Imbedding theorems for Sobolev spaces on domains with peak and on Hölder domains

Abstract

Abstract. Necessary and sufficient conditions are obtained for the continuity and compactness of the imbedding operators Gagliardo [3] in the case where the domain Ω ⊂ R n has the cone property. If l is a positive integer, 1 ≤ p < ∞, and lp < n, then the exponent q in the above imbedding takes the maximal possible value q = np/(n − lp).In [12], Maz ya obtained a necessary and sufficient condition for the continuity of the imbedding operatorn . These conditions are either isoperimetric (for p = 1), or capacitary (for p > 1) inequalities. As an application of these results, a maximal exponent q was found for which W

SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR Hölder-Α DOMAINS

“…Let us mention for example [2,12,34,35,36,37,38]. However, as far as we know, there are no results for the Stokes equations on this kind of domains.…”

Embedding of a weighted Sobolev space and properties of the domain

“…Labutin demonstrated [10] that condition (2) with θ = 1 is also necessary for the embedding under consideration. For a σ-regular domain whose boundary locally satisfies the Hölder condition with exponent 1/σ, the embedding theorem (1) is valid under conditions (2) with θ = σ, which was shown by Labutin [9] (for s = 1) and Maz'ya and Poborchi [12] (for an isolated degenerate cone and s ∈ N). In [2], in the class of σ-regular domains, we distinguished subclasses for which the embedding theorem (1) holds under conditions (2) with an arbitrary given θ ∈ [1, σ].…”

On a System of Equations of Evolution with a Non-Symmetrical Parabolic Part Occuring in the Analysis of Moisture and Heat Transfer in Porous Media

“…, (k) of applied differential operators (nonsymmetric in the parabolic part, but symmetric in the elliptic one) can be verified in practice. Of course, some additional assumptions on Ω must be accepted to ensure that the usual imbedding and trace theorems cannot be violated; the geometrical interpretation of such conditions has been discussed in great detail in [22], pp. 62, 220.…”

“…Through the whole paper we will apply the standard notation: all classes of special mappings applied here are introduced in [8] or [5], the notation of Lebesgue and Sobolev spaces is compatible with [22], the symbol * is reserved for adjoint spaces, the dot symbol (rarely) for time derivatives and Ê 0 is sometimes used instead of Ê + ∪ {0}.…”

“…Now we need a more precise estimate for v X in the last additive term: from [22], p. 222, we know that…”