Each of the 18 chapters of the book is divided into sections and most of the sections consist of subsections. The reader can get a general idea of the contents of the book from the Introduction.
Introduction
Here the necessary and sufficient conditions for the validity of embedding theorems stated in terms of the classesJαcharacterized by isoperimetric inequalities are found. 9 we present the De Giorgi–Federer theorem on conditions for the validity of the Gauss–Green formula.
Basic Properties of Sobolev Spaces
- Notation
- Absolute Continuity of Functions in L 1 p (Ω)
- Approximation of Functions in Sobolev Spaces by Smooth Functions in ΩFunctions inΩ
- Domains Starshaped with Respect to a Ball
- Domains of the Class C 0,1 and Domains Having the Cone PropertyProperty
- Generalized Poincar´ e Inequality
- Duals of Sobolev Spaces
- Removable Sets for Sobolev Functions
Note: The domain Ω, discussed at the beginning of this section, for which C∞( ¯Ω) has no density in Sobolev spaces, has the property ∂Ω=∂Ω. Since Gi is star-shaped with respect to any ball in B0∩B1, by the integral representation (1.1.8) and by the continuity of the integral operator with the kernel|x−y|l−k−n in Lp(G) we obtain .
Facts from Set Theory and Function Theory
- Two Theorems on Coverings
- Theorem on Level Sets of a Smooth Function
- Representation of the Lebesgue Integral as a Riemann Integral along a Halfaxis
- Formula for the Integral of Modulus of the Gradient
There exists a number M, depending only on the dimension of the space, such that each point of Rn belongs to at most M spheres in {Bm};. In the form of an inequality with a suitable definition of the modulus of the gradient of a function, theorem 1.2.4 can be extended to abstract metric spaces, cf.
Some Inequalities for Functions of One Variable
- Basic Facts on Hardy-type Inequalities
- Hardy-Type Inequalities with Indefinite Weights
- Three Inequalities for Functions on (0, ∞ )
- Estimates for Differentiable Nonnegative Functions of One VariableVariable
In the form of an inequality with a suitable definition of the modulus of the gradient of the function, theorem 1.2.4 can be extended to abstract metric spaces, cf. ii) The constant factor before the integral on the right-hand side of (1.3.1) is sharp. We begin with the proof of the following less general theorem about absolutely continuous measures μ and ν. Proof of Theorem 1. If we set f = 0 on the support of the singular part of measure ν, we obtain that (1.3.5) is equivalent.
The estimate B ≤ C can be derived in the same way as in the proof of Theorem 2, if |v(x)|p is replaced by dν∗/dxin. The next more general statement can be derived from Theorem 1 in the same way that Theorem 1.3.2/1 was derived from Theorem 1.3.2/2. Note that we must be careful here: in what follows, the two terms on the right-hand side of the previous equation cannot be evaluated separately, as this would lead to a restriction.
Then, the estimate involving the first term in (1.3.31) is determined using the weighted Hardy inequality.
Embedding Theorems of Sobolev Type
- Corollaries of Previous Results
- Generalized Sobolev Theorem
- Compactness Theorems
- Multiplicative Inequalities
We now proceed with the statement and proof of the basic theorem of this section. Since the coverage multiplicity {2Qj} is finite and depends only on no, it follows that. Lemma. Every bounded subset of the space of restrictions of functions in Vpl(Rn) to a bounded domain Ω is relatively compact in Vpl−1(Ω).
Then every subset of the space C∞( ¯Ω), bounded by Vpl(Ω), is relatively compact in the metric. Then it suffices to prove that every subset of the space C∞(Rn)∩Wpl(Rn), bounded by Wpl(Rn) =Vpl(Rn), is relatively compact in the metric (1.4.33). Since the cover set{B(i)} depends only on n, the right-hand side is large.
By applying the theorem in the present subsection to the function uηi and the measurement e→μ(e∩Q(i)), we get.
More on Extension of Functions in Sobolev Spaces
- Survey of Results and Examples of Domains
- Domains in EV p 1 which Are Not Quasidisks
- Extension with Zero Boundary Data
The boundary values u from the triangles tk and tk+1 coincide at their common vertex fork= 0.1,. Let Ω and G be planar domains such that Ω∈EVp1 and Ω⊂G, and let the origin be the only common point of intersection of the disc BR={eiθ : 0≤ < R} with G\Ω¯ (see Figure 9). Romanov [681] showed that the role of the critical value 2 in Theorem 1.5.2 is related to the special domain addressed by this theorem.
Suppose there exist constants C and θ such that the following condition is satisfied: for everyx, y∈Ω such that |x−y| ≤θ, there exists a rectifiable curveγ⊂Ω connecting xtoy such that. For a linear continuous expansion operator Vpl(Ω) → Vp,σl (Rn) to exist, it is sufficient and ifσ(x) depends only on |x| and is not decreasing near the origin, then this also necessary That. in a neighborhood of the origin, where c is a positive constant, independent of x. In the following cases there exists a linear continuous expansion operator Vpl(Ω) →Vql(Rn) forq <.
Different results on function expansions in Wpl(Ω) with class deterioration can be found in the research of Burenkov [156].
Inequalities for Functions with Zero Incomplete Cauchy DataCauchy Data
- Integral Representation for Functions of One Independent Variable
- Integral Representation for Functions of Several Variables with Zero Incomplete Cauchy Datawith Zero Incomplete Cauchy Data
- Embedding Theorems for Functions with Zero Incomplete Cauchy DataCauchy Data
Let L be an arbitrary ray drawn from the point x; let θ be a unit vector with origin atx and directed along L,y=x+τ θ,τ∈R1. Let γ be an arbitrary multi-index of order l−1 and let {Pγ(θ)} be the system of all homogeneous polynomials of degree l−1 in the variablesθ1. Sometimes statements similar to Theorems 1, 2, and 3 can be refined by replacing ∇luLp(Ω) with (−Δ)l/2uLp(Ω), where Δ is the Laplace operator.
This inequality arises from an obvious estimate for the Green function of the Dirichlet problem for the Laplace operator, which in turn follows from the maximum principle. Analogous estimates can be derived from pointwise estimates for the Green functionGm(x, s) of the Dirichlet problem for their harmonic operator in an n-dimensional domain (see Maz'ya. Here we show that the condition l≤2k does not can be weakened in the statements of the preceding section.
The restrictions onΩ that the Sobolev theorems hold for the space Vpl(Ω)∩˚Vpk(Ω), 2k < l, will be considered in Sect.
Density of Bounded Functions in Sobolev Spaces
- Functions with Bounded Gradients Are Not Always Dense in L 1 p (Ω)inL1p(Ω)
- Density of Bounded Functions in L 2 p (Ω) for Paraboloids in R n
Since functions in L1p(Ω) are absolutely continuous on almost all lines parallel to coordinate axes (Theorem 1.1.3/1), we have almost everywhere iΩ,. The convergence to zero of the last integral follows from the monotonic convergence theorem. According to Lemma 1.7.1, the subspace of bounded functions is dense in L1p(Ω) for an arbitrary domain ifp∈[1,∞).
Moreover, the following estimate holds for (x, y)∈Πi. here and in the following in this section it is a positive constant, independent of i). We saw in the previous section that bounded domains with a non-smooth boundary may not have the property that the set L2p(Ω)∩L∞(Ω) is compact inL2p(Ω). The following statement gives a necessary and sufficient condition for the spaceL2p(Ω)∩L∞(Ω) to be dense inL2p(Ω).
In particular, it gives sufficient conditions for the space Vpl(Ω)∩L∞(Ω) to be dense in Vpl(Ω), Ω ⊂ R2.
- Main Result
Once this estimate has been made, the result follows by integrating with the other variables and applying Hölder's inequality. If the first sum to the right of the inequality is greater than the second, we set i1=i. Suppose the endpoint ofi1 is the starting point of the next interval and repeat this process with the same k.
We stop it when the closed finite intervals i1, i2,. each of length at least |Δ|/k) forms a covering of the intervalΔ. Turning to the general case u∈Wpl∩L∞, we first assume that suppuis is bounded and consider a softening uh ofu with radiush. To complete the proof of the lemma, we remove the boundedness assumption of suppu.
So A is an algebra and therefore the space Wpl∩L∞ is the maximum algebra in Wpl.
Inequalities for Functions Vanishing at the Boundary
Conditions for Validity of Integral Inequalities (the Case p = 1)(the Casep= 1)
- Inequality Involving the Norms in L q (Ω, μ) and L r (Ω, ν ) (Case p = 1)(Casep= 1)
- Case q ∈ (0, 1)
- Inequality (2.1.10) Containing Particular Measures
Let N (x) denote the unit perpendicular to the boundary of the admissible set g at a point x directed to the interior of g. Let Φ(x, ξ) be a continuous function onΩ×Rn that is non-negative and positive homogeneous of first degree with respect to ξ. Let's start with the basic properties of the so-called non-incremental rearrangement of a function.
We construct a covering of the set {ζ :η = 0} of spheres Bj of radius j, equal to the distance of Bj from the hyperplane {ζ:ξ= 0}. Since (2.1.33) is also valid for α <1−m with the coefficient (1−m−α)−1, if u vanishes near the subspace η = 0, the arguments in the second and third parts of the proof follow 1 with obvious changes come we to the next sentence. Properties of the weighted area minimization function C introduced in Definition 2.1.4 were investigated under the assumption that Φ(x, ξ) does not depend on x and is convex.
Horiuchi [383] proved the sufficiency in Theorem 2.1.6/1 for an absolutely continuous measure ν and for a more general class of sets F depending on the behavior as ε→0 of the n-dimensional Lebesgue measure of the tube neighborhood of F, {z∈Rn+m: dist(z, F)< ε}.
2.2 (p, Φ)-Capacity
Definition and Properties of the (p, Φ)-Capacity
Tertikas and Tintarev [749] (see also Tintarev and Fieseler [753], Sect. 5.6, as well as Benguria, Frank and Loss [83]) proved the existence and nonexistence of optimizers in (2.1.37) studied and found sharp constants in some cases. In [277] Filippas, Maz'ya and Tertikas showed that for any convex domains Ω⊂Rn the inequality. From the same definition it follows that the (p, Φ)-capacity Relative toΩ does not increase under extension ofΩ.
Analogously, we prove the following property. iv) For every compact ⊂Ω and any ε >0 there exists an open set ω,. So, taking into account that the attenuation of the functions ϕ and ψ belong to P(K ∪ F, Ω) and P(K ∩ F, Ω), respectively, we get the required. If f is compact in Ω, then by property (iii), given ε >0, there exists an open set G such that.
It follows from the general theory of Choquet capacities that analytic groups, and in particular, Borel groups are (p, Φ) capable (see Choquet [186]).
Expression for the (p, Φ)-Capacity Containing an Integral over Level Surfacesover Level Surfaces
Remembering the property (2.2.1) of the (p, Φ)-capacity, note that incidentally we also proved the following lemma here. We divide both sides of the preceding inequality by (T−t)p/(p−1) and estimate the first factor on the right-hand side. From Lemma 2.2.2/3 and from the Lemma of this subsection we immediately get the following corollary.
If p > n, then the p-capacitance of the center of the ball BR with respect to BR is equal to ωn(pp−−n1)p−1Rn−p. Therefore, in the latter case, the capacitance of any compact with respect to any bounded open set containing that compact is positive. In the proof of the second part of Theorem 2.1.1, it was shown that the prior integral converges to σ(∂g), which gives the required upper bound.
We now recall the definition of the symmetrization of a compact group K in Rn with respect to the (n−s)-dimensional subspaceRn−s.
Conditions for Validity of Integral Inequalities (the Case p > 1)(the Casep >1)
- The (p, Φ)-Capacitary Inequality
- Capacity Minimizing Function and Its Applications
- Estimate for a Norm in a Birnbaum–Orlicz Space
- Best Constant in the Sobolev Inequality (p > 1)
- Estimate for the Norm in L q (Ω, μ) with q < p (First Necessary and Sufficient Condition)Necessary and Sufficient Condition)
The following application of the capacity minimization functionνpis is immediately derived from the capacitive inequality (2.3.6). The norm inLM(Ω, μ) of the characteristic function χE of the setE is χELM(Ω,μ)=μ(E)P−1. where P−1 is the inverse of the restriction of P to [0,∞). Then (2.3.16) can be replaced in the Theorem by the following more general estimate:. 2.3.17) As an illustration, note that Theorem 2.1.1 shows the equivalence of the inequality.
From the previous corollary and the isoperimetric inequality (2.2.12), we obtain the Sobolev (p >1)-Gagliardo (p= 1) inequality. Thus, we have reduced the question of the best constant in (2.3.21) to a one-dimensional variational problem, which was explicitly solved by Bliss [109] already in 1930 using classical methods of calculus of variations. The following theorem describes the conditions for the equivalence of the generalized Sobolev-type inequality (2.3.19) and the multiplicative integral inequality.
Remarks. The theorem just proved shows, in particular, the equivalence of the multiplicative inequality (2.3.25) and the Sobolev-type inequality (2.3.19).
