scholarly journals On the loss of compactness in the vectorial heteroclinic connection problem

2016 ◽  
Vol 146 (3) ◽  
pp. 595-608 ◽  
Author(s):  
Nikos Katzourakis
Keyword(s):  
A Priori ◽  
Functional Space ◽  
Alternative Proof ◽  
Local Minima ◽  
Minimizing Sequences ◽  
Connection Problem ◽  
Heteroclinic Connection ◽  
Image Position ◽  
Constraint Method ◽  
Loss Of Compactness

We give an alternative proof of the theorem of Alikakos and Fusco concerning existence of heteroclinic solutions U : ℝ → ℝN to the systemHere a± are local minima of a potential W ∈ C2(ℝN) with W(a±) = 0. This system arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of Alikakos and Fusco and establishes existence by analysing the loss of compactness in minimizing sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.

scholarly journals A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model.

2021 ◽  
Author(s):  
Serge Nicaise ◽  
Ismail Merabet ◽  
Rayhana REZZAG BARA
Keyword(s):  
Shell Model ◽  
A Priori ◽  
Functional Space ◽  
Finite Element Approximation ◽  
Posteriori Error ◽  
Error Estimator ◽  
Element Approximation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
A Priori Error Estimate

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized version is then considered. We obtain a non robust a priori error estimate of this penalized formulation, but a robust one is obtained for its mixed formulation. Moreover, we present a reliable and efficient a posteriori error estimator of the penalized formulation. Numerical tests are included that confirmthe efficiency of our residual a posteriori estimator.

The existence of a positive solution to asymptotically linear scalar field equations

2000 ◽  
Vol 130 (1) ◽  
pp. 81-105 ◽  
Author(s):  
G. Li ◽  
H.-S. Zhou
Keyword(s):  
Positive Solution ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Field Equations ◽  
Usual Condition ◽  
Image Position ◽  
Scalar Field Equations ◽  
Constraint Method ◽  
Linear Scalar

We consider the following elliptic equation: where m > 0, f(x, u)/u tends to a positive constant as u → + ∞. Here, the nonlinear term f(x, u) does not satisfy the usual condition, that is, for some θ > 0, which is important in using the mountain pass theorem. The aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition. Furthermore, if f(x, u) ≡ f(u), we also prove that the above problem has a ground state by using the artificial constraint method.

Asymptotics for Minimal Discrete Riesz Energy on Curves in ℝd

2004 ◽  
Vol 56 (3) ◽  
pp. 529-552 ◽  
Author(s):  
A. Martínez-Finkelshtein ◽  
V. Maymeskul ◽  
E. A. Rakhmanov ◽  
E. B. Saff
Keyword(s):  
Jordan Curve ◽  
Hausdorff Measure ◽  
Compact Sets ◽  
Minimizing Sequences ◽  
Point Sets ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Riesz Kernel ◽  
Riesz Energy ◽  
Image Position

AbstractWe consider the s-energy for point sets 𝒵 = {𝒵k,n: k = 0, …, n} on certain compact sets Γ in ℝd having finite one-dimensional Hausdorff measure,is the Riesz kernel. Asymptotics for the minimum s-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for s ≥ 1, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as n → ∞.

scholarly journals Two Enumerative Proofs of an Identity of Jacobi

1966 ◽  
Vol 15 (1) ◽  
pp. 67-71 ◽  
Author(s):  
C. Sudler
Keyword(s):  
Alternative Proof ◽  
Ferrers Graph ◽  
Image Position ◽  
Durfee Square

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namelyHere, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

XXIV.—On the Equation of Motion of a Free Particle in the Expanding Universe of Kinematical Relativity

1943 ◽  
Vol 61 (3) ◽  
pp. 288-297
Author(s):  
E. A. Milne
Keyword(s):  
Additional Assumption ◽  
A Priori ◽  
Free Particle ◽  
Equation Of Motion ◽  
Expanding Universe ◽  
Image Position ◽  
Special Case

In a recent paper in these Proceedings, Dr G. C. McVittie has published some criticisms of kinematical relativity. These criticisms are to a large extent based on his formula (4.10), namely,It must be stated at the outset that McVittie's interpretation of his derivation of (1) as a derivation of “Milne's formula for the acceleration of a ‘free particle moving in the presence of a substratum,’ for the special case of one spatial co-ordinate only” is wrong. McVittie does not derive the result, as he claims, from what he calls the “axioms of kinematical relativity” alone; he deduces it from these axioms together with an additional assumption, which is equivalent to begging the answer to the whole problem it was my object to solve. Instead of considering a free particle, as I did—that is, a particle whose motion we do not a priori know—he prescribes a priori the motion of his particle as being constrained to obey the rule, in his notation,

scholarly journals Irrationality proofs à la Hermite

2011 ◽  
Vol 95 (534) ◽  
pp. 407-413
Author(s):  
Li Zhou
Keyword(s):  
Recurrence Relation ◽  
Alternative Proof ◽  
Image Position

In [1] Niven used the integralto give a well-known proof of the irrationality of π. Recently Zhou and Markov [2] used a recurrence relation satisfied by this integral to present an alternative proof which may be more direct than Niven's.Niven did not cite any references in [1] and thus the origin or Hn seems rather mysterious and ingenious. However if we heed Abel's advice to ‘study the masters’, we find that Hn emerged much more naturally from the great works of Lambert [3] and Hermite [4].

Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions

1988 ◽  
Vol 110 (3-4) ◽  
pp. 183-198 ◽  
Author(s):  
R. Iannacci ◽  
M.N. Nkashama ◽  
P. Omari ◽  
F. Zanolin
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Periodic Solutions ◽  
Homogeneous Problem ◽  
A Priori ◽  
Critical Values ◽  
Liénard Equations ◽  
Jumping Nonlinearities ◽  
Existence Of Periodic Solutions ◽  
Image Position

SynopsisThis paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equationThe key assumptions relate the asymptotic behaviour as x →± ∞of g(t; x)/x to the “critical values” of the positively 1-homogeneous problemNo condition on f, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

Steady symmetric vortex pairs and rearrangements

1988 ◽  
Vol 108 (3-4) ◽  
pp. 269-290 ◽  
Author(s):  
G. R. Burton
Keyword(s):  
Elliptic Equation ◽  
A Priori ◽  
Vortex Pair ◽  
Semilinear Elliptic Equation ◽  
Symmetric Pair ◽  
Vorticity Field ◽  
Vortex Pairs ◽  
Semilinear Elliptic ◽  
Image Position ◽  
Increasing Function

SynopsisWe prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equationin a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.

A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation

1990 ◽  
Vol 114 (3-4) ◽  
pp. 243-258 ◽  
Author(s):  
J. G. B. Byatt-Smith ◽  
A. M. Davie
Keyword(s):  
Asymptotic Expansion ◽  
Integral Equations ◽  
Complex Plane ◽  
Existence And Uniqueness ◽  
Asymptotic Estimate ◽  
Contraction Mapping ◽  
Contraction Mapping Theorem ◽  
Oscillatory Solutions ◽  
Connection Problem ◽  
Image Position

SynopsisThe equationhas a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansionEach term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equationthe equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimatewhereThe proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansionswhere Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+

VIII.—The Schrödinger Equation with a Periodic Potential

1971 ◽  
Vol 69 (2) ◽  
pp. 125-131
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Floquet Theory ◽  
Periodic Potential ◽  
Eigenvalue Problems ◽  
Alternative Proof ◽  
Conditional Stability ◽  
Image Position

SynopsisThe differential equationin N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.

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