Representation Formula For Solutions Research Articles

Classification of global behavior of a system of rational difference equations

This paper deals with a system of rational difference equations xn+1=ayn+bcyn+d,yn+1=axn+bcxn+d,n=0,1,2,…,where a, b, c, d are real numbers with c≠0 and ad−bc≠0. We establish a representation formula of solutions of the system and classify global behavior of solutions when no initial values belong to the forbidden set of the system.

On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth "on each side" of the initial value hypersurface. A uniqueness result in Sobolev regularity $H^{1/2+\varepsilon}_\mathrm{loc}$ is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.

A Feynman–Kac formula for stochastic Dirichlet problems

A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.

Boundary value problems of statics of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with boundary value problems of statics of the thermoelasticity theory of isotropic microstretch materials with microtemperatures and microdilatations. For the system of differential equations of equilibrium the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

Mode stability on the real axis

A generalization of the mode stability result of Whiting [J. Math. Phys. 30, 1301–1305 (1989)] for the Teukolsky equation is proved for the case of real frequencies. The main result of the paper states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation Rhor,Rout, which are purely ingoing at the horizon and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogeneous Teukolsky equation and was recently used by Shlapentokh-Rothman [Ann. Henri Poincaré 16, 289–345 (2015)] for the scalar wave equation.

Mathematical problems of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with the linear theory of thermoelasticity for elastic isotropic microstretch materials with microtemperatures and microdilatations. For the differential equations of pseudo-oscillations the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

The vanishing discount problem and viscosity Mather measures. Part 2: Boundary value problems

In [17] (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems.

Heat equation in vector bundles with time-dependent metric

We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well. As a corollary we obtain a vanishing theorem for bounded ancient solutions under a curvature condition. Our results apply in particular to the case of differential forms.

Green’s Function for a Slice of the Korányi Ball in the Heisenberg GroupHn

We give a representation formula for solution of the inhomogeneous Dirichlet problem on the upper half Korányi ball and for the slice of the Korányi ball in the Heisenberg groupHnby obtaining explicit expressions of Green-like kernel when the given data has certain radial symmetry.

A Stability Criterion for High-Frequency Oscillations

— We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. In the unstable case, the key observation is that resonances correspond to weakly hyperbolic frequencies; the amplification proof then relies on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.

Cauchy integrals for the p-Laplace equation in planar Lipschitz domains

We construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for solutions.

A posteriori error estimation for the Lax–Wendroff finite difference scheme

In many application domains, the preferred approaches to the numerical solution of hyperbolic partial differential equations such as conservation laws are formulated as finite difference schemes. While finite difference schemes are amenable to physical interpretation, one disadvantage of finite difference formulations is that it is relatively difficult to derive the so-called goal oriented a posteriori error estimates. A posteriori error estimates provide a computational approach to numerically compute accurate estimates in the error in specified quantities computed from a numerical solution. Widely used for finite element approximations, a posteriori error estimates yield substantial benefits in terms of quantifying reliability of numerical simulations and efficient adaptive error control.The chief difficulties in formulating a posteriori error estimates for finite difference schemes is introducing a variational formulation–and the associated adjoint problem–and a systematic definition of residual errors. In this paper, we approach this problem by first deriving an equivalency between a finite element method and the Lax–Wendroff finite volume method. We then obtain an adjoint based error representation formula for solutions obtained with this method. Results from linear and nonlinear viscous conservation laws are given.

Time Decay of Scaling Critical Electromagnetic Schrödinger Flows

We obtain a representation formula for solutions to Schrodinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp \({L^1 \to L^\infty}\) time decay estimate for the 3D-inverse square and the 2D-Aharonov–Bohm potentials.

Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations

For autonomous linear integral equations, we establish an explicit asymptotic representation formula of solutions, developing the spectral analysis for the generator of the solution semigroup as well as the one for the formal adjoint operator associated with the generator.

A sup × inf-type inequality for conformal metrics on Riemann surfaces with conical singularities

A sup × inf-type inequality is proved for the regular part of conformal factors for Riemannian metrics on surfaces with conical singularities of positive order α>0. The proof is based on the explicit representation formula for solutions of the singular Liouville equation and generalizes to the singular case an argument by I. Shafrir.

Representation formulas for solutions of Isaacs integro-PDE

We prove suband super-optimality inequalities of dynamic programming for viscosity solutions of Isaacs integro-PDE associated with two-player, zero-sum stochastic differential game driven by a Levy type noise. This implies that the lower and upper value functions of the game satisfy the dynamic programming principle and they are the unique viscosity solutions of the lower and upper Isaacs integro-PDE. We show how to regularize viscosity suband super-solutions of Isaacs equations to smooth suband super-solutions of slightly perturbed equations.

Blow-up of Solutions to the Generalized Inviscid Proudman–Johnson Equation

For arbitrary values of a parameter $\lambda\in R$, finite-time blow-up of solutions to the generalized, inviscid Proudman-Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.

Representation formula for stochastic Schrödinger evolution equations and applications

We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L2 or in the energy space of model equations arising in Bose–Einstein condensation, Abdullaev et al (2001 Nonlinearity and Disorder: Theory and Applications (NATO Science Series vol 45) ed F Abdullaev et al (Dodrecht: Kluwer)), or in fiber optics, Abdullaev et al (2000 Physica D 135 369–86). Our results also give a justification of diffusion-approximation for stochastic nonlinear Schrödinger equations.

A representation formula for the wave equation revisited

We give an elementary, rigorous and self-contained derivation of a Kirchhoff representation formula for solutions of the wave equation for a bounded region in three-dimensional space for the case of non-homogeneous initial conditions. A special feature of the result is the role of spherical caps which seems not to be noticed previously. The formula is readymade for boundary integral formulations of initial/boundary-value problems for the wave operator.

Formal adjoint equations and asymptotic formula for solutions of Volterra difference equations with infinite delay

For linear Volterra difference equations with infinite delay, we obtain an explicit asymptotic representation formula of solutions by means of the formal adjoint equation associated with a certain bilinear form.