Semilinear Elliptic Partial Differential Equations Research Articles

Higher Order Triangular Mixed Finite Element Methods for Semilinear Quadratic Optimal Control Problems

In this paper, we investigate a priori error estimates for the quadratic opti- mal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods. The state and the co-state are ap- proximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k (k≥ 0). A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results. AMS subject classifications: 49J20, 65N30

Cesari-type conditions for semilinear elliptic equation with leading term containing controls

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. By studying the $G$-closure of the leading term, an existence result is established under a Cesari-type condition.

The existence of k-convex hypersurface with prescribed mean curvature

Using the strong maximum principle, we obtain a constant rank theorem for the k-convex solutions of semilinear elliptic partial differential equations. As an application we obtain an existence theorem of k-convex starshaped hypersurface with prescribed mean curvature in R n+1.

Uniqueness of positive radial solutions of [formula omitted] in [formula omitted], [formula omitted

We study the uniqueness of radially symmetric ground states for the semilinear elliptic partial differential equation Δ u + f ( u ) = 0 in R N , N ≥ 2 . Assuming that F ( t ) = ∫ 0 t f ( s ) d s is negative in ( 0 , u 1 ) and positive in ( u 1 , u ̄ ) , we obtain the uniqueness of nonnegative solutions with u ( 0 ) = sup u ∈ ( 0 , u ̄ ) in the case where S ( u ) = u f ′ ( u ) / f ( u ) is monotonically decreasing in [ u 1 , u ̄ ) .

The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: Δ u ( x ) + f ¯ ( x , u ( x ) , v ( ‖ x ‖ ) ) + g ( ‖ x ‖ ) x ⋅ ∇ u ( x ) = 0 , for x ∈ Ω r 0 ≔ { x ∈ R n , n ≥ 3 , ‖ x ‖ > r 0 } and v ∈ V , where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r 0 for which our problem possesses at least one positive evanescent solution in the exterior domain Ω r 0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.

Principal curvature estimates for the convex level sets of semilinear elliptic equations

We give a positive lower bound for the principal curvature of the strict convex level sets of harmonic functions in terms of the principal curvature of the domain boundary and the norm of the boundary gradient. We also extend this result to a class of semi-linear elliptic partial differential equations under certain structure condition.

Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space

The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls.

Asymptotic Expansion for the Solutions of Control Constrained Semilinear Elliptic Problems with Interior Penalties

In this work we consider the optimal control problem of a semilinear elliptic partial differential equation (PDE) with a Dirichlet boundary condition, where the control variable is distributed over the domain and is constrained to be nonnegative. The approach is to consider an associated family of penalized problems, parametrized by $\varepsilon>0$, whose solutions define a central path converging to the solution of the original problem. Our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem. This approach allows us to recover some known error bounds for the logarithmic barrier approximation and to obtain some new ones for a rather general class of barrier functions. In this manner, we generalize the results of [F. Alvarez et al., Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems, Math. Program. Ser. A, to appear], which were obtained in the ordinary differential equation (ODE) framework.

Uniqueness of source for a class of semilinear elliptic equations

Uniqueness of source is proved for an important class of semilinear elliptic partial differential equations useful for identification of doping profiles in semiconductor devices and for modeling of ion channels in biological systems.

Infinite horizon BSDEs with dissipative coefficients in Hilbert spaces and applications

In this paper we study a class of infinite horizon backward stochastic differential equations (BSDEs) of the form d Y ( t ) = λ Y ( t ) d t − f ( t , Y ( t ) , Z ( t ) ) d t + Z ( t ) d W ( t ) , 0 ⩽ t < ∞ , in a real separable Hilbert space, where λ is a given real parameter and the coefficient f is dissipative in y and Lipschitz in z . By Yosida approximation to dissipative mappings we show existence and uniqueness of the solutions for these equations. This result is applied to construct unique viscosity solutions to semilinear elliptic partial differential equations (PDEs).

Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls

An optimal control problem for semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing the control. Necessary conditions for optimal controls are established by the method of homogenized spike variation. The key to such a method is to modify the usual spike variational technique by taking into account the homogenization techniques for elliptic equations, together with an idea from the theory of relaxed controls. Problems with state constraints are also discussed by further adding some well-known penalty arguments involving the application of Ekeland's variational principle and finite codimensionality of certain sets.

Existence and uniqueness of solutions for higher order elliptic boundary value problems

A class of sufficient conditions are obtained for the existence and uniqueness of solutions to the boundary value problems of semi-linear elliptic partial differential equations, using a global inverse function theorem.

Uniqueness of radially symmetric positive solutions for [formula omitted] in an annulus

In this article we prove that the semi-linear elliptic partial differential equation − Δ u + u = u p in Ω , u > 0 in Ω , u = 0 on ∂ Ω possesses a unique positive radially symmetric solution. Here p > 1 and Ω is the annulus { x ∈ R N | a < | x | < b } , with N ⩾ 2 , 0 < a < b ⩽ ∞ . We also show the positive solution is non-degenerate.

Branching Points For a Class of Variational Equations Involving Potential With Parameter

Abstract In this paper, under the framework of Ambrosetti [1], using mountain-pass theorem and topological degree theory, we extend the result of Ambrosetti [1] (in which H is independent of λ and C1) to the equation Au + H(λ, u) = λu with H jointly continuous in (λ, u) and H(λ, ·) of class C0, 1. We give a description of the structure of local bifurcation from isolated eigenvalues of A: the set of bifurcation solutions at each isolated eigenvalue of A contains at least one connected branch. Applications to the existence of branching points to semilinear elliptic partial differential equations are given.

Numerical Verification of Optimality Conditions

A class of optimal control problems for a semilinear elliptic partial differential equation with control constraints is considered. It is well known that sufficient second-order conditions ensure the stability of optimal solutions, and the convergence of numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically). We will propose a new approach as follows: Starting with a numerical solution for a fixed mesh we will show the existence of a local minimizer of the continuous problem. Moreover, we will prove that this minimizer satisfies the sufficient second-order conditions.

Radial basis collocation method and quasi‐Newton iteration for nonlinear elliptic problems

AbstractThis work presents a radial basis collocation method combined with the quasi‐Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi‐Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi‐Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi‐Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Domain decomposition methods for linear and semilinear elliptic stochastic partial differential equations

In this paper, we study several overlapping domain decomposition methods for the numerical solutions of some linear and semilinear elliptic stochastic partial differential equations discretized by the finite element methods. In particular, we show that the algorithms converge and the convergence rates are independent of the finite element mesh parameter, as well as the number of subdomains used in the domain decomposition.

Finite element methods for semilinear elliptic stochastic partial differential equations

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results.

Infinite Horizon Stochastic Optimal Control Problems with Degenerate Noise and Elliptic Equations in Hilbert Spaces

Semilinear elliptic partial differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. These results are applied to a stochastic optimal control problem with infinite horizon. Applications to controlled stochastic heat and wave equations are given.

Existence of a semilinear elliptic system with exponential nonlinearities

We consider a system of semilinear elliptic partial differential equations with exponential nonlinearities in $R^2$. We construct a solution of the system viewing the system as a perturbation of the decoupled Liouville equations and applying suitable implicit function theorem. As a byproduct we obtain very precise information on the asymptotic behaviors of the solutions near infinity.