Garding Inequality Research Articles

A boundary element method for electromagnetic transmission problems

Time-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. We transform Maxwell's equations via an ansatz into a system of second kind integral equations on the interface. Uniqueness and Fredholm properties of the boundary integral operator are proven. A Garding inequality is satisfied and ensures the stability of the corresponding Galerkin scheme.

A gårding's inequality for variational problems with constraints

In this paper we introduce a generalization of the usual Garding inequality for bounded bilinear forms defined on Hilbert spaces. In addition, we give sufficient conditions for a variational problem with constraints to satisfy this generalized inequality, and show that this result induces a new procedure for the numerical treatment of such problems. More precisely, we prove that if a constrained variational problem is uniquely solvable and if its associated biliear form satisfies the generalized Garding inequality then an asymptotically convergent nonconforming Galerkin scheme for approximating the continuous solution can be derived.

Discretization of semicoercive variational inequalities

Consider the variational inequality: $$Find \hat x \varepsilon K such that \beta (\hat x, x - \hat x) \geqslant \lambda (x - \hat x) for all x \varepsilon K$$ and its discretization: $$Find x_h \varepsilon K_h such that \beta (x_h , x - x_h ) \geqslant \lambda (x - x_h ) for all x \varepsilon K_h .$$ Here, in a real reflexive separable Banach spaceX, β is a continuous bilinear form onX × X that is nonnegative on the diagonal,λ ∈ X* is a continuous linear form, and\(K \subseteq X, K_h \subseteq X_h \) are closed convex nonvoid sets, where the family{Xh}h >o of subspaces ofX describes a discretization scheme. Then under Glowinski's realistic assumptions on the approximation ofK by{Kh}h > o—not requiring that\(K_h \subseteq K - \) we prove norm convergence,\(\lim _{h \to 0} \left\| {x_h - \hat x} \right\| = 0\), provided the solution\(\hat x\) is unique andβ satisfies a Garding inequality: There exist a compact operatorT1:X→X* and a positive constant α such thatβ(x, x)+〈T1x, x〉 ⩾ α∥x∥2 for allx ∈ X.

Hypoellipticity of the stochastic partial differential operators

This work is devoted to the proof of the following result using the general results of our preceeding work on the nuclear space-valued semimartingales: Let u be a D ′-valued semimartingale satisfying the following SPDE: ; where p and qi are random partial differential operators with C ∞ coefficientsmi is the degree of qi and h is a semimartingale with values in the space of C ∞-functions. If p satisfies a Garding's inequality on the space of the semimartingales with values in D and indexed with the uniformly bounded random intervals, then u is a semimartingale with values in the space of C ∞-functions. Although the problem may seem of a particular type, the methods we use to solve it are general

Unitary Representations of Lie Groups and Garding's Inequality

Unitary Representations of Lie Groups and Garding's Inequality

On Gårding's inequality

In this paper we prove Garding's inequality for linear differential operators in generalized divergence form which satisfy a generalized Ehrling inequality, an ellipticity condition and a condition on the coefficients. Using a compact imbedding between certain anisotropic Sobolev spaces we substitute the first condition (Ehrling's inequality) by a simple condition on a set of multi-indices.

On uniqueness for the displacement problem in the dynamical theory of elasticity

By means of a weighted form of Garding's inequality, we prove a uniqueness theorem for the initial boundary value problem of finite elastodynamics in unbounded domains.

Boundary integral operators on curved polygons

The operators of the single layer potential, double layer potential, and the Hilbert transform, acting on plane curves with corners, are members of the class of integral operators which we define in this paper. We use expansions in homogeneous kernels of increasing degree to define lower order symbols via local Mellin transforms. In this way we can study the mapping properties in (augmented) Sobolev spaces including Garding inequalities and higher regularity, thus generalizing the results of [3]and [4]from polygons to curved polygons.

Boundary element method for membrane and torsion crack problems

Abstract Here we present as an application of [12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications. A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Garding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed. The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.

Solvability of Linear and Quasilinear Elliptic Boundary Value Problems via the A-Proper Mapping Theory

ABSTRACT In this paper we use the theory of A-proper mappings and their uniform limits to obtain in a simple way general variational approximation-solvability and/or existence theorems for quasilinear elliptic BVProblems in divergence form of order 2m. In Section 1 we first show how our simple approach is used to obtain constructive solvability of linear elliptic equations under a somewhat different definition of strong ellipticity condition which allows us to dispense with the usage of the Garding inequality. In view of this, the same approach is also used in Section 2 to establish the constructive solvability of nonlinear BVProblems when the leading part satisfies a nonlinear version of the strong ellipticity condition used here. This result is then used to establish a new existence result (see Theorem 2.3) for a not necessarily coercive quasilinear elliptic BVProblem which properly includes earlier results of Visik, Browder, Pohožayev, Leray and Lions and others. In Section 3 some special cases of our ...

Optimale Fehlerabsch�tzungen mit dualen Funktionalen bei semilinearen Randwertaufgaben gew�hnlicher Differentialgleichungen

In Shampine [7] it is shown how to obtain variational error bounds for approximate solutions of boundary value problems for semilinear ordinary differential equations. These bounds are depending on a certain constantK, the existence of which is assumed. Our paper aims at practical computation; in order to get applicableL ?-error bounds,K has to be computed explicitly. Using Garding's inequality, we obtainK=K(?) depending on a positive parameter ?. In order to make these bounds efficient,K(?) will be optimized. In application only the maximal zeros of three polynomials have to be computed. Some numerical examples are given to compare the error bounds with the actual errors.

�ber eine Koerzitivit�tsungleichung in W o m,p

Let G be a bounded open subset of Rn and let B be a Dirichlet bilinear form with bounded coefficients defined in G. If the generalized version of Garding's inequality in Lp holds for all u of the Sobolev space W0m,p(G), then we prove under additional assumptions that the form is uniformly elliptic in G and that for n=2 a root condition is satisfied. Further, we study the equivalence of some definitions of “properly elliptic”.

An Analogue of Garding's Inequality for Parabolic Operators and Its Applications

An Analogue of Garding's Inequality for Parabolic Operators and Its Applications