scholarly journals Singular sets and the Lavrentiev phenomenon

2015 ◽  
Vol 145 (3) ◽  
pp. 513-533 ◽  
Author(s):  
Richard Gratwick
Keyword(s):  
Variational Problem ◽  
Singular Set ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Strictly Convex ◽  
Superlinear Growth ◽  
Singular Sets

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subsetE⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactlyEbut still admit approximation in energy by smooth functions.

scholarly journals Singular Sets of Some Kleinian Groups (II)

1967 ◽  
Vol 29 ◽  
pp. 145-162 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Singular Set ◽  
Fundamental Domains ◽  
Dimensional Measure ◽  
Singular Sets ◽  
Discontinuous Groups ◽  
Positive Capacity ◽  
Automorphic Functions

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

scholarly journals Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

scholarly journals Cuspidal quintics and surfaces with , and 5-torsion

2016 ◽  
Vol 19 (1) ◽  
pp. 42-53
Author(s):  
Carlos Rito
Keyword(s):  
Computer Algebra ◽  
General Type ◽  
Free Action ◽  
The Other ◽  
Singular Set ◽  
Canonical Map ◽  
Singular Sets ◽  
Surface Of General Type ◽  
Quintic Threefold ◽  
Hyperplane Sections

If $S$ is a quintic surface in $\mathbb{P}^{3}$ with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover ${\it\phi}:X\rightarrow S$ branched only at the cusps such that $p_{g}(X)=4$, $q(X)=0$, $K_{X}^{2}=15$ and ${\it\phi}$ is the canonical map of $X$. We use computer algebra to search for such quintics having a free action of $\mathbb{Z}_{5}$, so that $X/\mathbb{Z}_{5}$ is a smooth minimal surface of general type with $p_{g}=0$ and $K^{2}=3$. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets $17\mathsf{A}_{2}$, $16\mathsf{A}_{2}$, $15\mathsf{A}_{2}+\mathsf{A}_{3}$ and $15\mathsf{A}_{2}+\mathsf{D}_{4}$.

Coercivity of integral functionals with non-everywhere superlinear lagrangians

2019 ◽  
Vol 12 (4) ◽  
pp. 447-458
Author(s):  
Cristina Marcelli
Keyword(s):  
Variational Problem ◽  
Sufficient Conditions ◽  
Integral Functionals ◽  
Superlinear Growth

AbstractWe consider the classical non-autonomous variational problem\text{minimize }\biggl{\{}F(v)=\int_{a}^{b}f(x,v(x),v^{\prime}(x))\,\mathrm{d}% x:v\in\Omega\biggr{\}},where {\Omega:=\{v\in W^{1,1}(a,b),\,v(a)=A,\,v(b)=B,\,v(x)\in I\}}, when the lagrangian f has non-everywhere superlinear growth, in the sense that it can vanish at some {x_{0}\in[a,b]}, or {s_{0}\in I}. We prove some sufficient conditions ensuring the coercivity of the functional F. As a consequence, when f is convex with respect to the last variable, the existence of the minimum can be immediately derived.

scholarly journals Rigid subsets of symplectic manifolds

2009 ◽  
Vol 145 (03) ◽  
pp. 773-826 ◽  
Author(s):  
Michael Entov ◽  
Leonid Polterovich
Keyword(s):  
Symplectic Manifold ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Smooth Functions ◽  
Torus Actions ◽  
Moment Maps ◽  
Quantum Homology ◽  
Singular Sets ◽  
Hamiltonian Torus Actions ◽  
Commutative Subalgebras

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Differentiable Montgomery-Samelson Fiberings with Finite Singular Sets

1969 ◽  
Vol 21 ◽  
pp. 1489-1495 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Fixed Points ◽  
Group Action ◽  
Singular Set ◽  
Differentiable Structure ◽  
Singular Sets ◽  
Complete Bibliography

In 1946 Montgomery and Samelson (11) introduced a generalization of the notion of a differentiable group action with one type of orbit besides fixed points. Such an object is essentially a locally trivial fibering except on a certain singular set over which fibres are pinched to points. In recent years there has been a fair amount of research on these MS-fiberings and similar singular fiberings. This paper is another effort in this direction. For a fairly complete bibliography of the literature, the reader should consult the references, and in particular, (5).Let f: Mn → Sp, with Mn a closed connected n-manifold and Sp the unit p-sphere with standard differentiable structure, be the projection map of a smooth MS-fibering with finite non-empty singular set.

scholarly journals Poincaré Theta Series and Singular Sets of Schottky Groups

1964 ◽  
Vol 24 ◽  
pp. 43-65 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Essential Role ◽  
Theta Series ◽  
Schottky Group ◽  
Singular Set ◽  
Schottky Groups ◽  
Linear Transformations ◽  
Convergence Problem ◽  
Discontinuous Group ◽  
Singular Sets ◽  
Metrical Property

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

scholarly journals Structure of singular sets of some classes of subharmonic functions

2021 ◽  
Vol 31 (4) ◽  
pp. 519-535
Author(s):  
B.I. Abdullaev ◽  
S.A. Imomkulov ◽  
R.A. Sharipov
Keyword(s):  
Harmonic Functions ◽  
Singular Set ◽  
Test Functions ◽  
Subharmonic Functions ◽  
Singular Sets ◽  
Basic Functions

In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.

Structure Theory for Montgomery-Samelson Fiberings between Manifolds. I

1969 ◽  
Vol 21 ◽  
pp. 170-179 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Total Space ◽  
Structure Theory ◽  
Homology Sphere ◽  
Singular Set ◽  
Connected Manifold ◽  
Singular Sets ◽  
Subsequent Publication

In (12), Montgomery and Samelson conjectured that an MS-fibering of polyhedra with total space an n-sphere must have a homology sphere as its singular set. Mahowald (11) has shown that, indeed, an orientable fibering with n ≧ 4 must have a Z2-cohomology sphere as its singular set, while Conner and Dyer (4) have shown this for n arbitrary provided the fiber itself is a Z2-cohomology sphere. We show that if the singular set is tame, then it is a Z-homology sphere if the fiber is also one. This result together with those of Stallings (15), Gluck (7), and Newman and Connell (13) are applied in the case where the singular sets are locally flat and tame. It is shown (Theorem 5.2) that MS-fiberings of spheres on spheres, with closed connected manifold fibers and singular sets, are topologically just suspensions of (Hopf) sphere bundles. In a subsequent publication, the case where the singular sets are finite shall be considered. The reader is invited to consult (3) and (18) in this case.

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