Semilinear Elliptic Partial Differential Equations Research Articles

单调迭代结合虚拟区域法求解非线性障碍问题

The numerical solution of obstacle problems with 2ndorder semilinear elliptic partial differential equations (PDEs) was addressed. The nonlinear obstacle problem was solved with the monotone iteration method, and the adjoint elliptic differential equations with the Dirichlet boundary conditions on irregular domains were solved with the fictitious domain method. In the calculation process, the conventional finite element discretization resulted in the trouble of computing integrals on the irregular body boundaries with the regular mesh of the extended domain. To overcome this difficulty, a new algorithm was designed based on the finite difference method allowing the use of fast solvers for PDEs. The proposed algorithm has a simple structure and is easily programmable. The numerical simulation of a steady state problem of the logistic population model with diffusion and obstacle to growth shows that the proposed method is feasible and efficient.

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017

Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator [Formula: see text] has a generalized drift. We investigate existence and uniqueness of generalized solutions of class [Formula: see text]. The generator [Formula: see text] is associated with a Markov process [Formula: see text] which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is [Formula: see text]. Since [Formula: see text] is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.

A reconstruction algorithm based on topological gradient for an inverse problem related to a semilinear elliptic boundary value problem

In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection of small inhomogeneities located inside a domain , where the coefficients of the equation are altered, starting from observations of the solution of the equation on the boundary . Exploiting theoretical results recently achieved in [], we implement a reconstruction procedure based on the computation of the topological gradient of a suitable cost functional. Numerical results obtained for several test cases finally assess the feasibility and the accuracy of the proposed technique.

A Novel Application of the Classical Banach Fixed Point Theorem

Using the classical Banach fixed point theorem, we propose a novel method to obtain existence and uniqueness result pertaining to the solutions of semilinear elliptic partial differential equation of the type \(\Delta u+f(x,u,Du)=0\), in \(\Omega \subset {\mathbb {R}}^n \) and \(u|_{\partial \Omega }=0\), in a suitable Sobolev space. Here \(f{:}\;\Omega \times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is either a linear or a non-linear Lipshitz continuous function. The approach attempted here can be used as an algorithm by the numerical analysts to determine a solution to a partial differential equation of the above type.

Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.

Qualitative analysis of the moving boundary problem for a biofilm reactor model

The work presents the qualitative analysis of the free boundary value problem related to a biofilm reactor model. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of nonlinear hyperbolic partial differential equations governing the microbial species growth, a system of semilinear elliptic partial differential equations describing the substrate trends, and a system of nonlinear differential equations for the mass balance in the reactor. The free boundary evolution is governed by a differential equation that also accounts for detachment and attachment. The main result is a uniqueness and existence theorem. By using the method of characteristics, the original differential system is converted to Volterra integral equations and then the fixed point theorem is used. The work is completed with numerical simulations describing the free boundary evolution and mainly focused on attachment/detachment process.

The existence and uniqueness of the solution for nonlinear elliptic equations in Hilbert spaces

The existence and uniqueness of the mild solution for semilinear elliptic partial differential equations associated with stochastic delay evolution equations are obtained by means of infinite horizon backward stochastic differential equations. Applications to optimal control for an infinite horizon are also given.

The effects of spatial heterogeneities on some multiplicity results

In [10], using a Theorem of Clark and in [1] several multiplicity results were obtained for families of semilinear elliptic partial differential equations. Here these results are extended so as to include more generalspatially heterogeneous models arising in population dynamics. The optimality of the general assumptions imposed to get some of these multiplicity results is also analyzed.

Fully Adaptive Newton--Galerkin Methods for Semilinear Elliptic Partial Differential Equations

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton--Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Newton-[formula omitted] method for the second-order semi-linear elliptic partial differential equations

Newton’s method with first-order system least squares (FOSLS) finite element method has been widely used to approximately solve a system of nonlinear partial differential equations (Adler et al., 2010 [9], Codd et al., 2003 [10], Manteuffel et al., 2006 [11]). In this paper, we propose to use the first order system LL∗ method to find a correction in each Newton’s iteration which provides an L2-approximation of the second-order semi-linear elliptic partial differential equations while the typical Newton-FOSLS method providesH1-approximations. The numerical tests have been conducted to validate the theory.

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.

Second-Order Necessary Optimality Conditions for a Class of Semilinear Elliptic Optimal Control Problems with Mixed Pointwise Constraints

This paper deals with first- and second-order necessary optimality conditions for a class of optimal control problems governed by semilinear elliptic partial differential equations with mixed pointwise state-control constraints.

Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the $L^1$-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing $L^2$-regularization parameter for the control. The associated convergence rate is estimated.

Relaxation of Optimal Control Problem Governed by 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. A relaxed problem is constructed by homogenization. By studying the G-closure problem, a local representation of admissible set of relaxed control is given. Finally, the maximum principle of relaxed problem is established via homogenization spike variation.

Spectral and stochastic properties of the $f$-Laplacian, solutions of PDEs at infinity and geometric applications

The aim of this paper is to suggest a new perspective to study qualitative properties of solutions of semilinear elliptic partial differential equations defined outside a compact set. The relevant tools in this setting come from spectral theory and from a combination of stochastic properties of the differential operators in question. Possible links between spectral and stochastic properties are analyzed in detail.

Three solutions for a Dirichlet problem with one-sided growth conditions on the nonlinearities

We consider a semilinear elliptic partial differential equation, depending on two positive parameters λ and μ, coupled with homogeneous Dirichlet boundary conditions. Assuming only one-sided growth conditions on the nonlinearities involved, we prove the existence of at least three weak solutions for λ and μ lying in convenient intervals. We employ techniques of nonsmooth analysis introduced by Degiovanni and Zani, and a theorem of Ricceri for multiplicity of local minimizers.

Weak lower semi-continuity of the optimal value function and applications to worst-case robust optimal control problems

Sufficient conditions ensuring weak lower semi-continuity of the optimal value function are presented. To this end, refined inner semi-continuity properties of set-valued maps are introduced which meet the needs of the weak topology in Banach spaces. The results are applied to prove the existence of solutions in various worst-case robust optimal control problems governed by semilinear elliptic partial differential equations.

Stable solutions of elliptic equations on Riemannian manifolds with Euclidean coverings

We investigate the rigidity properties of stable, bounded solutions of semilinear elliptic partial differential equations in Riemannian manifolds that admit a Euclidean universal covering, finding conditions under which the level sets are geodesics or the solution is constant.

A probabilistic approach to Dirichlet problems of semilinear elliptic PDEs with singular coefficients

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.