scholarly journals On the Gauss-Green theorem

1968 ◽  
Vol 8 (3) ◽  
pp. 385-396 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Lebesgue Measure ◽  
Inner Product ◽  
Open Set ◽  
The Individual ◽  
Image Position ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Dimensional Lebesgue Measure ◽  
Lebesgue Integration ◽  
Riemann Integration

In a previous paper [1], Green's theorem for line integrals in the plane was proved, for Riemann integration, assuming the integrability of Qx−Py, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx and Py. In the present paper, this result is extended to a proof of the Gauss-Green theorem for p-space (p ≥ 2), for Lebesgue integration, under analogous hypotheses. The theorem is proved in the form where Ω is a bounded open set in Rp (p-space), with boundary Ω; g(x) =(g(x1)…, g(xp)) is a p-vector valued function of x = (x1,…,xp), continuous in the closure of Ω; μv,(x) is p-dimensional Lebesgue measure; v(x) = (v1(x),…, vp(x)) and Φ(x) are suitably defined unit exterior normal and surface area on the ‘surface’ ∂Ω and g(x) · v(x) denotes inner product of p-vectors.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1981 ◽  
Vol 33 (1) ◽  
pp. 229-246 ◽  
Author(s):  
Calvin D. Ahlbrandt ◽  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operators ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Inner Product ◽  
Dimensional Vector ◽  
Selfadjoint Extension ◽  
Image Position ◽  
Vector Valued Function ◽  
Necessary And Sufficient ◽  
Vector Valued

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

scholarly journals On approximating Lebesgue integrals by Riemann sums

1991 ◽  
Vol 33 (2) ◽  
pp. 129-134
Author(s):  
Szilárd GY. Révész ◽  
Imre Z. Ruzsa
Keyword(s):  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Function ◽  
Riemann Sums ◽  
Function Classes ◽  
Open Set ◽  
Measurable Set ◽  
Image Position ◽  
Good Sequences

If f is a real function, periodic with period 1, we defineIn the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ ℝ is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of ℒp is ℒp [0, 1]. For all real xwe haveif f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ ℒ1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ ℒp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) towhere S ⊂ ℕ is some “good” increasing subsequence of ℕ. Naturally, for different function classes ℱ ⊂ ℒ1 we get different meanings of being good. That is, we introduce the class of ℱ-good sequences as

scholarly journals Almost periodic generalized functions

1983 ◽  
Vol 94 (1) ◽  
pp. 149-166
Author(s):  
H. Burkill ◽  
B. C. Rennie
Keyword(s):  
Generalized Functions ◽  
Inner Product ◽  
Almost Periodic ◽  
Test Functions ◽  
Complex Functions ◽  
Entire Complex ◽  
Order Of Magnitude ◽  
The Individual ◽  
Image Position ◽  
Derivatives Of

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be

The Measure of Plane Sets

1943 ◽  
Vol 39 (1) ◽  
pp. 51-53
Author(s):  
P. A. P. Moran
Keyword(s):  
Finite Number ◽  
Coordinate System ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Two Dimensional ◽  
Image Position ◽  
Bounded Sets ◽  
Dimensional Lebesgue Measure

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

Rearrangements and polar factorisation of countably degenerate functions

1998 ◽  
Vol 128 (4) ◽  
pp. 671-681 ◽  
Author(s):  
G. R. Burton ◽  
R. J. Douglas
Keyword(s):  
Level Sets ◽  
Smooth Boundary ◽  
Positive Lebesgue Measure ◽  
Nondegeneracy Condition ◽  
Open Set ◽  
Almost Everywhere ◽  
Alternative Hypotheses ◽  
Monotone Rearrangement ◽  
Vector Valued Function ◽  
Vector Valued

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

2015 ◽  
Vol 93 (2) ◽  
pp. 272-282 ◽  
Author(s):  
JAEYOUNG CHUNG ◽  
JOHN MICHAEL RASSIAS
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Cyclic Subgroup ◽  
Real Banach Space ◽  
Measure Zero ◽  
Stability Theorem ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces

2013 ◽  
Vol 56 (3) ◽  
pp. 853-871 ◽  
Author(s):  
Carlos Lizama ◽  
Rodrigo Ponce
Keyword(s):  
Infinite Delay ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Function Spaces ◽  
Linear Operators ◽  
Image Position ◽  
Vector Valued Function ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractLet A and M be closed linear operators defined on a complex Banach space X and let a ∈ L1(ℝ+) be a scalar kernel. We use operator-valued Fourier multipliers techniques to obtain necessary and sufficient conditions to guarantee the existence and uniqueness of periodic solutions to the equationwith initial condition Mu(0) = Mu(2π), solely in terms of spectral properties of the data. Our results are obtained in the scales of periodic Besov, Triebel–Lizorkin and Lebesgue vector-valued function spaces.

A harmonic quadrature formula characterizing open strips

1993 ◽  
Vol 113 (1) ◽  
pp. 147-151 ◽  
Author(s):  
D. H. Armitage ◽  
C. S. Nelson
Keyword(s):  
Harmonic Function ◽  
Lebesgue Measure ◽  
Quadrature Formula ◽  
Open Subset ◽  
Harmonic Functions ◽  
Mean Value ◽  
Open Ball ◽  
Mean Value Property ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, thenA converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such thatfor every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

The interaction between bulk energy and surface energy in multiple integrals

1994 ◽  
Vol 124 (4) ◽  
pp. 737-756 ◽  
Author(s):  
Andrea Braides ◽  
Alessandra Coscia
Keyword(s):  
Energy Density ◽  
Surface Energy ◽  
Special Functions ◽  
Lower Semicontinuous ◽  
Integral Functional ◽  
Open Set ◽  
Lower Semicontinuous Envelope ◽  
Bulk Energy ◽  
Image Position ◽  
Vector Valued

This paper is devoted to the study of integral functional denned on the spaceSBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the formWe suppose only thatfis finite at one point, and thatgis positively 1-homogeneous and locally bounded on the sets ℝk⊗vm, where {v1,…,vR} ⊂Sn−1is a basis of ℝn. We prove that the lower semicontinuous envelope ofFin theL1(Ω;ℝk)-topology is finite and with linear growth on the wholeBV(Ω;ℝk), and that it admits the integral representationA formula forϕis given, which takes into account the interaction between the bulk energy densityfand the surface energy densityg.

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

1996 ◽  
Vol 119 (1) ◽  
pp. 167-178 ◽  
Author(s):  
Michel L. Lapidus ◽  
Carl Pomerance
Keyword(s):  
Sobolev Space ◽  
Dirichlet Boundary ◽  
Zero Solution ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Smooth Functions ◽  
Open Set ◽  
Finite Dimensional ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

Sign in / Sign up

Export Citation Format

Share Document