Related papers: An analytical solution of Monge differential equat…
We investigate global H\"older gradient estimates for solutions to the Monge-Amp\`ere equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main…
Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries)…
The objective of this study is to ascertain the existence and forms of the finite order meromorphic and entire functions of several complex variables satisfying some certain Fermat-type partial differential-difference equations by…
The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = \lambda A(x)\mathbf{y}, \quad \ x \in [0, 1], $$ where $A(x) = diag\{a_1(x),…
We prove the existence of entire solutions of the Monge-Amp\`ere equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due…
The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space $M$, they are at most of the 2nd order.…
We leverage path differentiability and a recent result on nonsmooth implicit differentiation calculus to give sufficient conditions ensuring that the solution to a monotone inclusion problem will be path differentiable, with formulas for…
The aim of this paper is twofold. On one hand, the additive solvability of the system of functional equations \[d_{k}(xy)=\sum_{i=0}^{k}\Gamma(i,k-i) d_{i}(x)d_{k-i}(y) \qquad (x,y\in \R,\,k\in\{0,\ldots,n\}) \] is studied, where…
This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of…
By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Amp\`{e}re equation $\det D^2 u = 1$.
The elliptic Monge-Amp\`ere equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…
We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point.…
A sharp condition is provided to guarantee that the (nontrivial) solutions of a DDE of the form $\dot{x}(t)+F(t,x)=0$ $t\geq 0,$ (where $F(t,\cdot)$ is an odd-like causal operator) either oscillate, or converge monotonically to zero. The…
We begin by considering second order dynamical systems of the from $\ddot x(t) + \gamma(t)\dot x(t) + \lambda(t)B(x(t))=0$, where $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space ${\cal H}$,…
We introduce a non local analog to the Monge-Ampere operator and show some of its properties. We prove that a global problem involving this operator has C 1,1 solutions in the full space.
This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a…
The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the…
Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by…
The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving…
In this paper, we investigate the similarity solutions for a steady laminar incompressible boundary layer equations governing the Magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies.…