Search results for "Bounded"

O* - Dynamical Systems and * - Derivations of Unbounded Operator Algebras

1999

A spatial theory is developed for * - derivations of an algebra of unbounded operators, in terms of the concept of O*-dynamical systems. Three notions of spatiality emerge, depending on the nature of the corresponding generator. Special emphasis is put on O*-dynamical systems generated by one-parameter groups of *-automorphisms and their *-derivations.

The Navier–Stokes equations in exterior Lipschitz domains: L -theory

2020

Abstract We show that the Stokes operator defined on L σ p ( Ω ) for an exterior Lipschitz domain Ω ⊂ R n ( n ≥ 3 ) admits maximal regularity provided that p satisfies | 1 / p − 1 / 2 | 1 / ( 2 n ) + e for some e > 0 . In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on L σ p ( Ω ) for such p. In addition, L p - L q -mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions to the Navier–Stokes equations in the critical space L ∞ ( 0 , T ; L σ 3 ( Ω ) ) (locally in time and globally in time for small initial data).

Closure properties for integral problems driven by regulated functions via convergence results

2018

Abstract In this paper we give necessary and sufficient conditions for the convergence of Kurzweil–Stieltjes integrals with respect to regulated functions, using the notion of asymptotical equiintegrability. One thus generalizes several well-known convergence theorems. As applications, we provide existence and closure results for integral problems driven by regulated functions, both in single- and set-valued cases. In the particular setting of bounded variation functions driving the equations, we get features of the solution set of measure integrals problems.

2020

Abstract This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schrodinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity equation is put into relation with a long jump random walk with weights.

Hölder stability for Serrin’s overdetermined problem

2015

In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–93…

Sturmian words and overexponential codimension growth

2018

Abstract Let A be a non necessarily associative algebra over a field of characteristic zero satisfying a non-trivial polynomial identity. If A is a finite dimensional algebra or an associative algebra, it is known that the sequence c n ( A ) , n = 1 , 2 , … , of codimensions of A is exponentially bounded. If A is an infinite dimensional non associative algebra such sequence can have overexponential growth. Such phenomenon is present also in the case of Lie or Jordan algebras. In all known examples the smallest overexponential growth of c n ( A ) is ( n ! ) 1 2 . Here we construct a family of algebras whose codimension sequence grows like ( n ! ) α , for any real number α with 0 α 1 .

Singular integrals on regular curves in the Heisenberg group

2019

Let $\mathbb{H}$ be the first Heisenberg group, and let $k \in C^{\infty}(\mathbb{H} \, \setminus \, \{0\})$ be a kernel which is either odd or horizontally odd, and satisfies $$|\nabla_{\mathbb{H}}^{n}k(p)| \leq C_{n}\|p\|^{-1 - n}, \qquad p \in \mathbb{H} \, \setminus \, \{0\}, \, n \geq 0.$$ The simplest examples include certain Riesz-type kernels first considered by Chousionis and Mattila, and the horizontally odd kernel $k(p) = \nabla_{\mathbb{H}} \log \|p\|$. We prove that convolution with $k$, as above, yields an $L^{2}$-bounded operator on regular curves in $\mathbb{H}$. This extends a theorem of G. David to the Heisenberg group. As a corollary of our main result, we infer that all …

Conformal measures for multidimensional piecewise invertible maps

2001

Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow\ ]0,\infty[ , a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g}\ d\nu with a constant \lambda>0 . Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

The Liouville theorem and linear operators satisfying the maximum principle

2020

A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$ \mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z). $$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ i…

Dimension estimates for the boundary of planar Sobolev extension domains

2020

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.