Solutions Of Second Order Elliptic Equations Research Articles

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

Numerical Solutions of Second-Order Elliptic Equations with C-Bézier Basis

Entropy and renormalized solutions for a nonlinear elliptic problem in Musielak-Orlicz spaces

In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak-Orlicz functions and the right-hand side from the space L1(Ω). In nonreflexive Musielak-Orlicz-Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.

Asymptotic mean-value formulas for solutions of general second-order elliptic equations

Abstract We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators.

Gradient estimate for solutions of second-order elliptic equations

We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point $x=0$. In particular, we treat the case of solutions that are not Lipschitz continuous at $x=0$. We show that our estimate is sharp.

Anisotropic mesh quality measures and adaptation for polygonal meshes

Anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes. Three sets of alignment and equidistribution measures are developed, based on least squares fitting, generalized barycentric mappings, and the singular value decomposition, respectively. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given anisotropic metrics. Based on the second set of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second-order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.

On Mappings of Plane Domains by Solutions of Second-Order Elliptic Equations

In this paper, we study sufficient conditions for the one-to-one solvability of secondorder partial differential equations in a plane Jordan domain. For a continuous one-to-one and orientation-keeping map of the boundary of a Jordan domain to the rectifiable boundary of some other Jordan domain, we prove the following property: If the Cauchy integral whose measure is generated by this map is bounded by some constant in the exterior domain, then the solution to the corresponding Dirichlet problem in the domain with this boundary function maps these domains one-to-one. In the proof of the main result we use integral representations of equation solutions, particularly, properties of Fredholm-type integral equations on the domain boundary.

On Univalence of Solutions of Second-Order Elliptic Equations in the Unit Disk on the Plane

Sufficient conditions on a function continuous on the unit circle are found ensuring that the solution of the Dirichlet problem in the unit disk for a certain second-order partial differential equation with this boundary function is a homeomorphism of the unit disk onto a simply connected Jordan domain. Bibliography: 5 titles.

Mappings by the solutions of second-order elliptic equations

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.

Boundary control of elliptic solutions to enforce local constraints

We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy specific qualitative properties such as: (i) the norm of the gradient of one solution is bounded from below by a positive constant in the vicinity of a finite number of prescribed points; (ii) the determinant of gradients of n solutions is bounded from below in the vicinity of a finite number of prescribed points. Such constructions find applications in recent hybrid medical imaging modalities.The methodology is based on starting from a controlled setting in which the constraints are satisfied and continuously modifying the coefficients in the second-order elliptic equation. The boundary condition is evolved by solving an ordinary differential equation (ODE) defined via appropriate optimality conditions. Unique continuations and standard regularity results for elliptic equations are used to show that the ODE admits a solution for sufficiently long times.

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.

Local approximations by solutions of second-order elliptic equations and removable singularities

Local approximations by solutions of second-order elliptic equations and removable singularities

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H 1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h 3/2−ε) with ε > 0 if u ∈ H 3(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.

Uniform approximation by polynomial solutions of second-order elliptic equations, and the corresponding Dirichlet problem

Conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets of special form in ℝ2 are studied. The results obtained are of analytic character. Conditions of solvability and uniqueness for the corresponding Dirichlet problem are also studied. It is proved that the polynomial approximability on the boundary of a domain is not generally equivalent to the solvability of the corresponding Dirichlet problem.

Uniform approximability of functions by polynomial solutions of second-order elliptic equations on compact plane sets

We investigate conditions for the uniform approximability of functions by polynomial solutions of second-order elliptic equations with constant complex coefficients on compact sets in . Some new results of a reductive nature are obtained which ensure that a compact set is an approximation compactum if certain special subsets with a simpler topological structure have this property.

Uniform and -approximability of functions on compact subsets of by solutions of second-order elliptic equations

Several necessary and sufficient conditions for the existence of uniform or -approximation of functions on compact subsets of by solutions of elliptic systems of the form with constant complex coefficients , and are obtained. The proofs are based on a refinement of Vitushkin's localization method, in which one constructs localized approximating functions by “gluing together” some special many-valued solutions of the above equations. The resulting conditions of approximation are of a topological and metric nature.

Superconvergence of Mixed Finite Element Approximations over Quadrilaterals

A superconvergence result is established in this article for approximate solutions of second-order elliptic equations by mixed finite element methods over quadrilaterals. The superconvergence indicates an accuracy of ${\cal O}(h^{k+2})$ for the mixed finite element approximation if the Raviart--Thomas or Brezzi--Douglas--Fortin--Marini elements of order k are employed with optimal error estimate of ${\cal O}(h^{k+1})$. Numerical experiments are presented to illustrate the theoretical result.

Numerical solution of second-order elliptic equations on plane domains

On propose une methode generale de discretisation pour la solution numerique des equations de diffusion avec convection. Les regles pour la discretisation sont basees sur les formules d'integration pour les champs et possedent les avantages suivants: 1) elles restent efficaces aussi dans le cas de la dominance du terme de convection; 2) pour les solutions obtenues on peut verifier une loi de conservation discrete; 3) on obtient egalement un principe du maximum discret. On demontre la convergence de la solution des elements finis vers la solution exacte avec la vitesse O(h) concernant la norme en H 1

ON QUALITATIVE PROPERTIES OF SOLUTIONS OF A NONLINEAR EQUATION OF SECOND ORDER

A number of theorems are obtained on the behavior of solutions of second-order elliptic equations containing nonlinearities. Among them are a theorem on the uniqueness of the solutions of the exterior Dirichlet problem, a theorem on a removable singularity, and a theorem on the absence of entire solutions.Bibliography: 15 titles.

On positive solutions of second-order elliptic equations, stability results, and classification

Soit P un operateur uniformement elliptique d'ordre 2 defini dans un domaine ΩcR n . Soit p(Ω) le cone convexe de toutes les solutions positives de l'equation Pu=0 dans Ω. On etudie les proprietes generales de p(Ω) pour P et Ω generaux

Summability properties of solutions of second-order elliptic equations

Summability properties of solutions of second-order elliptic equations