First-order Partial Differential Operators Research Articles

Integro-Differential Equation with a Higher-Order Two-Dimensional Whitham Operator

We examine the unique solvability of an initial-value problem for a certain higher-order quasilinear partial integro-differential equation with a degenerate kernel. Expressing the higher-order partial integro-differential operator as the superposition of first-order partial differential operators, we represent the integro-differential equation considered as an ordinary integro-differential equations that describes the change of the unknown function along characteristics. Using the method of successive approximations, we prove the unique solvability of the initial-value problem and obtain an estimate for the convergence rate of the Picard iterative process.

Heat semigroups on Weyl algebra

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators ∇i± forming the Lie algebra [∇j±,∇k±]=iRjk± and [∇j+,∇k−]=i12(Rjk++Rjk−) with some anti-symmetric matrices Rij± and define the corresponding Laplacians Δ±=g±ij∇i±∇j± with some positive matrices g±ij. We show that the heat semigroups exp(tΔ±) can be represented as a Gaussian average of the operators expξ,∇± and use these representations to compute the product of the semigroups, exp(tΔ+)exp(sΔ−) and the corresponding heat kernel.

Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.

Начальная задача для нелинейного интегро-дифференциального уравнения с гиперболическим оператором высшего порядка и с отражением аргумента

In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.

Начальная задача для квазилинейного интегро-дифференциального уравнения в частных производных высшего порядка с вырожденным ядром

High-order partial differential equations are of great interest when it comes to physical applications. Many problems of gas dynamics, elasticity theory and the theory of plates and shells are reduced to the consideration of high-order partial differential equations. This paper studies the one-valued solvability of the initial value problem for a nonlinear partial integro-differential equation of an arbitrary order with a degenerate kernel. The expression of higher-order partial differential equations as a superposition of first-order partial differential operators has allowed us to apply methods for solving first-order partial differential equations. First-order partial differential equations can be locally solved by the methods of the theory of ordinary differential equations, reducing them to a characteristic system. The existence and uniqueness of the solution to this problem is proved by the method of successive approximation. An estimate of convergence of the iterative Picard process is obtained. The stability of the solution from the second argument of the initial value problem is shown.

Weak Lower Semicontinuity of Integral Functionals and Applications

Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. in 1952 and N.G. Meyers in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.

On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators

Abstract The contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.

Hodge-type Decomposition for Time-dependent First-order Parabolic Operators with Non-constant Coefficients: The Variable Exponent Case

In this paper we present a Hodge-type decomposition for variable exponent spaces. More concretely, we address some time-dependent parabolic firstorder partial differential operators with non-constant coefficients, where one of the components is the kernel of the parabolic-type Dirac operator. This decomposition is presented over different types of domains in the n-dimensional Euclidean space n-dimensional Euclidean space \({\mathbb{R}^{n}}\). The case of the time-dependent Schrodinger operator is included as a special case within this context.

Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients

In this paper, we present a Hodge decomposition for the \(L_p\)-space of some parabolic first-order partial differential operators with non-constant coefficients. This is done over different types of domains in Euclidean space \(\mathbb{R }^n\) and on some conformally flat cylinders and the \(n\)-torus associated with different spinor bundles. Initially, we apply a regularization procedure in order to control the non-removable singularities over the hyperplane \(t=0\). Using the setting of Clifford algebras combined with a Witt basis, we introduce some specific integral and projection operators. We present an \(L_p\)-decomposition where one of the components is the kernel of the regularized parabolic Dirac operator with non-constant coefficients. After that, we study the behavior of the solutions and the validity of our results when the regularization parameter tends to zero. To round off, we give some analytic solution formulas for the special context of domains on cylinders and \(n\)-tori.

Editorial

Editorial

Local Solvability in Lpof First-Order Linear Operators

We prove that first-order partial differential operators of principal type with smooth coefficients are locally solvable in L p , 1< p<∞, if they satisfy condition ( P ).

Intermittency-Type Estimates for Some Nondegenerate SPDE'S

In this paper we prove some intermittency-type estimates for the stochastic partial differential equation $du = \mathscr{L}u dt + \mathscr{M}_lu\circ dW^l_t$, where $\mathscr{L}$ is a strongly elliptic second-order partial differential operator and the $\mathscr{M}_l$'s are first-order partial differential operators. Here the $W^l$'s are standard Wiener processes and $\circ$ denotes Stratonovich integration. We assume for simplicity that $u(0,\cdot) \equiv 1$. Our interest here is the behavior of $\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ for large time and large $p$; more specifically, our interest is the growth of $(p^2t)^{-1}\log\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ as $t$, then $p$, become large.

Spectral properties of the nonstationary two-dimensional Dirac operator

We study the self-adjointness and spectral structure of general first-order partial differential operators (including the nonstationary two-dimensional Dirac operator) and the structure of the resolvent of the two-dimensional Dirac operator.

Nonlinear connections for curved twistor spaces

A holomorphic connection on (1, 0)-vector fields which is intrinsically defined on any curved twistor space is described. Although it is a local operator, it is given in terms of the nonlocal geometry of the twistor space corresponding to the local geometry of the spacetime. The connection is represented in local coordinates by a system ofnonlinear first-order partial differential operators. It has torsion but no curvature. A parallelism is given explicitly, and an example is computed.

Liouville's theorem for first-order partial differential operators

Liouville's theorem for first-order partial differential operators

Remarks on an inequality of Schulenberger and Wilcox

J. R. Schulenberger and C. H. Wilcox[1], [2], have proven a coerciveness inequality for a class of nonelliptic first-order partial differential operators of the form Λ = =E −1 AjDj, where Aj (j=1, ..., n) is a constant m × m Hermitian matrix, E=E(x) is uniformly positive definite, bounded, and uniformly differentiable Hermitian m × m matrix, and where the symbol Λ(p, x)=E(x) −1 Ajpj has constant rank for all p e Rn − {0} and x e Rn. They prove coerciveness on N(Λ)⊥, the orthogonal complement of the null space N(Λ) relative to the inner product $$\left\langle {u, v} \right\rangle = \int\limits_{R^n } {u(x) \cdot E(x)v(x)dx} $$ . Their proof is rather long, a simpler and shorter proof is given here. This proof leads naturally to a generalization of their results to the case where the Aj's need not be Hermitian.