Order Partial Differential Operators Research Articles

TheXsθspaces and unique continuation for solutions to the semilinear wave equation

The aim of this paper is twofold. First, we initiate a detailed study of the so-called Xs θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and Lp (Lq ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new Lp → Lq Carleman estimates, derived using the Xs θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.

Local Solvability of First Order Linear Operators with Lipschitz Coefficients in Two Variables

We prove that a first order partial differential operator of principal type with Lipschitz coefficients in the principal part and measurable locally bounded zero order term is locally solvable in R2 if it verifies condition (P).

The kernel of the neumann operator for a strictly diffractive analytic problem

We consider p a partial differential operator of order 2 and Rn= ω+ ∪ ∂ω ∪ ω− a partition of Rn , such that (p, ω+) admits a strictly diffractive point (in the sense of Friedlander and Melrose). We compute the trace and the trace of the normal derivative on ∂ω of the solution u of the diffraction problem pu= 0 in ω+ u satisfying a mixed boundary condition on ∂ω, ∂ω analytic. That is done using the construction by Lebeau of a Gevrey 3 parametrix in the neighborhood of the strictly diffractive point. This result generalizes, for a mixed boundary condition, the Gevrey 3 propogation result of Lebeau. We use this result to compute the leading term in the shadow region of the diffracted wave outside a strictly convex analytical obstacle with a mixed boundary condition and a given incoming wave.

Gevrey Hypoellipticity for Maximally Hypoelliptic Fourth Order Partial Differential Operators

Local C∞ and (sharp) Gevrey hypoellipticity are proved for maximally hypoelliptic fourth order partial differential operators of the form P = D|c|<4 aa(x)Xa in R where the independent real vector fields Xj,j<n - l, satisfy the Hormander condition that at most k brackets of the Xj spans the tangent space at each point.

On some geometry associated with a generalised Toda lattice

We define the notion of Darboux integrability for linear second order partial differential operators,.We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

On the Riesz Transforms for Gaussian Measures

Let B be an n × n positive-definite symmetric matrix, and L B the second order partial differential operator in R n defined by L Bu = 1 2 Δ − Bx · ∇ u. The operator L B is self-adjoint with respect to the Gaussian probability measure γ B n ( x) dx, where γ B n ( x) = C n, B exp(− Bx · x). In this paper a class of Riesz′s transforms naturally associated with L B is studied. It is shown that these transformations are bounded in the spaces L p γ B n ( R n ), p > 1, with a constant independent of the dimension an depending only on p and the number of different eigenvalues of the matrix B. The proof of this result is analytic and uses appropriate square-functions defined in terms of semigroups of operators related to L B and the Littlewood-Paley-Stein theory. The result contains as a particular case some inequalities proved by Meyer and Gundy using probabilistic methods.

The stability of rational approximations of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators.

On certain partial differential operators of finite odd type

Let P be a linear partial differential operator of order m > 1 with real-analytic coefficients defined in Q, an open set of Rn , and let y be in the cotangent space of Q minus the zero section. If P is of odd finite type k and if the Hormander numbers are 1 = ki < k2, k2 odd, then P is analytic hypoelliptic at y . These operators are not semirigid. INTRODUCTION In this paper we present a microlocal analytic hypoellipticity result for a class of linear partial differential operators of finite type with real-analytic coefficients. The assumption is that the type of the operator is an odd positive integer k and that there are two Hormander numbers k1, k2 with 1 = k, < k2 and k2 is odd. The main difference between this result and previous results obtained by the author is that the operators here may not be semirigid. Therefore we view it as a partial result of the more general problem of giving necessary and sufficient conditions for microlocal analytic hypoellipticity of operators of finite type. For CR structures a version of this problem has been recently solved by Tumanov [10] (sufficient conditions) and Baouendi-Rothschild [1] (necessary conditions). In the analytic case the necessity was proved in [2]. The condition used in [10] is a condition at a point in the base space called the minimality condition. An analytic CR structure is minimal at a point if it is of finite type at this point . For C? CR structures finite type implies minimality but not vice versa. The finite type condition used in this paper is microlocal. The new difficulty that arises in the proof of this result is that the phase function may contain terms of degree less than k which is the degree of the terms that result from the type of the operator. To make these lower degree terms negligible, we choose a special good contour and we use a more general sufficient condition than the one used in [6]. Received by the editors December 18, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 35H05; Secondary 58G15. The author was partially supported by NSF. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Determinants of Laplace-like operators on Riemann surfaces

We calculate determinants of second order partial differential operators defined on Riemann surfaces of genus greater than one using a relation between Selberg's zeta function and functional determinants. In addition, we perform a calculation of these determinants directly using Selberg's trace formula, and compare our results with previous computations which followed the latter route.

Causal fundamental solutions for the slow flow of a micropolar fluid

Causal Fundamental solutions for the slow two dimensional flow of a micropolar fluid are constructed. Explicit solution requires the factorisation of a fourth order partial differential operator into two quadratic operators which is achieved under a certain condition on the parameters of the problem. The use of the fundamental solutions in unsteady flow problems is indicated

Generalized Brownian functionals and the solution to a stochastic partial differential equation

A class of generalized Wiener functionals, related to those of Hida and Watanabe, is introduced and the Malliavin calculs associated with these functionals is developed. These notions are applied to the derivation of a solution to the stochastic partial differential equation ∂Y ∂t = LY + Y · η + ψ , where L is a second order partial differential operator in the n space variables, η denotes a white noise in the ( n + 1) space-time variables, Y · η denotes a Skorohod type stochastic integral, and ψ is a non-random function of the ( n + 1) independent variables.

Some aspects of differential geometry associated with hypoelliptic second order operators

I investigate some aspects of the geometry of the ^characteristics of a class of hypoelliptic second order partial differential operators. The resulting geometry looks quite a bit like Riemannian geometry, although with interesting differences. 1. Introduction. In this paper, we investigate the relationship between certain problems in the calculus of variations and the geometry of the bicharacteristics of a second order hypoelliptic operator. Let M denote a connected C°° manifold of dimension m. Let Δ denote a second order hypoelliptic partial differential operator on M. We assume that the set of second order zeros of the principal symbol of Δ is a smooth submanifold of T*M. Since Δ is second order it is always possible, at least locally, to find a function V(x) and vector fields {gi}ni=0 such that Δ = ]C?=i Cfgf + go + V{x) for some constants cι = ±1. Hόrmander [19] has shown that a sufficient condition that Δ be hypoelliptic is that the sign of the c, doesn't change and that the evaluation map on vector fields is at each point x e M onto TXM when restricted to the Lie algebra of vector fields (over R) generated by the vector fields {gi}ni=0. In this paper I will be interested in hypoelliptic operators which satisfy the stronger condition that the Lie algebra generated by {g/} =1 is onto TM. Now, because Δ is of second order, Δ defines a quadratic form G* on T*M: if fx, f2 e C°°(M) and fx (JC) = f2(x) = 0, then

Polygon discretization of the loop-space equation

Methods for studying equations on the loop space are developed using interpretation of the loop equation of many-color QCD as the functional (inhomogeneous) Laplace equation. We discretize the loop space by M-vertex polygons to approximate the functional laplacian by a (finite-dimensional) second order partial differential operator. A path-integral representation for the Green function of the functional laplacian is obtained as the M → ∞ limit of the Green function of the approximating operator. Finally, we speculate on the possibility of extending this approach to relativistic string.

On some classes of nonhypoelliptic second order partial differential operators

Soit ω un ensemble ouvert de R n , n≥1, contenant l'origine et soit T>0. On etudie la non-hypoellipticite d'operateurs differentiels d'ordre 2

Commutator Theory on Hilbert Space

Commutator theory has its origins in constructive quantum field theory. It was initially developed by Glirnm and Jaffe [7] as a method to establish self-adjointness of quantum fields and model Hamiltonians. But it has subsequently proved useful for a variety of other problems in field theory [17] [15] [8] [3], quantum mechanics [5], and Lie group theory [6]. Despite all these detailed applications no attempt appears to have been made to systematically develop the theory although reviews have been given in [22] and [9]. The primary aim of the present paper is to partially correct this situation. The secondary aim is to apply the theory to the analysis of first and second order partial differential operators associated with a Lie group.

On First Order Partial Differential Operators on Bounded Regions of the Plane

Mathematische NachrichtenVolume 131, Issue 1 p. 33-47 Article On First Order Partial Differential Operators on Bounded Regions of the Plane Jürgen Friedrich, Jürgen Friedrich Sektion Mathematik Karl-Marx-Universität Leipzig Karl-Marx-Platz DDR-Leipzig 7010, LeipzigSearch for more papers by this author Jürgen Friedrich, Jürgen Friedrich Sektion Mathematik Karl-Marx-Universität Leipzig Karl-Marx-Platz DDR-Leipzig 7010, LeipzigSearch for more papers by this author First published: 1987 https://doi.org/10.1002/mana.19871310104Citations: 4AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume131, Issue11987Pages 33-47 RelatedInformation

On a class of second order partial differential operators of non-principal type

On a class of second order partial differential operators of non-principal type

On the well-posedness of a 𝐶^{∞} Goursat problem for a partial differential operator of order greater than two

We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form \[ C 2 ( D x , D y ) D t + C 3 ( D x , D y ) {C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y}) \] to be C ∞ {C^\infty } -well posed, showing at the same time that a necessary and sufficient condition of Hasegawa cannot be extended. The result can be generalised to operators of higher orders but leads to cumbersome conditions; nevertheless, we show that the condition of Hasegawa is also not sufficient in this case.

Integral transforms by green's function on rn

For and for nonnegative integersWe set Let be the linear partial differential operator of order m satisfying p(ξ) ≠ 0 for all ξ ε Rn. Then, for the uniquely determined tempered fundamental solution , we consider the integral transform and investigate the relationship between f and u. In particular, we establish some isometrical identities and inverse formulae between f and u for functions f ε L2(Rn).

Singular Cauchy problems for second order partial differential operators with non-involutory characteristics

Soient (x,y) les variables de ( n+1 , Ou X∈EC et y=(y 1 ,y')∈C×C n−1 , et (ξ,η) les variables duales de (x,y). On considere des operateurs aux derivees partielles ecrits sous la forme suivante: P(x,y,δ'δx,δ'δy)=Σ i+K1≤2 x K(i, α ) a i α(x,y)(δ/δx) i (δ/δy)α