General Elliptic Equations Research Articles

On solvability of the Dirichlet problem with the boundary function in L 2 for a second-order elliptic equation

We consider Dirichlet problems in a bounded domain Q ⊂ Rn for a general secondorder elliptic equation with the boundary function in L2. In the author’s previous papers necessary and sufficient conditions for the existence of an (n − 1)-dimensionally continuous solution were obtained under some natural assumptions on the coefficients of equation. Those assumptions are formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. In the present paper we obtain necessary and sufficient conditions for the existence of a solution in terms of the original problem for a more narrow class of the right-hand sides. It is shown that if, in addition, the boundary function is assumed to be in the space W21/2 (∂Q), then the obtained conditions transform into solvability conditions in the space W21(Q).

Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

Abstract We prove the compactness of solutions of general fourth order elliptic equations which are L 1 L^{1} -perturbations of the Q-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.

On some transmission problems set in a biological cell, analysis and resolution

Some transmission problems are set in bodies with a crown of small thickness ε>0. For instance, those concerning the conductivity in the biological cell. By a natural change of variables, we transform them in transmission problems set in two cylindrical bodies ]−∞,0[×]−π,π[ and ]0,δ[×]−π,π[ (where δ=ln⁡(1+ε)) and then, in some general elliptic abstract differential equations (Pδ)δ>0. The goal of this work is to give a complete study of these problems (Pδ)δ>0 for every δ>0. Existence, uniqueness and maximal regularity results are obtained for the classical solutions essentially by using the semigroups theory.

On some transmission problems in a plane corner

We consider the transmission problem in Sobolev Slobodetskii spaces for two model general elliptic pseudo differential equations in a plane, where a one equation is given inside a corner, and another one is outside. Starting from general solutions of these equations we consider certain transmission conditions on angle sides. For some cases this transmission problem was reduced to an equivalent system of one-dimensional linear integral equations.

Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [Z. Cai et al., SIAM J. Numer. Anal., 39 (2001), pp. 1418--1445], in order to retain the full efficiency of the $L^2$ norm first-order system least-squares (FOSLS) approach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach of Cai et al. was applied to the div-curl system with added slack variables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite element approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator. We then propose an explicit residual error estimator and establish its reliability and efficiency bounds.

A Computational Study with Finite Element Method and Finite Difference Method for 2D Elliptic Partial Differential Equations

In this paper, we consider two methods, the Second order Central Difference Method (SCDM) and the Finite Element Method (FEM) with P1 triangular elements, for solving two dimensional general linear Elliptic Partial Differential Equations (PDE) with mixed derivatives along with Dirichlet and Neumann boundary conditions. These two methods have almost the same accuracy from theoretical aspect with regular boundaries, but generally Finite Element Method produces better approximations when the boundaries are irregular. In order to investigate which method produces better results from numerical aspect, we apply these methods into specific examples with regular boundaries with constant step-size for both of them. The results which obtained confirm, in most of the cases, the theoretical results.

Dirichlet-to-Neumann Mapping for the Characteristic Elliptic Equations with Symmetric Periodic Coefficients

AbstractBased on the numerical evidences, an analytical expression of the Dirichlet-to-Neumann mapping in the form of infinite product was first conjectured for the one-dimensional characteristic Schrödinger equation with a sinusoidal potential in [Commun. Comput. Phys., 3(3): 641-658, 2008]. It was later extended for the general second-order characteristic elliptic equations with symmetric periodic coefficients in [J. Comp. Phys., 227: 6877-6894, 2008]. In this paper, we present a proof for this Dirichlet-to-Neumann mapping.

Extended Harnack inequalities with exceptional sets and a boundary Harnack principle

Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Holder continuity.

Existence of sign changing solutions for an equation with a weighted [formula omitted]-Laplace operator

We consider radial solutions of a general elliptic equation involving a weighted p-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n−1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ⊂ ℝ n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W 2 1 (Q) under the additional requirement that the boundary function belongs to the space W 2 1/2 (∂Q).

Equivalence of three kinds of well-posed-ness of discontinuous Riemann-Hilbert problem for elliptic complex equation in multiply connected domains

In this article, we first introduce the general linear elliptic complex equation of first order with certain conditions, and then propose discontinuous Riemann-Hilbert problem and some kinds of modified well-posed-ness for the complex equation. Then we verify the equivalence of three kinds of well-posed-ness. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

Quantitative uniqueness estimates for the general second order elliptic equations

In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and bootstrapping arguments.

Local flux conservative numerical methods for the second order elliptic equations

A discontinuous Galerkin type nonconforming element method and a local flux matching nonconforming element method for the second order elliptic boundary value problems are presented. Both of these methods enjoy the local flux conservation property. The local flux matching method finds a numerical solution in the same solution space of the DG type nonconforming element method, but it achieves much faster iterative convergence speed by embedding continuity requirement in the approximation functions rather than using constraint equations that are used in the DG type nonconforming element method. The merits of the proposed local flux matching method are as follows: the formulation of the method is simple and the solution satisfies local flux conservation property. Moreover, it can be easily applied to general elliptic equations.

A kernel-free boundary integral method for implicitly defined surfaces

The kernel-free boundary integral (KFBI) method is a structured grid method for general elliptic partial differential equations. Unlike the standard boundary integral method, it avoids direct evaluation of volume and boundary integrals, which needs to know analytical expressions for the integral kernels. To evaluate a boundary or volume integral, the KFBI method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary. Selection of control points of the boundary plays a key role in the KFBI method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points. This work proposes a new point selection method, based on an overlapping surface decomposition of the boundary, which is implicitly defined by a level set function. The points selected are intersection points of the boundary with the grid lines of an underlying Cartesian grid. By the method, the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions, which allows efficient numerical differentiation and boundary reconstruction/representation. An additional equilibrating process of boundary data further guarantees stable numerical differentiation. Numerical results demonstrating the method with examples in both two and three space dimensions are presented.

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.

Error estimates for a multidimensional meshfree Galerkin method with diffuse derivatives and stabilization

A meshfree method with diffuse derivatives and a penalty stabilization is developed. An error analysis for the approximation of the solution of a general elliptic differential equation, in several dimensions, with Neumann boundary conditions is provided. Theoretical and numerical results show that the approximation error and the convergence rate are better than the diffuse element method.

Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and \(L^2\) norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach \(h^{r+2}\) and even \(h^{2r}\), comparing with \(h^{r+1}\) rate of the counterpart finite element method. Here \(r\) is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.

Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.

On non-local reflection for elliptic equations of the second order in $\mathbb{R}^{2}$ (the Dirichlet condition)

Point-to-point reflection holding for harmonic functions subject to the Dirichlet or Neumann conditions on an analytic curve in the plane almost always fails for solutions to more general elliptic equations. We develop a non-local, point-to-compact set, formula for reflecting a solution of an analytic elliptic partial differential equation across a real-analytic curve on which it satisfies the Dirichlet conditions. We also discuss the special cases when the formula reduces to the point-to-point forms.

Maximum Principles for $P1$-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical partial differential equation theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^\alpha)$-acute in the sense that each angle $\alpha_{ij}$ (for the triangle) or interior dihedral angle $\alpha_{ij}$ (for the tetrahedron) must satisfy $\alpha_{ij}\le \pi/2-\gamma h^\alpha$ for some $\alpha\ge 0$ and $\gamma>0$. For the Poisson problem where the differential operator is given by the Laplacian, the angle requirement is the same as the existing ones: either all the triangles are nonobtuse or each interior edge is nonnegative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.