Order Linear Partial Differential Operators Research Articles

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

On the solutions of Fermat type quadratic trinomial equations in ℂ2 generated by first order linear C-SHIFT and partial differential operators

On essential self-adjointness for first order differential operators on domains in $\mathbb R^d$

We consider general symmetric systems of first order linear partial differential operators on domains \Omega \subset \mathbb R^d , and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to C^1 (\Omega) and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for certain wave propagation problems of classical physics.

Sufficient conditions for local solvability of some degenerate pdo with complex subprincipal symbol

We will show a local solvability result for a class of degenerate second order linear partial differential operators with a complex subprincipal symbol. Due to the form of the operators in the class the subprincipal symbol is invariantly defined and we shall give sufficient conditions for the local solvability to hold involving the real and the imaginary part of the latter. Under suitable conditions we will prove that the class under consideration is L^2 to L^2 locally solvable.

On the Stability of the 𝐶^{∞}-Hypoellipticity under Perturbations

We study the problem of perturbations of C ∞ C^\infty -hypoelliptic operators by lower order terms. After giving several examples which show many different possibilities, we then prove a stability result which shows that a hypoelliptic linear partial differential operator P P which loses finitely many derivatives and whose formal adjoint P ∗ P^* is still hypoelliptic (but with no assumption on the loss of derivatives) remains hypoelliptic with the same loss of derivatives after perturbation by a lower order linear partial differential operator (whose order depends on the loss of derivatives).

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.

Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated with second order linear degenerate elliptic partial differential operators X with rough coefficients, of the form X=−div(P∇)+HR+S′G+F, where the n×n matrix function P=P(x) is nonnegative definite and allowed to degenerate, R, S are families of subunit vector fields, G, H are vector valued functions and F is a scalar function. We operate in a geometric homogeneous space setting and we assume the validity of certain Sobolev and Poincaré inequalities related to a symmetric nonnegative definite matrix of weights Q=Q(x) that is comparable to P; we do not assume that the underlying measure is doubling. We give a maximum principle for weak solutions of Xu≤0, and we follow this with a result describing a relationship between compact projection of the degenerate Sobolev space QH1,p, related to the matrix of weights Q, into Lq and a Poincaré inequality with gain adapted to Q.

A remark on the stability of $$C^\infty $$ C ∞ -hypoellipticity under lower-order perturbations

We study the problem of perturbations of $$C^\infty $$ -hypoelliptic operators by lower order terms. We show that hypoellipticity with a finite loss of derivatives of a linear partial differential operator $$P$$ (with no assumption on the characteristic set), along with its formal adjoint $$P^*$$ , is stable under perturbations by lower order linear partial differential operators whose order depends on the loss of derivatives.

Solvability and stationarity for the optimal control of variational inequalities with point evaluations in the objective functional

AbstractMotivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non‐convex and non‐differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C‐stationarity in the associated Sobolev space setting. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let Ω be a domain in ℝ N and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup \((e^{-t A_{V}})_{t\geqslant0}\) on L 2(Ω,dx), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p -contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.

On Global Hypoellipticity

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

On a class of hypoelliptic operators with unbounded coefficients in $R^N$

We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Holder-) continuous functions. As a consequence, we obtain Holder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.

Formal Gevrey Class of Formal Power Series Solution for Singular First Order Linear Partial Differential Operators

Formal Gevrey Class of Formal Power Series Solution for Singular First Order Linear Partial Differential Operators

Reciprocal diffusions and symmetries

We prove that the partition of Markov diffusions into reciprocal classes is equivalent to the partition of the one parameter symmetry groups of second order linear partial differential operators under gauge equivalence. In order to do so we explicit the system of Determining Equations which characterize the symmetries of a second order linear p.d.e. and we prove that the set of compatibility conditions for this system only depends on the reciprocal class. We then show that the reciprocal class of a Markov diffusion is preserved under the symmetries of the associated partial differential operator

Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Chebyshev polynomials of the first and the second kind in n n variables z 1 , z 2 , … , z n {z_1},{z_2}, \ldots ,{z_n} are introduced. The variables z 1 , z 2 , … , z n {z_1},{z_2}, \ldots ,{z_n} are the characters of the representations of S L ( n + 1 , C ) SL(n + 1,{\mathbf {C}}) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates z 1 , z 2 , … , z n {z_1},{z_2}, \ldots ,{z_n} and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.

A duality theorem for solutions of elliptic equations

Let L be a second order linear partial differential operator of elliptic type on a domain Ω of ℝm with coefficients in C∞(Ω). We consider the linear space of all solutions of the equation Lu = 0 on Ω with the topology of uniform convergence on compact subsets and describe the topological dual of this space. It turns out that this dual may be identified with the space of solutions of an adjoint equation near the boundary modulo the solutions of this adjoint equation on the entire domain.

Factored Product Expansions of Solutions of Nonlinear Differential Equations

Lie transformations are used to represent solutions of initial value problems for systems of nonlinear ordinary differential equations as exponentials of first order linear partial differential operators. These exponentials are then expanded using an analog of the usual exponential identities. This expansion is called the factored product expansion. Such expansions have been found useful in the study of magnetic and optical lenses.

The geometry of second order linear partial differential operators

The geometry of second order linear partial differential operators

On a parametrix in some weak sense of a first order linear partial differential operator with two independent variables

On a parametrix in some weak sense of a first order linear partial differential operator with two independent variables

On a parametrix in some weak sense of a first order linear partial differential operator with two independent variables

On a parametrix in some weak sense of a first order linear partial differential operator with two independent variables

Note on Hypoellipticity of a First Order Linear Partial Differential Operator

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).