scholarly journals On generalized a priori estimates for quasi-elliptic differential operators

1979 ◽  
Vol 4 ◽  
pp. 227-252 ◽  
Author(s):  
Veikko T. Purmonen
Keyword(s):  
Differential Operators ◽  
A Priori Estimates ◽  
A Priori ◽  
Priori Estimates ◽  
Elliptic Differential Operators

The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness

1992 ◽  
Vol 121 (3-4) ◽  
pp. 279-301 ◽  
Author(s):  
John Locker
Keyword(s):  
Differential Operator ◽  
Spectral Theory ◽  
Differential Operators ◽  
A Priori Estimates ◽  
A Priori ◽  
Second Order ◽  
Boundary Values ◽  
Horizontal Strip ◽  
Priori Estimates

SynopsisThis paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L2[0, 1] determined by a second order formal differential operator l = −D2 + pD + q and by independent boundary values B1, B2. The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B1, B2. For Cases 1–4 it is shown that if λ = ρ2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L2[0, 1].

scholarly journals Elliptic Problems with Additional Unknowns in Boundary Conditions and Generalized Sobolev Spaces

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 292
Author(s):  
Anna Anop ◽  
Iryna Chepurukhina ◽  
Aleksandr Murach
Keyword(s):  
Boundary Conditions ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Generalized Solutions ◽  
Bounded Operators ◽  
Priori Estimates

In generalized inner product Sobolev spaces we investigate elliptic differential problems with additional unknown functions or distributions in boundary conditions. These spaces are parametrized with a function OR-varying at infinity. This characterizes the regularity of distributions more finely than the number parameter used for the Sobolev spaces. We prove that these problems induce Fredholm bounded operators on appropriate pairs of the above spaces. Investigating generalized solutions to the problems, we prove theorems on their regularity and a priori estimates in these spaces. As an application, we find new sufficient conditions under which components of these solutions have continuous classical derivatives of given orders. We assume that the orders of boundary differential operators may be equal to or greater than the order of the relevant elliptic equation.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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