Order Semilinear Partial Differential Equation Research Articles

On a novel approach for the investigation and approximation of solutions to the systems of higher order nonlinear PDEs

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.

REMARKS ON THE DERIVATION OF SEVERAL SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS FROM A GENERALIZATION OF THE EINSTEIN EQUATIONS

A generalization of the Einstein equations with the cosmological constant is considered for complex line elements. Several second order semilinear partial differential equations are derived from them as semilinear field equations in homogeneous and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The properties of spatial expansion and contraction are studied based on energy estimates of the field equations. Several dissipative and anti-dissipative properties are remarked.

On polar actions invariant solutions of semi-linear equations on manifolds

In this paper we put together some tools from differential geometry and analysis to study second order semi-linear partial differential equations on a Riemannian manifold M. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [10] we are able to write a (reduced) second order semi-linear problem on a submanifold Σ. This submanifold is, in a sense, transversal to the orbits of the group actions and its existence is assumed. We describe precise conditions on the Riemannian Manifold M and the submanifold Σ in order to be able to write the reduced equation on Σ. These conditions are satisfied by several particular cases including some examples treated separately in the literature such as the sphere, surfaces of revolution and others. Our framework also includes the setup of polar actions or exponential coordinates. Using this procedure, we are left with a second order semi-linear equation posed on a submanifold. In particular, if the submanifold Σ is one-dimensional, we can use suitable tools from analysis to derive existence and properties of solutions.

Effects of a smartphone-based attention bias modification training on body satisfaction and self-esteem – A randomized controlled pilot study

In this paper we put together some tools from differential geometry and analysis to study second order semi-linear partial differential equations on a Riemannian manifold M. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [10] we are able to write a (reduced) second order semi-linear problem on a submanifold Σ. This submanifold is, in a sense, transversal to the orbits of the group actions and its existence is assumed. We describe precise conditions on the Riemannian Manifold M and the submanifold Σ in order to be able to write the reduced equation on Σ. These conditions are satisfied by several particular cases including some examples treated separately in the literature such as the sphere, surfaces of revolution and others. Our framework also includes the setup of polar actions or exponential coordinates. Using this procedure, we are left with a second order semi-linear equation posed on a submanifold. In particular, if the submanifold Σ is one-dimensional, we can use suitable tools from analysis to derive existence and properties of solutions.

Derivatives pricing with market impact and limit order book

This paper investigates the derivatives pricing under the existence of liquidity costs and market impact for the underlying asset in continuous time. First, we formulate the charge for the liquidity costs and the market impact on the derivatives prices through a stochastic control problem that aims to maximize the mark-to-market value of the portfolio less the quadratic variation multiplied by a risk aversion parameter during the hedging period and the liquidation cost at maturity. Then, we obtain the derivatives price by reduction of this charge from the premium in the Bachelier model. Second, we consider a second order semilinear partial differential equation (PDE) of parabolic type associated with the control problem, which is analytically solved or approximated by an asymptotic expansion around a solution to an explicitly solvable nonlinear PDE. Finally, we present the numerical examples of the pricing for a variance option and a European call option, and show comparative static analyses.

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than those in which the delay appears only in the state. This is particularly true when we look at the associated Hamilton--Jacobi--Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong [SIAM J. Control Optim., 19 (1981), pp. 139--153] for the deterministic case) the HJB equation is an infinite-dimensional second order semilinear partial differential equation that does not satisfy the so-called structure condition, which substantially means that the control can act on the system, modifying its dynamics at most along the same directions upon which the noise acts. The absence of such a condition---together with the lack of smoothing properties, a common feature of problems with delay---prevents the use of known techniques (based on backward stochastic differential equations or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation, and so no results in this direction have been proved till now. In this paper we provide a result on existence of regular solutions of HJB equations of this type. This opens the door to proving existence of optimal feedback controls, a task that will be accomplished in the companion paper [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 3013--3038]. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specific class of equations and data which arise naturally in many applied problems.

Finite Element Methods for One Dimensional Fourth Order Semilinear Partial Differential Equation

In this paper, we consider one dimensional fourth order semilinear partial differential equation. Some a priori bounds using Lyapunov functional are derived and existence and uniqueness results for the weak solution are proved. We discuss the finite element Galerkin methods and establish optimal error estimates for the semidiscrete case. Crank–Nicolson scheme is used in the temporal direction and optimal error estimates are derived. Finally, we discuss some numerical experiments and validate with the theoretical results.

Formal Gevrey theory for singular first order semi-linear partial differential equations

This paper is concerned with the existence, the uniqueness, convergence and divergence of formal power series solutions of singular first order quasi-linear partial differential equations. Firstly we give the condition under which the formal solution exists uniquely. However, this formal power series solution does not necessarily converge. So we characterize the rate of divergence of the formal solution via the Gevrey order of formal power series. The Gevrey orders of formal solutions are determined by the Newton polyhedrons of nonlinear partial differential operators.

A Boundary Value Problem for Symmetric Elliptic Systems of First Order Semilinear Partial Differential Equations in Open Sets of Class C 1

A Boundary Value Problem for Symmetric Elliptic Systems of First Order Semilinear Partial Differential Equations in Open Sets of Class C 1

A boundary value problem for symmetric elliptic systems of first order semilinear partial differential equations in open sets of class $C\sp{1}$

Si dimostrano un teorema di unicità ed un teorema di esistenza per sistemi ellittici simmetrici a coefficienti costanti di equazioni alle derivate parziali del primo ordine semilineari in aperti di classe ${C^1}$ e con dati al contorno in ${L^p}$.

A Note on First Order Semi-Linear Partial Differential Equations

A Note on First Order Semi-Linear Partial Differential Equations