Differential Operator Of Order 2m Research Articles

On the spectral instability for weak intermediate triharmonic problems

We define the weak intermediate boundary conditions for the triharmonic operator −Δ3. We analyse the sensitivity of this type of boundary conditions upon domain perturbations. We construct a perturbation (Ωϵ)ϵ > 0 of a smooth domain Ω of for which the weak intermediate boundary conditions on ∂Ωϵ are not preserved in the limit on ∂Ω, analogously to the Babuška paradox for the hinged plate. Four different boundary conditions can be produced in the limit, depending on the convergence of ∂Ωϵ to ∂Ω. In one particular case, we obtain a ‘strange’ boundary condition featuring a microscopic energy term related to the shape of the approaching domains. Many aspects of our analysis could be generalised to an arbitrary order elliptic differential operator of order 2m and to more general domain perturbations.

A differential operator of order 2m and its fundamental solution

A differential operator of order 2m and its fundamental solution

Regularity of solutions to Dirichlet boundary value problem in domains on a manifold

The regularity of solutions to the Dirichlet boundary value problem is studied for elliptic differential operators of order 2m defined on subdomains of a manifold. Relationships between the smoothness of the right hand side, the boundary, and the solutions are obtained.

Topological sensitivity analysis for elliptic differential operators of order 2m

The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m⩾1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m − 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.

Green′s Function and Convergence of Fourier Series for Elliptic Differential Operators with Potential from Kato Space

We consider the Friedrichs self‐adjoint extension for a differential operator A of the form A = A0 + q(x)·, which is defined on a bounded domain Ω ⊂ ℝn, n ≥ 1 (for n = 1 we assume that Ω = (a, b) is a finite interval). Here A0 = A0(x, D) is a formally self‐adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth coefficients and a potential q(x) is a real‐valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients of A and for positive λ large enough we obtain the existence of Green′s function for the operator A + λI and its estimates up to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω of Fourier series in eigenfunctions of this operator. In particular, these results can be applied for the basis of the Fourier method which is usually used in practice for solving some equations of mathematical physics.

Regularized traces of higher-order singular differential operators

We consider singular differential operators of order 2m, m ∈ ℕ, with discrete spectrum in L2[0, + ∞). For self-adjoint extensions given by the boundary conditions y(0) = y″(0) = ⋯ = y(2m−2)(0) = 0 or y′(0) = y‴(0) = ⋯ = y(2m−1)(0) = 0, we obtain regularized traces. We present the explicit form of the spectral function, which can be used for calculating regularized traces.

On the Root Functions of General Elliptic Boundary Value Problems

We consider a boundary value problem for an elliptic differential operator of order 2m in a domain \({\mathcal{D}} \subset {\mathbb{R}}^{n}\). The boundary of \({\mathcal{D}}\) is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and \(\partial {\mathcal{D}}\) bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of \(\partial {\mathcal{D}}\). The corresponding spaces are weighted Sobolev spaces \(H^{{s,\gamma }} {\left( {\mathcal{D}} \right)}\), and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in \(L^{2} ({\mathcal{D}})\). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $$ u_{t} {\left( {t,x} \right)} + Au{\left( {t,x} \right)} = f{\left( {t,x,u, \ldots \nabla ^{m} u} \right)},\quad {\left( {t,x} \right)} \in {\left( {0,T} \right)} \times \Omega . $$ is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in u,... , ∇ m u but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity. We also give a partial generalization of previous work of the second author and von Wahl to Navier boundary conditions.

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system $$ u_{t} {\left( {t,x} \right)} + Au{\left( {t,x} \right)} = f{\left( {t,x,u, \ldots \nabla ^{m} u} \right)},\quad {\left( {t,x} \right)} \in {\left( {0,T} \right)} \times \Omega . $$ is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in u,... , ∇ m u but with “bad” coefficients and by means of a continuity method, where the time serves as parameter of continuity.

On the solvability of the Lyapunov equation for nonselfadjoint differential operators of order 2m with nonlocal boundary conditions

This paper is devoted to the solvability of the Lyapunov equation $A^*U+UA=I$, where $A$ is a given nonselfadjoint differential operator of order $2m$ with nonlocal boundary conditions, $A^*$ is its adjoint, $I$ is the identity operator and $U$ is the sel

Approximation mittels Linearkombinationen von Fundamentallösungen elliptischer Differentialoperatoren

AbstractLet Ω ⊂ Rn be a bounded domain with the smooth boundary Γ, L an elliptic differential operator of order 2m with constant coefficients, E a fundamental solution of L and B = (B1, …, Bm) a normal system of boundary operators on Γ. Furtehr let (yk)Γ1 ⊂ Rn/Ω be a sequence of points and Dj(Γ) (j = 1, …, m) suitable function spaces over Γ (e.g. Cs‐spaces or SOBOLEV spaces).It is investigated, under which conditions on the sequence (yk)x1 ⊂ Rn/Ω the set

Variational formulation of higher order elliptic boundary value problems

Let A be an elliptic (partial) differential operator of order 2m on a compact manifold with boundary Г. Let B be a normal system of m differential boundary operators on Г. Assume all manifolds and coefficients are arbitrarily smooth. We construct sesquilinear forms J in terms of which there are equivalent variational formulations of the natural boundary value problems determined by A and B with solutions in Sobolev spaces HS (M), 0 &lt; s &lt; 2m. Such forms are also constructed for problems with mixed boundary conditions. The variational formulation permits localization of a priori estimates and the interchange of existence and uniqueness questions between the boundary value problem and an associated adjoint problem.

ON A COERCIVE INEQUALITY FOR AN ELLIPTIC OPERATOR IN INFINITELY MANY INDEPENDENT VARIABLES

In the present paper a coercive inequality is established for an infinite-dimensional elliptic differential operator of order 2m, and a theorem on smoothness of a generalized solution of the Dirichlet problem for an equation containing such an operator is established. Bibliography: 7 items

Uber das dirichletsche rand- und anfangswertproblem fur hyper-boiische differentialgleichungen†

We study the- initial and boundary value problem for the equation utt = L[u] with Dirichlet boundary data where L is the sum of a strongly elliptic formally self-adjoint differential operator of order 2m and of an arbitrary differential operator of order q ≦m. We give a weak formulation of this problem in an appropriate Hilbert space. The main ideas of the existence, uniqueness and regularity theory are sketched. Detailed proofs will appear in two subsequent papers. Our existence argument makes essential use of L2-spaces of vector-valued functions with values in a Hilbert space. The theory of these spaces can be developed by employing methods of L. Schwartz's theory of distributions. In particular, our analysis avoids Lebesgue and Bochner integration

Existence Theorems for Some Non-Linear Equations of Evolution

In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form(1-1)where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by(1-2)with

Ellipticity and regularity for periodic nonlinear equations

The composition of two strongly elliptic linear operators with suitably differentiable coefficients is also strongly elliptic. This fact has been used [3, pp. 178-181] to yield a particularly simple proof of the regularity of weak solutions of periodic strongly elliptic linear equations. In this paper we prove a similar regularity theorem for periodic nonlinear equations under the assumption of strong ellipticity for the composition of the given nonlinear operator with certain linear operators. Let A be a nonlinear differential operator of order 2m given by