Weighted Inequalities and Degenerate Elliptic Partial Differential Equations

Stredulinsky, Edward W.

doi:10.1007/bfb0101268

Cited by 59 publications

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“…In these situations the classical methods fail to be applied directly so that the existence and the multiplicity results (which hold in the nondegenerate case) may become a delicate matter that is closely related to some phenomena due to the degenerate character of the differential equation. These problems have been intensively studied starting with the pioneering paper by Murthy and Stampacchia [15] (we also refer to [8,13,16], as well as to the monograph [22]). …”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Singular elliptic problems with lack of compactness

Ghergu

Rădulescu

2005

Annali di Matematica

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Abstract. We consider the following nonlinear singular elliptic equationwhere g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli-KohnNirenberg critical exponent associated to a, b, and N . Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above problem has at least two distinct solutions provided that λ ∈ (0, λ 0 ). The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem without the Palais-Smale condition, combined with a weighted variant of the Brezis-Lieb Lemma.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Singular elliptic problems with lack of compactness

Ghergu

Rădulescu

2005

Annali di Matematica

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“…For example, one can derive various Sobolev embedding theorems in the L p setting, which can then be used to prove the existence of solutions of the Cauchy problem for elliptic and parabolic PDEs (see e.g. [9,22,36,15,44,55,39]), to study the asymptotic behaviour of solutions [2,57], as well as their stability [10,11]. They are present in probability theory (see e.g.…”

Section: Introductionmentioning

confidence: 99%

On a variant of the Hardy inequality between weighted Orlicz spaces

Kałamajska

Pietruska-Pałuba

2009

Studia Math.

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Abstract. Let M be an N -function satisfying the ∆2-condition, and let ω, ϕ be two other functions, with ω ≥ 0. We study Hardy-type inequalitieswhere u belongs to some set R of locally absolutely continuous functions containing C ∞ 0 (R+). We give sufficient conditions on the triple (ω, ϕ, M ) for such inequalities to be valid for all u from a given set R. The set R may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

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“…Also, when a = 1, p = 2, and w = v, (8.1) and (8.2) agree with the notion of thinness (irregularity) introduced in [16] for boundary regularity of solutions to degenerate elliptic second-order divergence-form partial differential equations. The present definition is motivated by the above results as well as those of Stredulinsky [34] and Bauman [9]. Our main observation is Theorem 8.2.…”

Section: Qjmentioning

confidence: 91%

“…See e.g. [34,18,16,13]. The interesting thing is that our technique basically returns to the methods established in the early 1970s for treating the one-side weight case as opposed to the now "two-weight" case.…”

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confidence: 98%

“…In particular, and especially of interest to us in this paper, is the question of the boundary regularity of such solutions to Dirichlet problems for certain degenerate divergence form hnear and nonlinear operators as well as for linear nondivergence uniformly elliptic operators. See [16,34,9]. The condition that must be met at the boundary is, of course, some analogue of the Wiener criteria.…”

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Weighted Nonlinear Potential Theory

Adams¹

1986

Transactions of the American Mathematical Society

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Abstract. The potential theoretic idea of the " thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting-the weights representing in general singularities/degeneracies-and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials'<<:(//'*«&) ', 1 < p < q < oo, which at least in the first-order case (a = 1) have found some use in a number of places in analysis.1. Introduction. In recent years a number of papers have appeared pointing out the connection between the theory of weights a la Muckenhoupt-Wheeden-FeffermanCoifman et al. and the pointwise behavior of solutions to certain elliptic second order partial differential equations in bounded domains of Euclidean yV-space. In particular, and especially of interest to us in this paper, is the question of the boundary regularity of such solutions to Dirichlet problems for certain degenerate divergence form hnear and nonlinear operators as well as for linear nondivergence uniformly elliptic operators. See [16,34,9]. The condition that must be met at the boundary is, of course, some analogue of the Wiener criteria. These criteria can be expressed as a Dini type integral involving a capacity set function built out of the differential operator. In the simplest case, these capacities can be viewed as a slight modification of the Newtonian capacity (associated with the Laplace operator) using the structure coefficients of the differential operator. In the nondivergence linear equation case, these capacities may well involve the use of the Green's function for the differential operator on some fixed ball. However, in any case the idea is to study these capacities and their corresponding Wiener criteria in order to obtain useful sufficient conditions that will imply when two differential operators have the same boundary regular points.

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Weighted Inequalities and Degenerate Elliptic Partial Differential Equations

Cited by 59 publications

References 0 publications

Singular elliptic problems with lack of compactness

Singular elliptic problems with lack of compactness

On a variant of the Hardy inequality between weighted Orlicz spaces

Weighted Nonlinear Potential Theory

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