Powers of Distances to Lower Dimensional Sets as Muckenhoupt Weights

Aimar, Hugo; Carena, Marilina; Durán, Ricardo G.; Toschi, Marisa

doi:10.1007/s10474-014-0389-1

Cited by 36 publications

(44 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem is our main result concerning the A p -properties of distance weights. In [2], corresponding results were obtained in metric spaces, but using a completely different approach and under the much stronger assumption that both X and E satisfy Ahlfors regularity conditions; see e.g. [2, Theorems 6 and 7].…”

Section: Powers Of Distance Functions As Weightsmentioning

confidence: 99%

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

et al. 2018

View full text Add to dashboard Cite

Abstract. Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x, E) −α , where E is a closed set in X and α ∈ R. We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt's A p classes of weights, 1 ≤ p < ∞. With the help of general A p -weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.

show abstract

Section: Powers Of Distance Functions As Weightsmentioning

confidence: 99%

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

et al. 2018

View full text Add to dashboard Cite

show abstract

“…Here α ∈ (n − 2, n) and d z (x) = |x − z| denotes the Euclidean distance to z. Since α ∈ (n − 2, n) ⊂ (−n, n), the weight ρ belongs to the Muckenhoupt class A 2 [2]. Consequently, H 1 (ρ, Ω), defined by (7), is a Hilbert space endowed with the norm (8).…”

Section: The Optimal Control Problem With Point Sourcesmentioning

confidence: 99%

“…Since the state equation (2) contains a linear combination of l Dirac measures as a forcing term and n > 1 the state y does not belong to H 1 (Ω). Consequently, the error analysis involved in the finite element approximation of problem (2) is not standard. We refer the reader to [10,40,47] for sub-optimal error analyses on quasi-uniform meshes and [8,31] for quasi-optimal results based on graded meshes.…”

Section: Introductionmentioning

confidence: 99%

Ana posteriorierror analysis for an optimal control problem with point sources

et al. 2018

View full text Add to dashboard Cite

We propose and analyze a reliable and efficient a posteriori error estimator for a controlconstrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposed a posteriori error estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform. AN OPTIMAL CONTROL PROBLEM WITH POINT SOURCES3 polytope, we prove the global reliability and local efficiency of our proposed error estimator. The analysis is delicate since it involves the interaction of L ∞ (Ω), R l and weighted Sobolev spaces, combined with having to deal with the first-order necessary and sufficient optimality condition that characterizes the optimal controlū. It is important to comment that this work exploits the ideas developed in [4] for the a posteriori error analysis of the so-called pointwise tracking optimal control problem. Although the mathematical techniques are similar, the a posteriori error analysis of our control problem does not follow directly from [4]; it requires its own analysis. This is mainly due to the following reasons:• The optimal control variableū belongs to R l , while the one of the problem studied in [4] belongs to L 2 (Ω).This in a sense simplifies the analysis. For instance, as opposed to [4,30], we can obtain local efficiency estimates that do not require convexity of Ω. Nevertheless, it comes with its own set of complications. In particular, the low regularity of the state equation. • The adjoint problem is a Poisson equation with a forcing term y − y d , which does not belong to L ∞ (Ω).Consequently, we must consider a pointwise error indicator that accounts for unbounded right hand sides.Since we were not able to locate one in the literature, in section 4, we propose such an error indicator and provide its analysis on the basis of [9,15]. Notice that, thanks to the structure of the control problem of [4], such an estimator was not needed there. The outline of this paper is as follows. In section 2, we introduce the notation and functional framework we shall work with. Section 3 contains the description of our control problem and reviews the a priori error analysis developed in [7]. In section 4, we propose and analyze a pointwise a posteriori error estimator for the Laplacian that allows for unbounded right hand sides. Combining this estimator and another one based on Muckenhoupt weighted Sobolev spaces, in section 5 we devise an a posteriori error estimator for our optimal control problem. We show in sections 5.2 and 5.3, its ...

show abstract

“…where L denotes L q (Ω, −q ). The constant C in the previous inequality is independent of Ω, and K is the constant introduced in (18). Now, we are ready to construct the -decomposition.…”

Section: Lemma 44mentioning

confidence: 99%

“…where K is the geometric constant introduced in (18) and C n is a constant that depends only on n. Otherwise, if x ∉ ∪ s∈Γ B s or x ∈ B s , where s ≠ t and s p ≠ t, then…”

Section: Lemma 45mentioning

confidence: 99%