Coefficients of singularities of the biharmonic problem of Neumann type: case of the crack.
Coefficients of the singularities on domains with conical points
As a model for elliptic boundary value problems, we consider the Dirichlet problem for an elliptic operator. Solutions have singular expansions near the conical points of the domain. We give formulas for the coefficients in these expansions.
Concentration phenomena for fourth-order elliptic equations with critical exponent.
Concentration phenomena for Liouville's equation in dimension four
Convergence of -cycle and -cycle multigrid methods for the biharmonic problem using the Morley element.
Convex shape optimization for the least biharmonic Steklov eigenvalue
The least Steklov eigenvalue d1 for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the L2-norm of harmonic functions. These applications suggest to address the problem of minimizing d1 in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular, the existence of a convex domain...
Covariant differential operators and Green's functions
The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green’s functions. For instance, we offer a new proof of a formula for Green’s function of the mth power of the ordinary Laplace’s operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green’s functions associated with mth powers of the Poincaré invariant Laplace operator . It turns...
Curved triangular finite -elements
Curved triangular -elements which can be pieced together with the generalized Bell’s -elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the...
Discrete forms of Friedrichs' inequalities in the finite element method
Discrete Sobolev spaces and regularity of elliptic difference schemes
Eigenvalues of polyharmonic operators on variable domains
We consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are...
Elliptic and degenerate-elliptic operators in unbounded domains
Elliptic boundary value problems with nonvariational perturbation and the finite element method
Equations de von Kármán. I. Résultat d'existence pour les problèmes aux limites non homogènes.
Dans l'article, on a défini une équation d'operateur équivalent à la formulation variationnelle du problème. Les solutions de cette équation sont des points critiques de la fonctionnelle qu'elle porte le nom d'énergie totale de déformation. La fonctionnelle est coercive et faiblement séquentiellement semi-continue inférieure. Par le théorème de l'analyse fonctionnelle, on a obtenu le résultat d'existence pour le problème.
Équations de von Kármán. II. Approximation de la solution
Dans l'article, on a donné quelques conditions suffisantes pour l'unicité locale et globale de la solution du problème. On a construit une solution variationnelle du problème par la méthode de Newton-Kantorovitch et la méthode du prolongement continu avec ces conditions suffisantes pour l'unicité.
Error Bounds for the Galerkin Method Applied to Singular and Nonsingular Boundary Value Problems.
Error estimates for the assumed stresses hybrid methods in the approximation of 4th order elliptic equations
Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace
Estimates for the Green function and characterization of a certain Kato class by the Gauss semigroup in the half space.
Estimates for the Green function and existence of positive solutions for higher-order elliptic equations.