Lagrange formula for differential operators on a tree-graph and the resolvents of well-posed restrictions of operator

2017 ◽  
Author(s):  
Nurbol Koshkarbayev ◽  
Baltabek Kanguzhin
Keyword(s):  
Differential Operators ◽  
Tree Graph ◽  
Lagrange Formula ◽  
Well Posed

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

2014 ◽  
Vol 11 (01) ◽  
pp. 185-213 ◽  
Author(s):  
TATSUO NISHITANI
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Energy Estimates ◽  
Characteristic Set ◽  
Finite Speed ◽  
Hyperbolic Operators ◽  
New Energy ◽  
The Cauchy Problem ◽  
Well Posed

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

scholarly journals On the Cauchy problem for Dt2 − Dx(b(t)a(x))Dx

2020 ◽  
Vol 17 (01) ◽  
pp. 75-122
Author(s):  
Ferruccio Colombini ◽  
Tatsuo Nishitani
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Second Order ◽  
Gevrey Class ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Well Posed ◽  
Two Independent Variables

We consider the Cauchy problem for second-order differential operators with two independent variables [Formula: see text]. Assuming that [Formula: see text] is a nonnegative [Formula: see text] function and [Formula: see text] is a nonnegative Gevrey function of order [Formula: see text], we prove that the Cauchy problem for [Formula: see text] is well-posed in the Gevrey class of any order [Formula: see text] with [Formula: see text].

Nonlinear diffusion of scalar images using well-posed differential operators

1994 ◽  
Author(s):  
Niessen ◽  
ter Haar Romeny ◽  
Florack ◽  
Salden ◽  
Viergever
Keyword(s):  
Nonlinear Diffusion ◽  
Differential Operators ◽  
Well Posed

Boundary value problems for the Laplace tidal wave equation

1990 ◽  
Vol 428 (1874) ◽  
pp. 157-180 ◽  
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Eigenvalue Problem ◽  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Tidal Wave ◽  
Transformation Theory ◽  
Sturm Liouville ◽  
Well Posed

This paper discusses the eigenvalue problem associated with the Laplace tidal wave equation (LTWE) given, for μ ϵ (—1,1), by 1 − μ 2 μ 2 − τ 2 y ′ ( μ ) ′ + 1 μ 2 − τ 2 s τ μ 2 + τ 2 μ 2 − τ 2 + s 2 1 + μ 2 y ( μ ) = λ y ( μ ) , ( LTWE ) where s and τ are parameters, with s an integer and 0 < τ < 1, and λ determines the eigenvalues. This ordinary differential equation is derived from a linear system of partial differential equations, which system serves as a mathematical model for the wave motion of a thin layer of fluid on a massive, rotating gravitational sphere. The problems raised by this differential equation are significant, for both the analytic and numerical studies of Sturm-Liouville equations, in respect of the interior singularities, at the points ± τ , and of the changes in sign of the leading coefficient (1 - μ 2 )/( μ 2 - τ 2 ) over the interval (-1, 1). Direct sum space methods, quasi-derivatives and transformation theory are used to determine three physically significant, well-posed boundary value problems from the Sturm-Liouville eigenvalue problem (LTWE), which has singular end-points ± 1 and, additionally, interior singularities at ± τ . Self-adjoint differential operators in appropriate Hilbert function spaces are constructed to represent each of the three well-posed boundary value problems derived from LTWE and it is shown that these three operators are unitarily equivalent. The qualitative nature of the common spectrum is discussed and finite energy properties of functions in the domains of the associated differential operators are studied. This work continues the studies of LTWE made by earlier workers, in particular Hough, Lamb, Longuet-Higgins and Lindzen.

scholarly journals ON THE ESSENTIAL SPECTRA OF GENERAL DIFFERENTIAL OPERATORS

1999 ◽  
Vol 30 (2) ◽  
pp. 105-126
Author(s):  
SOBHY EL-SAYED IBRAHIM
Keyword(s):  
Discrete Spectrum ◽  
Differential Operators ◽  
Integral Operators ◽  
Essential Spectra ◽  
Limit Circle ◽  
All Solutions ◽  
Limit Circle Case ◽  
Complex Coefficients ◽  
Well Posed ◽  
Circle Case

In this paper, it is shown in the cases of one and two singular end-points and when all solutions of the equation $M[u]-\lambda uw=0$, and its adjoint $M^+[v] -\lambda wv = 0$ are in $L_w^2 (a, b)$ (the limit circle case) with $f\in L^2_w(a,b)$ for $M[u]-\lambda wu=wf$ that all well-posed extensions of the minimal operator $T_0(M)$ generated by a general ordinary quasi-differential expression $M$ of $n$-th order with complex coefficients have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly slovable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression $M$ studied in [1] and [12], and also extend those proved in [8] in the case of one singular end-point of the interval [a,b).

Well-posed problems in a layer with differential operators in boundary conditions

1992 ◽  
Vol 44 (8) ◽  
pp. 983-989 ◽  
Author(s):  
L. V. Fardigola
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Well Posed

LAGRANGE FORMULA FOR DIFFERENTIAL OPERATORS AND SELF-ADJOINT RESTRICTIONS OF THE MAXIMAL OPERATOR ON A TREE

2019 ◽  
Vol 10 (1) ◽  
pp. 16-29
Author(s):  
Baltabek Kanguzhin, ◽  
◽  
Lyailya Zhapsarbayeva ◽  
Zhumabay Madibaiuly ◽  
◽  
...  
Keyword(s):  
Maximal Operator ◽  
Differential Operators ◽  
Lagrange Formula

scholarly journals Fractional Evolution Equations Governed by Coercive Differential Operators

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Fu-Bo Li ◽  
Miao Li ◽  
Quan Zheng
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Evolution Equations ◽  
Differential Operators ◽  
Fractional Evolution Equations ◽  
Resolvent Family ◽  
Well Posed

This paper is concerned with evolution equations of fractional orderDαu(t)=Au(t);u(0)=u0,u′(0)=0,whereAis a differential operator corresponding to a coercive polynomial taking values in a sector of angle less thanπand1<α<2. We show that such equations are well posed in the sense that there always exists anα-times resolvent family for the operatorA.

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