Elliptic Equations Research Articles

On the regularity of optimal potentials in control problems governed by elliptic equations

Abstract In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV \mathrm{BV} one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV \mathrm{BV} regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.

Global Solution for a Fourth-Order Nonlinear Elliptic Equation

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

In this work, we present a conservative Allen–Cahn (CAC) equation and investigate its unconditionally maximum principle-preserving linear numerical scheme. The operator splitting strategy is adopted to split the CAC model into a conventional AC equation and a mass correction equation. The standard finite difference method is used to discretize the equations in space. In the first step, the temporal discretization of the AC equation is performed by using the energy factorization technique. The discrete version of the maximum principle-preserving property for the AC equation is unconditionally satisfied. In the second step, we apply mass correction by using an explicit Euler-type approach. Without the constraint of time step, we estimate that the absolute value of the updated solution is bounded by 1. The unique solvability is analytically proved. In each time step, the proposed method is easy to implement because we only need to solve a linear elliptic type equation and then correct the solution in an explicit manner. Various computational experiments in two-dimensional and three-dimensional spaces are performed to confirm the performance of the proposed method. Moreover, the experiments also indicate that the proposed model can be used to simulate two-phase incompressible fluid flows.

Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a highly nonlinear way on a parameter y ∈ [−1, 1]N. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove ℓp-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

New weighted Trudinger-Moser inequality for functions not necessarily radially symmetric and applications

In this paper, we prove some new inequality of Trudinger-Moser' type defined in some weighted Sobolev space whose norm is a combination of the norms of the N-Laplacian and the p-Laplacian with 1<p<N in RN,N≥2. The presence and the behavior of the weights prevent us to use of the symmetrization arguments. This inequality holds for functions which are not required to be radially symmetric. The strategy used in order to establish our inequality mainly consists of a proper change of variables which permits to return to the non-weighted case. Various improvements of this inequality are also proved. Finally, one of these improvements is used to study some elliptic equation involving a weighted (N,p)-Laplacian operator.

Convergence of Cubic Renormalized FEM for Linear Elliptic Equations with L1-Data

Convergence of Cubic Renormalized FEM for Linear Elliptic Equations with L1-Data

Continuous finite elements satisfying a strong discrete Miranda–Talenti identity

Abstract This article introduces continuous $H^{2}$-nonconforming finite elements in two and three space dimensions that satisfy a strong discrete Miranda–Talenti inequality in the sense that the global $L^{2}$ norm of the piecewise Hessian is bounded by the $L^{2}$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^{1}$ continuity on the vertices (2D) or edges (3D). As an application, these finite elements are used to approximate uniformly elliptic equations in nondivergence form under the Cordes condition without additional stabilization terms. For the biharmonic equation in three dimensions, the proposed methods has less degrees of freedom than existing nonconforming schemes of the same order. Numerical results in two and three dimensions confirm the practical feasibility of the proposed schemes.

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .

Boundary Domain Integral Equations for Variable Coefficient Mixed BVP in 2D Unbounded Domain

In this paper, the direct segregated Boundary Domain Integral Equations (BDIEs) for the Mixed Boundary Value Problems (MBVPs) for a scalar second order elliptic Partial Differential Equation (PDE) with variable coefficient in unbounded (exterior) 2D domain is considered. Otar Chkadua, Sergey Mikhailov and David Natroshvili formulated both the interior and exterior 3D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order elliptic PDE with a variable coefficients. On the other hand Sergey Mikhailov and Tamirat Temesgen formulated only the interior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. However, in this paper we formulated the exterior 2D domain of the direct segregated systems of BDIEs for the MBVPs for a scalar second order divergent elliptic PDE with a variable coefficients. The aim of this work is to reduce the MBVPs to some direct segregated BDIEs with the use of an appropriate parametrix (Levi function). We examine the characteristics of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original MBVPs. This analysis is based on the corresponding properties of the MBVPs in weighted Sobolev spaces that are proved as well.

On an inverse problem for an elliptic equation

On an inverse problem for an elliptic equation

An efficient and accurate mapping method for elliptic equations in irregular annular domains

In this paper, we introduce a coordinate transformation, which transforms the irregular annular domain to a unit disk. We present its basic properties. As examples, we consider Poisson type equation and Cauchy–Navier elastic equations with variable coefficients in two-dimensional irregular annular domains, and prove the existence and uniqueness of weak solutions. We also construct the mixed Fourier–Legendre spectral schemes, and derive the optimal convergence of numerical solutions under the H1-norm. The numerical results indicate that the suggested method achieves high-order accuracy.

Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

Given a linear elliptic equation \sum a_{ij}u_{ij}=0 in \mathbb{R}^{3} , it is a classical problem to determine if its 1-homogeneous solutions u are linear. The answer is negative in general, by a construction of Martinez-Maure. In contrast, the answer is affirmative in the uniformly elliptic case, by a theorem of Han, Nadirashvili and Yuan, and it is a known open problem to determine the degenerate ellipticity condition on (a_{ij}) under which this theorem still holds. In this paper we solve this problem. We prove the linearity of u under the following degenerate ellipticity condition for (a_{ij}) , which is sharp by Martinez-Maure’s example: if \mathcal{K} denotes the ratio between the largest and smallest eigenvalues of (a_{ij}) , we assume \mathcal{K}|_{\mathcal{O}} lies in L_{\rm loc}^{1} for some connected open set \mathcal{O}\subset \mathbb{S}^{2} that intersects any configuration of four disjoint closed geodesic arcs of length \pi in \mathbb{S}^{2} . Our results also give the sharpest possible version under which an old conjecture by Alexandrov, Koutroufiotis and Nirenberg (disproved by Martinez-Maure’s example) holds.

Elliptic equations in divergence form with discontinuous coefficients in domains with corners

Elliptic equations in divergence form with discontinuous coefficients in domains with corners

A strong maximum principle for quasilinear elliptic differential equations in a variable exponent Sobolev space

A strong maximum principle is an important property for elliptic and parabolic equations. In this paper, we consider a strong maximum principle for a class of quasilinear elliptic equations containing p(⋅)-Laplacian and mean curvature operator.

A New Second-order Maximum-principle-preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients

A new development of Finite Volumes (FV, for short) and its theoretical analysis are the purpose of this work. Recall that FV are known as powerful tools to address equations of conservation laws (mass, energy, momentum,...). Over the last two decades investigators have succeeded in putting in place a mathematical framework for the theoretical analysis of FV. A perfect illustration of this progress is the design and mathematical analysis of Discrete Duality Finite Volumes (DDFV, for short). We propose now a new class of DDFV for 2nd order elliptic equations involving discontinuous diffusion coefficients or nonlinearities. A one-dimensional linear elliptic equation is addressed here for illustrating the ideas behind our numerical strategy. The algebraic structure of the discrete system we have got is different from that of standard DDFV. The main novelty is that the so-called diamond mesh elements are confined in homogeneous zones for flow problems governed by piecewise constant coefficients. This is got from our judicious definition of the primal mesh. The gain is that there is no need to compute homogenized coefficients to be allocated to the so-called diamond cells as required to conventional DDFV. Notice that poor homogenized permeability allocated to diamond elements leads to poor approximations of fluxes across grid-block interfaces. Moreover for 1-D flow problems in a porous medium involving permeability discontinuities (piecewise constant permeability for instance) the proposed FV scheme leads to a symmetric positive-definite discrete system that meets the discrete maximum principle; we have shown its second order convergence under relevant assumptions.

Classification of solutions to some elliptic equations with Robin boundary condition and finite Morse index

We deal here with the non-existence of weak finite Morse index solutions to the following nonlinear elliptic equation −Δu=(1+|x|α)|u|p−1uinRN, or in half space with Robin boundary condition, where N≥2, α>−2 and p>1.

A finite difference method for elliptic equations with the variable-order fractional derivative

A finite difference method for elliptic equations with the variable-order fractional derivative

Nonlinear stability of smooth multi-solitons for the Dullin-Gottwald-Holm equation

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δH1/δu=m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work.

Modified Neumann–Neumann methods for semi- and quasilinear elliptic equations

The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators.

Schauder and Cordes–Nirenberg estimates for nonlocal elliptic equations with singular kernels

AbstractWe study integro‐differential elliptic equations (of order ) with variable coefficients, and prove the natural and most general Schauder‐type estimates that can hold in this setting, both in divergence and non–divergence form. Furthermore, we also establish Hölder estimates for general elliptic equations with no regularity assumption on , including for the first‐time operators like , provided that the coefficients have “small oscillation.”