the hypersurface M = graph u over \varOmega{\varOmega} and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of ?\varOmega{\partial \varOmega} that guarantees the existence of smooth solutions even for not necessarily convex domains.     

Exact number of solutions of a prescribed mean curvature equation

We study the exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equation