Garding Inequality Research Articles

First-Kind Boundary Integral Equations for the Dirac Operator in 3-Dimensional Lipschitz Domains

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain's boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H-1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.

Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions

This work combines results from operator and interpolation theory to show that elliptic systems in divergence form admit maximal elliptic regularity on the Bessel potential scale $$ H ^{s-1}_D(\varOmega )$$ for $$s>0$$ sufficiently small, if the coefficient in the main part satisfies a certain multiplier property on the spaces $$ H ^{s}(\varOmega )$$ . Ellipticity is enforced by assuming a Garding inequality, and the result is established for spaces incorporating mixed boundary conditions with very low regularity requirements for the underlying spatial set. To illustrate the applicability of our results, two examples are provided. Firstly, a phase-field damage model is given as a practical application where higher differentiability results are obtained as a consequence to our findings. These are necessary to show an improved numerical approximation rate. Secondly, it is shown how the maximal elliptic regularity result can be used in the context of quasilinear parabolic equations incorporating quadratic gradient terms.

Trigonometric integrators for quasilinear wave equations

Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical Garding inequality.

The Coercivity of Functional Differential Equations

New sufficient conditions for the Garding inequality for variable-coefficient functional differential equations with expanded and contracted arguments of higher derivatives of the unknown function are found. These sufficient conditions generalize earlier known sufficient conditions.

Gårding inequality in large dimension

We give a lower bound for a pseudodifferential operator in a large dimensional setting, with a nonnegative real-valued symbol. The lower bound is explicitly written as a function of the semiclassical parameter, and of parameters used for the estimations of some derivatives of the symbol, without any constant depending on the dimension. This is an analog, in large dimension, of the sharp Garding inequality.

On a class of sectorial functional-differential operators

In a bounded domain containing the origin, we consider a partial differential equation whose leading terms contain transformations of arguments of the unknown function in the form of contractions and dilations. We study algebraic conditions under which the operator occurring in the equation satisfies the Garding inequality. A criterion obtained earlier for constant coefficients cannot be generalized to the case of variable coefficients. We suggest a new approach to the solution of the problem in the case of variable coefficients based on the pseudodifferential operator calculus.

Garding inequality on unbounded domains

This note presents a simple method to generalize the Garding inequality to unbounded domains. By introducing a special partition of unity associated to some covering of unbounded domains, we show that the Garding inequality, known in the literature on bounded domains (see Garding, 1953), holds for more general domains. The method proposed here only requires weak smoothness assumptions which allows coverage of a large class of elliptic operators. It also gives a new tool to extend some local properties to global ones, in particular when we deal with functions vanishing on the boundary.

Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems

We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric second-order quasi-linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in $\mathbb{R}^d$, subject to mixed Dirichlet-Neumann boundary conditions. Optimal-order asymptotic bounds are derived on the discretization error in each case without requiring the global Lipschitz continuity or uniform monotonicity of the stress tensor. Instead, only local smoothness and a Garding inequality are used in the analysis.

Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces

Let s w p be the set of all a ∈ l such that a w (x, D) is Schatten p-operator on L 2. Then we prove the following: \(S(m,g)\subseteq s_p^w\) iff \(m\in L^p\). Furthermore, \(L^p\cap S(m,g)\subseteq s_p^w\) when \(h_g^{N/2}m\in L^p\). Consequently, \(S^r_{\rho, \delta}\cap L^\infty \subseteq s^w_\infty\) when \(0\le \delta <\rho \le 1\); if \(g(z,\zeta )=\sum \lambda _j(z_j^2+\zeta _j^2)\), then \(G(z,\zeta )=\sum \lambda _j^\alpha (z_j^2+\zeta _j^2)\) is symplectically invariantly defined. Moreover, if \(-1\le \alpha \le 1\) and \( g\le g^{\sigma}\) is slowly varying (and σ-temperate), then the same is true for G; a generalization of sharp Garding's inequality.

The Berezin and Garding Inequalities

Let ϕ be a convex function on ℂ, let ℒ(σ) be a pseudodifferential operator with symbol σ, let Λσ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λσ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that $$\sum {_{\lambda \in \Lambda _\sigma } m(\lambda )} \varphi (\lambda ) \leqslant \operatorname{Tr} L(\varphi (\sigma )) + R$$ , where R is an error term of the same order as the remainder term in the Garding inequality.

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Garding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C∞well-posedness of the Cauchy problem.

Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

We introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1ε, but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock.

The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $

We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on $ \Bbb R^n $ . More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a Garding inequality. We identify the domain of their square roots as the natural Sobolev space $ H^m(\Bbb R^n,\Bbb C^N) $ . We also make some remarks on the relation between various ellipticity conditions and Garding inequality.

Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ℝ n+1×ℝ n+1, and then develop a periodisation procedure for the calculus of symbols on the cylinder $$\mathbb{T}^n $$ ×ℝ. We show Garding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.

Weak Solutions of Fluid-Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Garding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.

Convergence of Galerkin method solutions of the integral equation for thin wire antennas

In this paper we consider the Pocklington integro–differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We show that this problem is well posed in suitable fractional order Sobolev spaces and obtain a coercive or Garding type inequality for the associated operator. Combining this coercive inequality with a standard abstract formulation of the Galerkin method we obtain rigorous convergence results for Galerkin type numerical solutions of Pocklington's equation, and we demonstrate that certain convergence rates hold for these methods.

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schrodinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schrodinger-type equations. In particular, the sharp Garding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof.

Non-local approach in mathematical problems of fluid-structure interaction

Three-dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω 1 ⊆ R 3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω 2 = R 3 \Ω 1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady-state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω 1 . The problems are studied by the so-called non-local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional-variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov-Poincare type operators and on the basis of the Garding inequality and the Lax-Milgram theorem. In particular, it is shown that the physical fluid-solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter.

Coerciveness of functional-differential equations

We consider functional-differential equations with the Dirichlet conditions and with contraction and dilatation of the arguments. Necessary and sufficient conditions are obtained under which a Garding type inequality holds. These results allow us to verify coerciveness by using a special “symbol” of the equation considered.

Dilation Invariant Estimates and the Boundary Garding Inequality for Higher Order Elliptic Operators

Dilation Invariant Estimates and the Boundary Garding Inequality for Higher Order Elliptic Operators