Abstract
We consider the
$L^{p}$
-regularity of the Szegö projection on the symmetrised polydisc
$\mathbb {G}_{n}$
. In the setting of the Hardy space corresponding to the distinguished boundary of ...the symmetrised polydisc, it is shown that this operator is
$L^{p}$
-bounded for
$p\in (2-{1}/{n}, 2+{1}/{(n-1)})$
.
We establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system:{ut-divð'oe(x,t,∇u)=div|F|p-2F+finΩT,u=u0onΩ×{0}, Image omitted ...\left\{\begin{aligned} \displaystyle{}u_{t}-\operatorname{div}\mathcal{A}(x,t,% \nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&% \displaystyle\phantom{}\text{in}\ \Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on}\ \Omega% \times\{0\},\end{aligned}\right.by proving that, for given Image omitted, there exists Image omitted depending on δ and the structural data such that|∇u0|p+ε∈Lloc1(Ω) and |F|p+ε,|f|(δp(n+2)n)′+ε∈L1(0,T;Lloc1(Ω))⟹|∇u|p+ε∈L1(0,T;Lloc1(Ω)). Image omitted \lvert\nabla u_{0}\rvert^{p+\varepsilon}\in L^{1}_{\operatorname{loc}}(\Omega)% \quad\text{and}\quad\lvert F\rvert^{p+\varepsilon},\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1}_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{% \operatorname{loc}}(\Omega)).Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with Image omitted and we provide an optimal regularity theory in the literature.
We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial regularity theorem for the Navier-Stokes equations. The key idea is to find a quantitative counterpart for the absolute ...continuity of the dissipation energy using a pigeonholing argument. Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending exponentially on the natural local energies. Then, we give two applications of the latter result in the axially symmetric case. The first one is a local regularity criterion for suitable weak solutions with small swirl. The second one is a slightly improved one-point CKN criterion which implies all known (slightly supercritical) Type I regularity results in the literature.
We prove higher Hölder regularity for solutions of equations involving the fractional p-Laplacian of order s, when p≥2 and 0<s<1. In particular, we provide an explicit Hölder exponent for solutions ...of the non-homogeneous equation with data in Lq and q>N/(sp), which is almost sharp whenever sp≤(p−1)+N/q. The result is new already for the homogeneous equation.
We establish sharp regularity estimates for solutions to Lu=f in Ω⊂Rn, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on Sn−1, ...called the spectral measure.
First, we study the interior regularity of solutions to Lu=f in B1. We prove that if f is Cα then u belong to Cα+2s whenever α+2s is not an integer. In case f∈L∞, we show that the solution u is C2s when s≠1/2, and C2s−ϵ for all ϵ>0 when s=1/2.
Then, we study the boundary regularity of solutions to Lu=f in Ω, u=0 in Rn∖Ω, in C1,1 domains Ω. We show that solutions u satisfy u/ds∈Cs−ϵ(Ω‾) for all ϵ>0, where d is the distance to ∂Ω.
Finally, we show that our results are sharp by constructing two counterexamples.
In this paper, we introduce a new notion in a semigroup
as an extension of Mary's inverse. Let
. An element
is called left (resp. right) invertible along
if there exists
such that
(resp.
) and
(resp.
...). An existence criterion of this type inverse is derived. Moreover, several characterizations of left (right) regularity, left (right)
-regularity and left (right)
-regularity are given in a semigroup. Further, another existence criterion of this type inverse is given by means of a left (right) invertibility of certain elements in a ring. Finally, we study the (left, right) inverse along a product in a ring, and, as an application, Mary's inverse along a matrix is expressed.
Set regularities and feasibility problems Kruger, Alexander Y.; Luke, D. Russell; Thao, Nguyen H.
Mathematical programming,
03/2018, Letnik:
168, Številka:
1-2
Journal Article
Recenzirano
Odprti dostop
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. ...Several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms.
Abstract Let H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$ -cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $ ...-regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $ -regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$ -regular element of a normal Hall subgroup H is $\alpha $ -regular in $G.$
Abstract Let $\alpha $ be a complex-valued $2$ -cocycle of a finite group $G.$ A new concept of strict $\alpha $ -regularity is introduced and its basic properties are investigated. To illustrate the ...potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible $\alpha _N$ -characters of N equals the number of $\alpha $ -regular conjugacy classes of G contained in $N,$ where N is a normal subgroup of $G.$
This paper is devoted to the study of the normal (tangential) regularity of a closed set and the subdifferential (directional) regularity of its distance function in the context of Riemannian ...manifolds. The Clarke, Fréchet and proximal subdifferentials of the distance function from a closed subset in a Riemannian manifold are represented by corresponding normal cones of the set.
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