Global Sobolev inequalities and Degenerate P-Laplacian equations
aa r X i v : . [ m a t h . A P ] J a n GLOBAL SOBOLEV INEQUALITIES AND DEGENERATEP-LAPLACIAN EQUATIONS
DAVID CRUZ-URIBE, OFS, SCOTT RODNEY, AND EMILY ROSTA
Abstract.
We prove that a local, weak Sobolev inequality implies a global Sobolevestimate using existence and regularity results for a family of p -Laplacian equations.Given Ω ⊂ R n , let ρ be a quasi-metric on Ω, and let Q be an n × n semi-definitematrix function defined on Ω. For an open set Θ ⋐ Ω, we give sufficient conditionsto show that if the local weak Sobolev inequality (cid:16) B | f | pσ dx (cid:17) pσ ≤ C h r ( B ) B | p Q ∇ f | p dx + B | f | p dx i p holds for some σ >
1, all balls B ⊂ Θ, and functions f ∈ Lip (Θ), then the globalSobolev inequality (cid:16) ˆ Θ | f | pσ dx (cid:17) pσ ≤ C (cid:16) ˆ Θ | p Q ∇ f ( x ) | p dx (cid:17) p also holds. Central to our proof is showing the existence and boundedness of solu-tions of the Dirichlet problem ( X p,τ u = ϕ in Θ u = 0 in ∂ Θ , where X p,τ is a degenerate p -Laplacian operator with a zero order term: X p,τ u = div (cid:16)(cid:12)(cid:12)p Q ∇ u (cid:12)(cid:12) p − Q ∇ u (cid:17) − τ | u | p − u. Introduction
Given an open set Θ ⊂ R n , the classical Sobolev inequality, (cid:18) ˆ Θ | f | pσ dx (cid:19) pσ ≤ C (cid:18) ˆ Θ |∇ f | p dx (cid:19) p , Date : December 28, 2017.1991
Mathematics Subject Classification.
Key words and phrases. degenerate Sobolev spaces, Sobolev inequality, p -Laplacian.D. Cruz-Uribe is supported by NSF Grant DMS-1362425 and research funds from the Deanof the College of Arts & Sciences, the University of Alabama. S. Rodney is supported by theNSERC Discovery Grant program. E. Rosta is a graduate of the undergraduate honours programin mathematics at Cape Breton University; her work was supported through the NSERC USRAprogram. holds for 1 ≤ p < n , σ = nn − p >
1, and all functions f ∈ Lip (Θ) (that is, Lipschitzfunction such that supp( f ) ⋐ Ω). For this result and extensive generalizations, see[E, GT, HK].We are interested in determining sufficient conditions for a degenerate Sobolevinequality,(1.1) (cid:18) ˆ Θ | f | pσ dx (cid:19) pσ ≤ C (cid:18) ˆ Θ | p Q ∇ f | p dx (cid:19) p , to hold, where 1 < p < ∞ , σ >
1, and Q is an n × n matrix of measurable functions de-fined on Θ such that for almost every x ∈ Θ, Q ( x ) is semi-definite . Such inequalitiesarise naturally in the study of degenerate elliptic PDEs: a global Sobolev inequalityis necessary to prove the existence of weak solutions (see, for instance [CMN, MR])and to prove compact embeddings of (degenerate) Sobolev spaces (see [CRW]).Our goal is to show that such global estimates can be derived from weaker, localSobolev inequalities. Definition 1.1.
Given ≤ p < ∞ and σ > , a local Sobolev property of order p withgain σ holds in Ω if there is a constant C > and a positive, continuous function r : Ω → (0 , ∞ ) such that for any y ∈ Ω , < r < r ( y ) , and f ∈ Lip ( B ( y, r )) , (1.2) (cid:18) B ( y,r ) | f | pσ dx (cid:19) pσ ≤ C r (cid:18) B ( y,r ) | p Q ∇ f | p dx (cid:19) p + C (cid:18) B ( y,r ) | f | p dx (cid:19) p . Local Sobolev estimates arise naturally in the study of regularity of degenerateelliptic equations (see [SW1, SW2]), but they are not sufficient for proving the exis-tence of solutions. So it is natural to ask if local inequalities imply global ones. Theobvious approach is to use a partition of unity argument, but this does not work. Ifthe local Sobolev property of order p with gain σ holds on Ω, then given any openset Θ ⋐ Ω a partition of unity argument shows that there is a constant C (Θ) suchthat(1.3) (cid:18) ˆ Θ | f | pσ dx (cid:19) pσ ≤ C (Θ) (cid:20)(cid:18) ˆ Θ | p Q ∇ f | p dx (cid:19) p + (cid:18) ˆ Θ | f | p dx (cid:19) p (cid:21) holds for every f ∈ Lip (Θ). However, we cannot remove the second term on theright of (1.3) even when the second term on the right of (1.2) is not present.Nevertheless, with some additional assumptions we are able to pass from a localto a global Sobolev inequality. To set the stage for our main result, we first state animportant special case. Theorem 1.2.
Fix < p < ∞ and let Ω ⊂ R n be an open set. Suppose that Q is a semi-definite matrix function in L ∞ (Ω) . Suppose that the local Sobolev property LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 3 of order p with gain σ > holds, and suppose further that a local Poincar´einequality (cid:18) B ( y,r ) | f − f B ( y,r ) | p dx (cid:19) p ≤ Cr (cid:18) B ( y,βr ) | p Q ∇ f | p dx (cid:19) p holds for y ∈ Ω , β ≥ , < βr < r ( y ) , and f ∈ Lip (Ω) . Then given any open set Θ ⋐ Ω , there exists a constant C (Θ) such that (1.1) holds. Our main result, Theorem 2.7, generalizes Theorem 1.2 in several ways. First,we remove the assumption that Q is bounded, allowing it to be singular as wellas degenerate. Second, we can change the underlying geometry by replacing theEuclidean metric with a quasi-metric, and defining balls with respect to this metric.The statement is rather technical and requires some additional hypotheses, which iswhy we have deferred the statement until below.The remainder of the paper is organized as follows. In Section 2 we give thenecessary assumptions and definitions and then state Theorem 2.7. In Section 3 wegive an application of Theorem 2.7 to a family of Lipschitz vector fields. Such vectorfields are a natural example of where degenerate p -Laplacians arise.A central and somewhat surprising part of our proof of Theorem 2.7 is to provethe existence and boundedness of solutions of the Dirichlet problem ( X p,τ u = ϕ in Θ u = 0 in ∂ Θ , where X p,τ is a degenerate p -Laplacian operator with a zero order term: X p,τ u = div (cid:16)(cid:12)(cid:12)p Q ∇ u (cid:12)(cid:12) p − Q ∇ u (cid:17) − τ | u | p − u. We prove the existence of solutions using Minty’s theorem in Section 4 and we proveboundedness using ideas from [CRW, MRW1] in Section 5. Finally, in Section 6 weprove Theorem 2.7.Throughout this paper, Ω will be a fixed open, connected subset of R n . We sayan open set Θ is compactly contained in Ω and write Θ ⋐ Ω if Θ is bounded and¯Θ ⊂ Ω. The set
Lip (Ω) consists of all Lipschitz functions f such that supp( f ) ⋐ Ω.A constant C may vary from line to line; if necessary we will denote the dependenceof the constant on various parameters by writing, for instance, C ( p ).2. The main result
In order to state Theorem 2.7 we need to make some technical assumptions and givesome additional definitions. We begin with the topological framework. Fix Ω ⊂ R n and let ρ : Ω × Ω → R be a symmetric quasimetric on Ω: that is, there is a constant κ ≥ x, y, z ∈ Ω: DAVID CRUZ-URIBE, OFS, SCOTT RODNEY, AND EMILY ROSTA (1) ρ ( x, y ) = 0 if and only if x = y ;(2) ρ ( x, y ) = ρ ( y, x );(3) ρ ( x, y ) ≤ κ (cid:0) ρ ( x, z ) + ρ ( x, y ) (cid:1) .Given x ∈ Ω and r > ρ -ball of radius r centered at x by B ( x, r ); that is B ( x, r ) = { y ∈ Ω : ρ ( x, y ) < r } . We will assume that the balls B ( x, r )are Lebesgue measurable. We will also use D ( x, r ) to denote the correspondingEuclidean ball { x ∈ Ω : | x − y | < r } . We will always assume that the quasi-metric ρ and the Euclidean distance satisfy the following:(2.1) given x, y ∈ Ω , | x − y | → ρ ( x, y ) → . Equivalently, we may assume that given x ∈ Ω and any ǫ >
0, there exists δ > D ( x, δ ) ⊂ B ( x, ǫ ), and that given any δ > γ > B ( x, γ ) ⊂ D ( x, δ ). Either of these conditions hold if we assume that the topologygenerated by the balls B ( x, r ) is equivalent to the Euclidean topology on Ω. Remark 2.1.
This assumption on the topology of (Ω , ρ ) is taken from [MRW1] ; itis closely related to a condition first assumed by C. Fefferman and Phong [FP] (seealso [SW1] ). Let S n denote the collection of all positive, semi-definite n × n self-adjoint matrices;fix a function Q : Ω → S n whose entries are Lebesgue measurable. For a.e. x ∈ Ω andfor all ξ, η ∈ R n (where ξ ′ denotes the transpose of ξ ) define the associated quadraticform, Q ( x, ξ ) = h Q ( x ) ξ, ξ i = ξ ′ Q ( x ) ξ and inner product h Q ( x ) ξ, η i = η ′ Q ( x ) ξ . Thesesatisfy 0 ≤ h Q ( x ) ξ, ξ i , |h Q ( x ) ξ, η i| ≤ h Q ( x ) η, η i h Q ( x ) ξ, ξ i . We define the operator norm of Q ( x ) by | Q ( x ) | op = sup | ξ | =1 | Q ( x ) ξ | . Since Q is semi-definite a.e., √ Q is well defined, and | p Q ( x ) | op = | Q ( x ) | op . We willwrite Q ∈ L p (Ω), 1 ≤ p ≤ ∞ , if | Q | op ∈ L p (Ω).Besides the local Sobolev inequality given in Definition 1.2, and which we willhereafter assume holds for ρ -balls B , we will also need a local Poincar´e property.In stating it, we assume that r : Ω → (0 , ∞ ) is the same function that appears inDefinition 1.2. Definition 2.2.
Given ≤ p < ∞ and t ′ ≥ , a local Poincar´e property of order p with gain t ′ ≥ holds in Ω if there are constants C > and β ≥ such that for any y ∈ Ω and r such that < βr < r ( y ) , (2.2) (cid:18) B ( y,r ) | f − f B ( y,r ) | pt ′ dx (cid:19) pt ′ ≤ C r (cid:18) B ( y,βr ) | p Q ∇ f | p dx (cid:19) p LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 5 holds for every f ∈ Lip ( B ( y, βr )) such that √ Q ∇ f ∈ L p ( B ( y, βr )) , and where f B ( y,r ) = ffl B ( y,r ) f ( x ) dx . Remark 2.3.
Inequality (2.2) will be used to establish that the embedding of thedegenerate Sobolev space b H ,pQ, (that we will define below) into L p is compact. We willalso use it to prove a product rule for functions in this Sobolev space. The parameter t ′ is determined by the regularity of Q : the more regular Q is, the smaller t ′ is permittedto be. In fact, if Q is locally bounded in Ω , (2.2) is not required to establish the productrule: see the proof of Lemma 6.1. Remark 2.4.
The problem of determining sufficient conditions on the matrix Q forthe Poincar´e inequality (2.2) to hold has been considered in a somewhat different formin [CIM, CRR] . It is interesting to note that in this case the condition involves thesolution of a Neumann problem for a degenerate p -Laplacian operator. Our final definition is a technical assumption on the geometry of (Ω , ρ ). Thiscondition, which we refer to as the “cutoff” condition, ensures the existence of accu-mulating sequences of Lipschitz cutoff functions on ρ -balls. Again, the function r isassumed to be the same as in Definition (1.2). Definition 2.5.
Given (Ω , ρ ) and a matrix Q , a cutoff condition of order ≤ s ≤ ∞ holds if there exist constants C , N > and < α < such that given x ∈ Ω and < r < r ( x ) there exists a sequence { ψ j } ∞ j =1 ⊂ Lip ( B ( x, r )) such that for all j ∈ N , (2.3) ≤ ψ j ≤ , supp ψ ⊂ B ( x, r ) ,B ( x, αr ) ⊂ { y ∈ B ( x, r ) : ψ j ( y ) = 1 } , supp ψ j +1 ⊂ { y ∈ B ( x, r ) : ψ j ( y ) = 1 } , (cid:18) B ( x,r ) | p Q ( y ) ∇ ψ j ( y ) | s dy (cid:19) s ≤ C N j r . Definition 2.5 first appeared in [SW1], though it is a generalization of a conceptthat has appeared previously in the literature; see [SW1] for further references. If ρ is the Euclidean metric and Q is bounded, then this sequence of cutoff functions canbe taken to be the standard Lipschitz cutoff functions. More generally, it was shownin [SW1] that with our assumptions on ρ , if Q is continuous, then such a sequenceexists with s = ∞ . This cutoff condition holds for a wide variety of geometries,including ρ which produce highly degenerate balls. See, for example, [M]. Remark 2.6.
There is a close connection between the cutoff condition and doubling.In [KMR] it was shown that if the local Sobolev property of order p and gain σ andthe cutoff condition (2.3) both hold, then Lebesgue measure is locally doubling for the DAVID CRUZ-URIBE, OFS, SCOTT RODNEY, AND EMILY ROSTA collection of ρ -balls { B ( x, r ) } x ∈ Ω; r> . That is, there exists a positive constant C sothat given any x ∈ Ω and < r < r ( x ) then (cid:12)(cid:12) B ( x, r ) (cid:12)(cid:12) ≤ C | B ( x, r ) | . Consequently,for any < r ≤ s < r ( x ) , (2.4) (cid:12)(cid:12) B ( x, s ) (cid:12)(cid:12) ≤ ˜ C (cid:0) sr (cid:1) d (cid:12)(cid:12) B ( x, r ) | , where d = log ( C ) . We will use this fact to prove Proposition 5.3 below. We can now state our main result.
Theorem 2.7.
Given a set Ω ⊂ R n , let ρ be a quasi-metric on Ω . Fix < p < ∞ and < t ≤ ∞ , and suppose Q is a semi-definite matrix function such that Q ∈ L pt loc (Ω) .Suppose further that that the cutoff condition of order s > pσ ′ , the local Poincar´eproperty of order p with gain t ′ = ss − p , and the local Sobolev property of order p withgain σ ≥ hold. Then, given any open set Θ ⋐ Ω there is a positive constant C (Θ) such that the global Sobolev inequality (2.5) (cid:18) ˆ Θ | f | pσ dx (cid:19) pσ ≤ C (Θ) (cid:18) ˆ Θ | p Q ∇ f | p dx (cid:19) p holds for all f ∈ Lip (Θ) . Remark 2.8. If Q ∈ L ∞ loc (Ω) , then we can take t = ∞ and t ′ = 1 , so that s = ∞ .Thus, we only need to assume a local Poincar´e inequality of order p without gain. Ifwe assume that ρ is the Euclidean metric, then as we noted above, the cutoff conditionholds with s = ∞ . Thus, Theorem 1.2 follows immediately from Theorem 2.7. Example: diagonal Lipschitz vector fields
In this section we give an illustrative example of the application of Theorem 2.7.Let Ω be any bounded domain in R n and let 1 < p < ∞ . Fix a vector function a = ( a , . . . , a n ) where a = 1 and a , . . . , a n : Ω → R are such that for 2 ≤ j ≤ n , a j is bounded, nonnegative and Lipschitz continuous. Further, assume that the a j satisfy the RH ∞ condition in the first variable x uniformly in x , . . . , x n : there existsa constant C for each interval I and x = ( x , . . . , x n ) ∈ Ω, a j ( x , . . . , x n ) ≤ C I a ( z , x , . . . , x n ) dz . For instance, we can take a j ( x , . . . , x n ) = | x | α j b j ( x , . . . , x n ), where α j ≥ b j is a non-negative Lipschitz function. (For more on the RH ∞ condition, see [CN].)Now let X j = a j ∂∂x j and ∇ a = ( X , ..., X n ), and define the associated p -Laplacian(3.1) L p,a u = div a (cid:0) |∇ a u | p − ∇ a u (cid:1) . LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 7
If we let Q ( x ) = diag(1 , a , ..., a n ), then we have that L p,a = div (cid:0) |√ Q ∇ u | p − Q ∇ u (cid:1) .It is shown in [MRW2] that each of Definitions 1.1, 2.2, and 2.5 hold with respectthe family of non-interference balls A ( x, r ) defined as in [SW1], and there exists aquasi-metric ρ such that the non-interference balls are equivalent to the ρ -balls. Infact, setting B ( x, r ) = A ( x, r ) and r ( x ) = δ ′ dist( x, ∂ Ω) for δ ′ > k Q k ∞ , condition (2.3) holds with s = ∞ , (2.2) holds with t ′ = 1 and(1.2) holds with σ = d d − p where d is the doubling exponent associated to Lebesguemeasure and the collection of balls A ( x, r ), as in (2.4). As a result, both [MRW1,(1.15) and (1.16)] hold with t = ∞ and t ′ = 1. Therefore, we can apply Theorem 2.7to get that for any open subdomain Θ ⋐ Ω, there exists a constant C (Θ) such thatthe Sobolev inequality(3.2) (cid:18) ˆ Θ | f | pσ dx (cid:19) pσ ≤ C (Θ) (cid:18) ˆ Θ (cid:12)(cid:12)(cid:12)p Q ∇ f (cid:12)(cid:12)(cid:12) p dx (cid:19) p holds for all f ∈ Lip (Θ). Moreover, by the doubling property (2.4), if we let Θ = A ( x, r ) for 0 < r < δ dist( x, ∂ Ω), then we get (cid:18) A ( x,r ) | f | pσ dx (cid:19) pσ ≤ Cr (cid:18) A ( x,r ) (cid:12)(cid:12)(cid:12)p Q ∇ f (cid:12)(cid:12)(cid:12) p dx (cid:19) p for any f ∈ Lip ( A ( x, r )).As a consequence, when p = 2 inequality (3.2) is suffiencent to use [R, Theorem3.10] to prove the existence of a unique weak solution of the linear Dirichlet problem ( div ( Q ∇ u ) = f in Θ u = 0 on ∂ Θ . Weak solutions of degenerate p -Laplacians A key step in the proof of Theorem 2.7 is to prove the existence and boundednessof solutions of the Dirichlet problem(4.1) ( X p,τ u = ϕ in Θ u = 0 in ∂ Θ , where X p,τ is a degenerate p -Laplacian operator with a zero order term:(4.2) X p,τ u = div (cid:16)(cid:12)(cid:12)p Q ∇ u (cid:12)(cid:12) p − Q ∇ u (cid:17) − τ | u | p − u. In this section we will define weak solutions to this equation and prove that theyexist.As the first step we define the degenerate Sobolev spaces related to Q . Detaileddiscussions of these spaces can be found in [CMN, CRR, CRW, MRW1, MRW2, SW2]; DAVID CRUZ-URIBE, OFS, SCOTT RODNEY, AND EMILY ROSTA here we will sketch the key ideas and refer the reader to these references for furtherinformation. Fix 1 ≤ p < ∞ and a matrix function Q such that √ Q ∈ L p loc (Ω). Fixan open set Θ ⋐ Ω, and for 1 ≤ p < ∞ define L pQ (Θ) to be the collection of allmeasurable R n valued functions f = ( f , ..., f n ) that satisfy(4.3) k f k L pQ (Θ) = (cid:18) ˆ Θ (cid:12)(cid:12)p Q f (cid:12)(cid:12) p dx (cid:19) /p < ∞ . More properly we define L pQ (Θ) to be the normed vector space of equivalence classesunder the equivalence relation f ≡ g if k f − g k L pQ (Θ) = 0. Note that if f ( x ) = g ( x )a.e., then f ≡ g , but the converse need not be true, depending on the degeneracyof Q .Let Lip Q (Θ) be the collection of all functions f ∈ Lip loc (Θ) such that f ∈ L p (Θ)and ∇ f ∈ L pQ (Θ). We now define the corresponding degenerate Sobolev space b H ,pQ (Θ) to be the formal closure of Lip Q (Θ) with respect to the norm k f k b H ,pQ (Θ) = (cid:20) ˆ Θ | f | p dx + ˆ Θ Q ( x, ∇ f ) p dx (cid:21) p = (cid:20) ˆ Θ | f | p dx + ˆ S | p Q ∇ f | p dx (cid:21) p . Similarly, define b H ,pQ, (Θ) ⊂ b H ,pQ (Θ) to be the formal closure of Lip (Θ) with respectto this norm.Because of the degeneracy of Q , we cannot represent either b H ,pQ, (Θ) or b H ,pQ (Θ)as spaces of functions except in special situations. But, since L p (Θ) and L pQ (Θ)are complete, given an equivalence class of b H ,pQ (Θ) there exists a unique pair ~ f =( f, g ) ∈ L p (Θ) × L pQ (Θ) that we can use to represent it. Such pairs are unique andso we refer to elements of b H ,pQ (Θ) using their representative pair. However, becauseof the classical example in [FKS], g need not be uniquely determined by the firstcomponent f of the pair: if we think of g as the “gradient” of f , then there existnon-constant functions f whose gradient is 0.On the other hand, since √ Q ∈ L ploc (Ω) and since constant sequences are Cauchy,if f ∈ Lip Q (Θ), then ( f, ∇ f ) ∈ b H ,pQ (Θ) where ∇ f is the classical gradient of f in Θ:see [GT].We need one structural result about these Sobolev spaces. The following result isproved in [CRR] for the space H ,pQ (Θ), which is the closure of C ( ¯Θ) with respect tothe b H ,pQ (Θ) norm, but the proof is identical in our case. Lemma 4.1.
Given ≤ p < ∞ , Θ ⊂ Ω , and a matrix Q , b H ,pQ (Θ) and b H ,pQ, (Θ) areseparable Banach spaces. If p > , they are reflexive. We now define the weak solution of the Dirichlet problem (4.1) for equation (4.2).We will assume that 1 < p < ∞ , τ ≥ ϕ ∈ L p ′ loc (Ω), and √ Q ∈ L p loc (Ω). Associated LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 9 to the Dirichelt problem is the non-linear form A p,τ : b H ,pQ (Θ) × b H ,pQ (Θ) → R , defined for ~ u = ( u, ~g ) and ~ v = ( v, ~h ) by(4.4) A p,τ ~ u ( ~ v ) = ˆ Θ h Q~g, ~g i p − h Q~g, ~h i dx + τ ˆ Θ | u | p − uv dx ;we use the convention that A p,τ ~ ( · ) = 0 if 1 < p <
2. The notation used on theleft-hand side of (4.4) is meant to suggest that for each fixed ~ u = ( u, ~g ) ∈ b H ,pQ (Θ),the operator A p,τ ~ u ( · ) ∈ (cid:0) b H ,pQ (Θ) (cid:1) ′ ; see Lemma 4.7 below.We use this form to define a weak solution. Definition 4.2.
A weak solution to the Dirichlet problem (4.1) is an element ~ u =( u, ~g ) ∈ b H ,pQ, (Θ) such that the equality (4.5) A p,τ ~ u ( v ) = ˆ Θ h Q~g, ~g i p − h Q~g, ∇ v i dx + τ ˆ Θ | u | p − uv dx = − ˆ Θ ϕv dx holds for every v ∈ Lip (Θ) . Remark 4.3. If ~ u ∈ b H ,pQ, (Θ) is a weak solution, then by a standard density argumentwe have that (4.5) holds if we replace ( v, ∇ v ) with any ~ v = ( v, ~h ) ∈ b H ,pQ, (Θ) . We can now state and prove our existence result.
Proposition 4.4.
Let Ω ⊂ R n be open. Given < p < ∞ and τ > , suppose Q ∈ L p loc (Ω) . Then, for any open set Θ ⋐ Ω the Dirichlet problem (4.1) with ϕ ∈ L p ′ (Θ) has a weak solution ~ u = ( u, ~g ) ∈ b H ,pQ, (Θ) . To prove Proposition 4.4 we will use Minty’s theorem as found in [Sh]; this resultis a generalization of the Lax-Milgram theorem to general Banach spaces. To state itwe fix some notation. Let X be a separable, reflexive Banach space with norm k · k X ,and let X ∗ denote its dual space. Given a map T : X → X ∗ and u, v ∈ X , we willwrite T ( u )( v ) = h T ( u ) , v i . Theorem 4.5. (Minty)
Let X be a separable, reflexive Banach space and fix Γ ∈ X ∗ .Let T : X → X ∗ be an operator that is: • bounded: T maps bounded subsets of X to bounded subsets of X ′ ; • monotone: h T ( u ) − T ( v ) , u − v i ≥ for all u, v ∈ X ; • hemicontinuous: for z ∈ R , the mapping z T [ u + zv ]( v ) is continuous forall u, v ∈ X ; • almost coercive: there exists β > such that h T v, v i > h Γ , v i for all v ∈ X such that k v k X > β . Then the set u ∈ X such that T ( u ) = Γ is non-empty. To apply Theorem 4.5 to solve the Dirichlet problem (4.1), let X = b H ,pQ, (Θ); thenby Lemma 4.1, X is a separable, reflexive Banach space. Fix ϕ ∈ L p ′ (Θ); given ~ v = ( v, ~h ), define Γ ∈ X ∗ by(4.6) Γ( ~ v ) = − ˆ Θ ϕv dx. Let T = A p,τ ; then ~ u ∈ b H ,pQ, (Θ) is a weak solution if A p,τ ~ u ( ~ v ) = Γ( ~ v ) for every ~ v =( v, ~h ) ∈ b H ,pQ, (Θ). By Minty’s theorem, such a ~ u exists if A p,τ is bounded, monotone,hemicontinuous, and almost coercive. To complete the proof of Proposition 4.4, wewill prove each of these properties in turn.We begin with three useful inequalities which we record as a lemma. For theirproof, see [PL, Ch. 10]. Lemma 4.6.
For all s, r ∈ R n , (4.7) h| s | p − s − | r | p − r, s − r i ≥ if p ≥ , (4.8) (cid:12)(cid:12) | s | p − s − | r | p − r (cid:12)(cid:12) ≤ c ( p ) (cid:0) | s | p − + | r | p − (cid:1) | s − r | ; if < p ≤ , (4.9) (cid:12)(cid:12) | s | p − s − | r | p − r (cid:12)(cid:12) ≤ c ( p ) | s − r | p − . Lemma 4.7. A p,τ is bounded on b H ,pQ, (Θ) for all < p < ∞ and τ ∈ R .Proof. Fix 1 < p < ∞ and τ ∈ R . Let ~ u = ( u, ~g ) , ~ v = ( v, ~h ) ∈ b H ,pQ, (Θ). If we applyH¨older’s inequality twice, then we have that |A p,τ ~ u ( ~ v ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ h Q~g, ~g i p − h Q~g, ~h i dx (cid:12)(cid:12)(cid:12)(cid:12) + | τ | (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ | u | p − uvdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ Θ h Q~g, ~g i p − h Q~h, ~h i dx + | τ |k u k p − L p (Θ) k v k L p (Θ) ≤ k p Q ~g k p − L p (Θ) k p Q ~h k L p (Θ) + | τ |k u k p − L p (Θ) k v k L p (Θ) ≤ (1 + | τ | ) k ~ u k p − b H ,pQ, (Θ) k ~ v k b H ,pQ, (Θ) . It follows at once from this inequality that A p,τ is bounded. (cid:3) Lemma 4.8. A p,τ is monotone for all < p < ∞ and τ ≥ . LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 11
Proof.
Fix p and τ , and let ~ u , ~ v ∈ b H ,p (Θ) be as in Lemma 4.7. Then, hA p,τ ~ u − A p,τ ~ v , ~ u − ~ v i = A p,τ ~ u ( ~ u − ~ v ) − A p,τ ~ v ( ~ u − ~ v )= ˆ Ω h Q~g, ~g i p − h Q~g, ~g − ~h i dx − ˆ Ω h Q~h, ~h i p − h Q ∇ ~h, ~g − ~h i dx + τ (cid:18) ˆ Ω ( | u | p − u − | v | p − v )( u − v ) dx (cid:19) = I + τ I . We estimate I and I separately, beginning with I . By inequality (4.7), I = ˆ Ω h| u | p − u − | v | p − v, u − v i dx ≥ . To estimate I note that by the symmetry of Q we have that h Q~g, ~g i = |√ Q~g | .Hence, I = ˆ Θ | p Q~g | p − h Q~g, ~g − ~h i dx − ˆ Θ | p Q~h | p − h Q~h, ~g − ~h i dx = ˆ Θ h| p Q~g | p − p Q~g − | p Q~h | p − p Q~h, p Q~g − p Q~h i dx. With s = √ Q ~g and r = √ Q ~h , the integrand is again of the form (4.7) and sonon-negative. Thus I ≥ (cid:3) Lemma 4.9. A p,τ is hemicontinuous for all < p < ∞ and τ ∈ R .Proof. Fix ~ u , ~ v ∈ b H ,pQ, (Θ) as in the previous lemmas. For z ∈ R , let z~ v = z ( v, ~h ) =( zv, z~h ) ∈ b H ,pQ, (Θ); we will show that the function z
7→ A p,τ ( ~ u + z~ v )( ~ v ) is continuous.By the definition of A p,τ we can split this mapping into the sum of two parts: z
7→ G p ( ~ u + z~ v )( ~ v ) = ˆ Θ h Q ( ~g + z~h ) , ( ~g + z~h ) i p − h Q ( ~g + z~h ) , ~h i d x,z
7→ H p,τ ( ~ u + z~ v )( ~ v ) = ˆ Θ τ | u + zv | p − ( u + zv ) v d x. We will show each part is continuous in turn.To show that the mapping z
7→ G p ( ~ u + z~ v )( ~ v ) is continuous, we modify an argumentfrom the proof of [CMN, Proposition 3.15]. Fix z, w ∈ R ; then (since Q = √ Q √ Q is symmetric), |G p ( ~ u + z~ v )( ~ v ) − G p ( ~ u + w~ v )( ~ v ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ | p Q ( ~g + z~h ) | p − (cid:10)p Q ( ~g + z~h ) , p Q~h (cid:11) dx − ˆ Θ | p Q ( ~g + w~h ) | p − (cid:10)p Q ( ~g + w~h ) , p Q~h (cid:11) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ˆ Θ (cid:12)(cid:12) | p Q ( ~g + z~h ) | p − p Q ( ~g + z~h ) − | p Q ( ~g + w~h | p − p Q ( ~g + w~h ) (cid:12)(cid:12) | p Q~h | dx ;if p ≥
2, then by (4.8) and H¨older’s inequality with exponent pp − , ≤ C ( p ) ˆ Θ (cid:0) | p Q ( ~g + z~h ) | p − + | p Q ( ~g + w~h ) | p − (cid:1) | z − w || p Q~h | dx ≤ C ( p ) (cid:18) ˆ Θ (cid:0) | p Q ( ~g + z~h ) | p − + | p Q ( ~g + w~h ) | p − (cid:1) pp − dx (cid:19) p − p × | z − w | (cid:18) ˆ Θ | p Q~h | p dx (cid:19) p ≤ C ( p ) | z − w | (cid:0) k ~g k L pQ (Θ) + ( | z | + | w | ) k ~h k L pQ (Θ) (cid:1) p − k ~h k L pQ (Θ) . Since the norms in the final term are all finite, we see that this term tends to 0 as w → z ; thus the mapping z
7→ G p ( ~ u + z~ v )( ~ v ) is continuous. when p ≥ < p <
2, we can essentially repeat the above argument but instead ap-ply (4.9) to get that |G p ( ~ u + z~ v )( ~ v ) − G p ( ~ u + w~ v )( ~ v ) | ≤ C ( p ) ˆ Θ | z − w | p − | p Q~h | p dx, and the desired continuity again follows.To show that the mapping z
7→ H p,τ ( ~ u + z~ v )( ~ v ) is continuous, again fix z, w ∈ R .Then (cid:12)(cid:12) H p,τ ( ~ u + z~ v )( ~ v ) − H p,τ ( ~ u + w~ v )( ~ v ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ τ | u + zv | p − ( u + zv ) − τ | u + wv | p − ( u + wv ) v dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | τ | ˆ Θ (cid:12)(cid:12) | u + zv | p − ( u + zv ) − | u + wv | p − ( u + wv ) (cid:12)(cid:12) | v | dx. The integrand in the final term tends to 0 pointwise as w → z , so the desiredcontinuity will follow by the dominated convergence theorem if we can prove that the LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 13 integrand is dominated by an integrable function. But we have that (cid:12)(cid:12) | u + zv | p − ( u + zv ) − | u + wv | p − ( u + wv ) (cid:12)(cid:12) | v |≤ | u + zv | p − | v | + | u + wv | p − | v |≤ C ( p )( | u | p − | v | + ( | z | p − + | w | p − ) | v | p ) ≤ C ( p, | z | )( | u | p − | v | + | v | p ) . By H¨older’s inequality, the final term is in L (Θ). Hence, we have that the mapping z
7→ H p,τ ( ~ u + z~ v )( ~ v ) is continuous and this completes the proof. (cid:3) Lemma 4.10.
Given < p < ∞ and ϕ ∈ L p ′ (Θ) , define Γ by (4.6) . Then for all τ > , A p,τ is almost coercive.Proof. Let ~ u = ( u, ~g ) ∈ b H Q, (Θ) and ϕ ∈ L p ′ (Θ). Then we have that A p,τ ~ u ( ~ u ) = ˆ Θ h Q~g, ~g i p − h Q~g, ~g i dx + τ ˆ Θ | u | p − u dx = ˆ Θ h Q~g, ~g i p dx + τ ˆ Θ | u | p dx ≥ η (cid:18) ˆ Θ h Q~g, ~g i p dx + ˆ Θ | u | p dx (cid:19) = η k ~ u k p b H ,pQ, , where η = min { , τ } >
0. Therefore, if we let β = (cid:0) η − k ϕ k p ′ (cid:1) p ′ − , then by H¨older’sinequality we have that for all k ~ u k b H ,pQ, (Θ) > β , | Γ( v ) | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ ϕu dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k p ′ k u k p ≤ k ϕ k p ′ k ~ u k b H ,pQ, (Θ) < η k ~ u k p b H ,pQ, (Θ) ≤ A p,τ ~ u ( ~ u ) . Thus, A p,τ is almost coercive. (cid:3) Boundedness of solutions to degenerate p -Laplacians In this section we will prove that solutions of the Dirichlet problem (4.1) arebounded. The proof is quite technical, as it relies on a very general result from [MRW1]and much of the work in the proof is checking the hypotheses.
Proposition 5.1.
Given a set Ω ⊂ R n , let ρ be a quasi-metric on Ω . Fix
pσ ′ , thelocal Poincar´e property of order p with gain t ′ = ss − p , and the local Sobolev propertyof order p with gain σ ≥ hold. Given any Θ ⋐ Ω and q ∈ [ p ′ , ∞ ) ∩ ( pσ ′ , ∞ ) , if ϕ ∈ L q (Θ) , then there exists a positive constant C such that for all τ ∈ (0 , , thecorresponding weak solution ~ u τ = ( u τ , ~g τ ) ∈ b H Q, (Θ) of (4.1) satisfies (5.1) ess sup x ∈ Θ | u τ ( x ) | ≤ C k ϕ k p − L q (Θ) . The constant C is independent of ϕ , ~ u τ , and τ . Remark 5.2.
The hypotheses of Proposition 5.1 are the same as those of Theorem 2.7except that we do not require higher integrability on Q : Q ∈ L p loc (Ω) is sufficient forthis result. The proof of Proposition 5.1 requires that the mapping I : b H ,pQ, (Θ) → L p (Θ), I (( u, ~g )) = u , is compact. This is a consequence of the following result. Proposition 5.3.
Given a set Ω ⊂ R n , let ρ be a quasi-metric on Ω . Fix < p < ∞ and < t ≤ ∞ , and suppose Q is a semi-definite matrix function such that Q ∈ L p loc (Ω) . Suppose further that that the cutoff condition of order s > pσ ′ , the localPoincar´e property of order p with gain t ′ = ss − p , and the local Sobolev property oforder p with gain σ ≥ hold. Fix Θ ⋐ Ω ; then the mapping I : b H ,pQ, (Θ) → L p (Θ) , I (( u, ~g )) = u , is compact.Proof. Proposition 5.3 is a particular case of a general imbedding result from [CRW,Theorem 3.20]. So to prove it we only need to show that the hypotheses of thisresult are satisfied. We will go through these in turn but for brevity we have omittedrestating the precise form of each hypothesis as given there and instead refer to themas they are stated in the theorem and the preliminaries in [CRW, Section 3]. Werefer the reader to this paper for complete details.Since (Ω , ρ ) is a quasi-metric space, ρ satisfies (2.1), and by Remark 2.6 Lebesguemeasure satsfies a local doubling property for metric balls, the topological assump-tions of Section 3A and condition (3-12) in [CRW] hold. In particular, since weassume that the function r in Remark 2.6 is continuous, the local geometric dou-bling condition in [CRW, Definition 3.3] holds.In the definition of the underlying Sobolev spaces, and in the Poincar´e and Sobolevinequalities [CRW, Definitions 3.5, 3.16], we let the measures µ, ν, ω all be theLebesgue measure. Since the local Poincar´e inequality of order p , Defintion 2.2,holds, and since r is assumed to be continuous, [CRW, Definition 3.5] holds. (Seealso [CRW, Remark 3.6].) Similarly, since we assume that the local Sobolev inequalityof order p with gain σ , Definition (1.2), holds, [CRW, Definition 3.16] holds.The existence of an accumulating sequence of cut-off functions, Definition 2.5, letsus prove the cut-off property of order s ≥ pσ ′ in [CRW, Definition 3.18]. Fix acompact subset K of Ω and let R > r on K . Given x ∈ K and 0 < r < R , since r < r ( x ), the cutoff condition of order s ≥ pσ ′ gives ψ ∈ Lip ( B ( x, r )) such that(1) 0 ≤ ψ ( x ) ≤ ψ ( x ) = 1 on B ( y, αr ),(3) ∇ ψ ∈ L sQ (Ω). LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 15
This yields the desired function in [CRW, Definition 3.18].Finally, we show that the weak Sobolev inequality, [CRW, Inequality (3.33)], holdswith t ′ = ss − p . Since s ≥ pσ ′ , t = sp ≥ σ ′ , and so 1 < t ′ ≤ σ . Again let K be acompact subset of Ω and R > r on K . If x ∈ K and0 < r < R , then by the local Sobolev property with B = B ( x, r ) we have that forany u ∈ Lip ( B ), (cid:18) ˆ B | u | pt ′ dy (cid:19) pt ′ ≤ C ( B ) (cid:18) ˆ B | u | pσ dy (cid:19) pσ ≤ C ( B ) k ( u, ∇ u ) k b H ,pQ, (Ω) , which gives us inequality (3.33).Thus, we have shown that we satisfy the necessary hypotheses, and so Proposi-tion 5.3 follows from [CRW, Theorem 3.20]. (cid:3) Proof of Proposition 5.1.
Let Θ ⋐ Ω and q ∈ [ p ′ , ∞ ) ∩ ( pσ ′ , ∞ ) with ϕ ∈ L q (Θ). Notethat since q ≥ p ′ and Θ is bounded, it follows that ϕ ∈ L p ′ (Θ). Fix 0 < τ <
1; then byProposition 4.4 there exists a weak solution ~ u τ = ( u τ , ~g τ ) ∈ b H ,pQ, (Θ) of the Dirichletproblem (4.1). To complete the proof, we will first use [MRW1, Theorem 1.2] to showthat ~ u τ satisfies (5.1). Then we will show using Proposition 5.3 that the constant isindependent of τ . (By [MRW1, Theorem 1.2] we have that it is independent of ϕ and ~ u τ .)To apply [MRW1, Theorem 1.2], first note that (Ω , ρ ) is a quasi-metric spaceand (2.1) holds, we satisfy the topological assumptions of this paper, including [MRW1,(1.9)]. As a result, if 0 < βr < r ( y ), the local Sobolev condition, Definition 1.1,the Poincar´e inequality, Definition 2.2, and the cutoff condition, Definition 2.5, hold.This shows that assumptions [MRW1, (1.13), (1.14), (1.15), (1.16)] hold with t ′ = ss − p .(For condition (1.16), see also [MRW1, Remark 1.1].)We now show that ~ u τ is the solution of an equation with the appropriate properties.Define A, ˜ A : Θ × R × R n → R n , and B : Θ × R × R n → R by(1) A ( x, z, ξ ) = h Q ( x ) ξ, ξ i p − Q ( x ) ξ ,(2) ˜ A ( x, z, ξ ) = h Q ( x ) ξ, ξ i p − p Q ( x ) ξ ,(3) B ( x, z, ξ ) = ϕ ( x ) + τ | z | p − z .Given these functions, the differential equation (4.2) can be rewritten asdiv (cid:0) A ( x, u, ∇ u ) (cid:1) = B ( x, u, ∇ u ) , which is [MRW1, (1.1)]. Furthermore, we have that A ( x, z, ξ ) = p Q ( x ) ˜ A ( x, z, ξ ),and for all ( z, ξ ) ∈ R × R n and a.e. x ∈ Θ,(1) ξ · A ( x, z, ξ ) = h Q ( x ) ξ, ξ i p − h Q ( x ) ξ, ξ i = |√ Q ξ | p ,(2) | ˜ A ( x, z, ξ ) | = | p Q ( x ) ξ | p − , (3) | B ( x, z, ξ ) | ≤ | ϕ ( x ) | + τ | z | p − < | ϕ ( x ) | + | z | p − .Therefore, the structural conditions [MRW1, (1.3)] are satisfied with the exponents δ = γ = ψ = p and the coefficients a = 1 , b = 0 , c = 0 , d = 1 , e = 0 , f = | ϕ | , g = 0,and h = 0.The above shows that we satisfy the hypotheses of [MRW1, Theorem 1.2]. There-fore, fix ǫ ∈ (0 ,
1] such that p − ǫ > k > u τ = | u τ | + k .Then for each y ∈ Θ and 0 < βr < r ( y ), we have the L ∞ -estimate(5.2) ess sup x ∈ B ( y,αr ) | u τ ( x ) | ≤ CZ (cid:20) | B ( y, r ) | ˆ B ( y,r ) | u τ | s ∗ p dx (cid:21) s ∗ p , where s ∗ is the dual exponent of s > σ ′ , define by s = s p . (In (5.2), α is the constantfrom (2.3); note that in [MRW1] it is called τ .) The term Z on the right-hand sideis defined by Z = " (cid:18) r p | B ( y, r ) | − p − ǫpσ ′ k k − p | ϕ |k L pσ ′ p − ǫ ( B ( y,r )) (cid:19) ǫ s ∗ σ − s ∗ . Since pσ ′ p − ǫ < q and ϕ ∈ L q (Θ), Z is bounded with a bound independent of τ for any k > B ( y, r ).If ϕ = 0, let k = k ϕ k p − L q (Θ) >
0; then by Minkowski’s inequality and the localSobolev inequality (1.2), (5.2) becomesess sup x ∈ B ( y,αr ) | u τ ( x ) | ≤ C "(cid:18) | B ( y, r ) | ˆ B ( y,r ) | u τ | pσ dx (cid:19) pσ + k ϕ k p − L q (Θ) ≤ C " | B ( y, r ) | p k ~ u τ k b H ,pQ, (Θ) + k ϕ k p − L q (Θ) . The constant C depends on B ( y, r ) but not on τ or ϕ . The case when ϕ = 0 issimilar and left to the reader.We now extend this estimate to all of Θ using the fact that Θ is compact. Byassumption (2.1), the balls B ( y, r ) are open, so we can find a finite cover B j =˜ B ( y j , r j ), 1 ≤ j ≤ N , with 0 < βr j < r ( y j ). Hence, we have that(5.3) ess sup y ∈ Θ | u τ ( x ) | ≤ C (cid:20) k ~ u τ k b H ,pQ, (Θ) + k ϕ k p − L q (Θ) (cid:21) , where the constant C depends on min { r j : 1 ≤ j ≤ N } > τ or ϕ .To complete the proof we will show that(5.4) k ~ u τ k b H ,pQ, (Θ) ≤ C k ϕ k p − L q (Θ) LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 17 with a constant C independent of τ . To do so we will use Proposition 5.3. Supposeto the contrary that (5.4) is false. Then there exists a sequence { τ k } ⊂ (0 ,
1) andcorresponding sequence of weak solutions { ~ u τ k } = { u τ k , ~g τ k } ⊂ b H ,pQ, (Θ) of (4.1) suchthat k ~ u τ k b H ,p (Θ) → ∞ as k → ∞ . We must have that τ k → k → ∞ . To see this, note that since ~ u τ k isa valid test function in the definition of a weak solution, we have that τ k k ~ u τ k k p b H ,p (Θ) = τ k (cid:20) ˆ Θ | u τ k | p − u τ k u τ k dx + ˆ Θ h Q~g τ k , ~g τ k i p − h Q~g τ k , ~g τ k i dx (cid:21) ≤ (cid:12)(cid:12)(cid:12)(cid:12) A p,τ k ~ u τ k ( ~ u τ k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ u τ k ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ~ u τ k k b H ,p (Θ) k ϕ k L p ′ (Θ) . Since k ~ u τ k k b H ,p (Θ) = 0,this inequality implies that k ~ u τ k k b H ,p (Θ) ≤ (cid:18) τ k k ϕ k L p ′ (Θ) (cid:19) p − which in turn implies that τ k → k ∈ N define ~ v τ k = k ~ u τ k k − b H ,pQ, (Θ) ~ u τ k . Then, ~ v τ k = ( v τ k , ~h τ k ) ∈ b H ,pQ, (Θ), k ~ v τ k k b H ,pQ, (Θ) = 1, and ~ v τ k is a weak solution of the Dirichlet problem ( div (cid:0) h Q ∇ w, ∇ w i p − Q ∇ w (cid:1) − τ k | w | p − w = ϕ k in Θ w = 0 on ∂ Θwhere ϕ k = k ~ u τ k k − p b H ,pQ, (Θ) ϕ . By Proposition 5.3, b H ,pQ, (Θ) is compactly embeddedin L p (Θ). Therefore, since { ~ v τ k } is a bounded sequence in b H ,pQ, (Θ), by passingto a subsequence (renumbered for simplicity of notation) we have that there exists v ∈ L p (Θ) such that v τ k → v in L p (Θ). Furthermore, arguing as we did above toprove τ k →
0, we have a Caccioppoli-type estimate: ˆ Θ h Q~h τ k , ~h τ k i p dx = ˆ Θ h Q~h τ k , ~h τ k i p − h Q ~h τ k , ~h τ k i dx ≤ A p,τ k ~ v τ k ( ~ v τ k ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ v τ k ϕ k dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k k L p ′ (Θ) . By definition, k ϕ k k L p ′ (Θ) → k → ∞ ; hence, ~ v τ k → ~ v = ( v,
0) in b H ,pQ, (Θ) norm.But then we have that k v k L p (Θ) = k ~ v k b H ,pQ, (Θ) = 1 . By the Poincar´e inequality (2.2) we have that for any y ∈ Θ and r > ˆ B ( y,r ) | v − v B ( y,r ) | p dx = 0 . Hence v is constant a.e. (or, more properly, constant on each connected componentof Θ). We claim that v = 0 a.e., which would contradict the fact that k v k L p (Θ) = 1.To show this, extend ~ v to all of Ω as follows. Let { ψ j } ⊂ Lip (Θ) be such that( ψ j , ∇ ψ j ) → ~ v in b H ,pQ, (Θ) norm. Define η j ∈ Lip (Ω) so that η j = ψ j in Θ and η j = 0 on Ω \ Θ. Then { ( η j , ∇ η j ) } is Cauchy in the b H ,pQ (Ω) norm and so convergesto some ~ w = ( w, ∈ b H ,pQ, (Ω). If we again apply Poincar´e’s inequality, we see that¯ v is constant in Ω (since Ω is connected). However, w = v in Θ and w = 0 in Ω \ Θ,and so we must have that v = 0 a.e.From this contradiction we have that our assumption is false and so (5.4) holdswith a constant independent of τ . This completes our proof. (cid:3) Proof of Theorem 2.7
Before proving our main result, we give one more lemma, a product rule for degener-ate Sobolev spaces. The proof is adapted from the proof of [MRW1, Proposition 2.2].
Lemma 6.1.
Given ≤ p < ∞ and < t ≤ ∞ , suppose √ Q ∈ L ptloc (Ω) and thatthe local Poincar´e property of order p with gain t ′ holds. Let Θ ⋐ Ω . If ~ u = ( u, ~g ) ∈ b H ,pQ (Θ) and v ∈ Lip (Θ) , then ( uv, v~g + u ∇ v ) ∈ b H ,pQ, (Θ) .Proof. By assumption, there exists a sequence { ψ j } ⊂ Lip Q (Θ) such that ψ j → u in L p (Θ) and ∇ ψ j → ~g in L pQ (Θ) as j → ∞ . For each j ∈ N , define φ j = ψ j v . Then { φ j } ⊂ Lip (Θ) and k uv − φ j k L p (Θ) ≤ k v k L ∞ (Θ) k u − ψ j k L p (Θ) ;hence, φ j → uv in L p (Θ) as j → ∞ .To complete the proof, we will show that ∇ φ j → v~g + u ∇ v in L pQ (Θ). Since ∇ φ j = v ∇ ψ j + ψ j ∇ v , we have that ˆ Θ | p Q ( v~g + u ∇ v − v ∇ ψ j − ψ j ∇ v ) | p dx ≤ C (cid:20) ˆ Θ | p Q ( ~g − ∇ ψ j | p | v | p dx + ˆ Θ | p Q ∇ v | p | u − ψ j | p dx (cid:21) ≤ C (cid:20) k v k pL ∞ (Θ) ˆ Θ | p Q ( ~g − ∇ ψ j ) | p dx + k p Q ∇ v k pL pt (Θ) k u − ψ j k pL pt ′ (Θ) (cid:21) . The first term on in the last line goes to 0 by our choice of ψ j and if t ′ = 1, then thesecond term does as well. If t ′ >
1, then to estimate the second term, note first that
LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 19 it follows from the local Poincar´e inequality (2.2) that for all y and r > f ∈ Lip (Ω), (cid:18) B ( y,r ) | f | pt ′ dx (cid:19) pt ′ ≤ Cr (cid:18) B ( y,βr ) | p Q ∇ f | p dx (cid:19) p + (cid:18) B ( y,r ) | f | p dx (cid:19) p Therefore, by a partition of unity argument like that used to prove the weak globalSobolev inequality (1.3) from the weak local Sobolev inequality (1.2), we have that(6.1) (cid:18) ˆ Θ | f | pt ′ dx (cid:19) pt ′ ≤ C (cid:18) ˆ Θ | p Q ∇ f | p dx (cid:19) p + (cid:18) ˆ Θ | f | p dx (cid:19) p . But then, in the second term we have that k√ Q ∇ v k L pt (Θ) < ∞ since ∇ v ∈ L ∞ (Θ)and Q ∈ L ptloc (Ω). Moreover, by (6.1) we have that k u − ψ j k L pt ′ (Θ) ≤ k u − ψ j k b H ,pQ (Θ) , and the right-hand term goes to 0 as j → ∞ . Therefore, we have shown that( uv, v~g + u ∇ v ) ∈ b H ,pQ, (Θ). (cid:3) Proof of Theorem 2.7.
Fix v ∈ Lip (Θ). Our goal is to show that the global Sobolevestimate (2.5) holds. It will be enough to show that for some η , 1 < η < σ ,(6.2) (cid:18) ˆ Θ | v | pη dx (cid:19) pη ≤ C (cid:18) ˆ Θ | p Q ∇ v | p dx (cid:19) p . For given this, by the weak global Sobolev inequality (1.3), we have that (cid:18) ˆ Θ | v | pσ dx (cid:19) pσ ≤ C (cid:20)(cid:18) ˆ Θ | p Q ∇ v | p dx (cid:19) p + (cid:18) ˆ Θ | v | p dx (cid:19) p (cid:21) ≤ C (cid:20)(cid:18) ˆ Θ | p Q ∇ v | p dx (cid:19) p + (cid:18) ˆ Θ | v | pη dx (cid:19) pη (cid:21) ≤ C (cid:18) ˆ Θ | p Q ∇ v | p dx (cid:19) p , and this is the desired inequality.To prove (6.2), fix q ∈ [ p ′ , ∞ ) ∩ ( pσ ′ , ∞ ) and let η = q ′ >
1; note that 1 < η < σ since q > pσ ′ > σ ′ . Then by duality we have that(6.3) (cid:18) ˆ Θ | v | pη dx (cid:19) pη = (cid:20)(cid:18) ˆ Θ ( | v | p ) η dx (cid:19) η (cid:21) p = sup (cid:20) ˆ Θ ϕ | v | p dx (cid:21) p , where the supremum is taken over all non-negative ϕ ∈ L q (Θ), k ϕ k L q (Θ) = 1.Fix a non-negative function ϕ ∈ L q (Θ), k ϕ k L q (Θ) = 1; we estimate the last integral.Fix τ ∈ (0 , τ will be determined below. Since q ≥ p ′ and Θ isbounded, ϕ ∈ L p ′ (Θ), and so by Proposition 4.4, there exists ~ u τ = ( u τ , ~g τ ) ∈ b H ,pQ, (Θ) that is a weak solution of the Dirichlet problem (4.1). Since | v | p ∈ Lip (Θ), we canuse it as a test function in the definition of weak solution. This yields ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ∇ ( | v | p ) i dx + τ ˆ Θ | u τ | p − u τ | v | p dx = − ˆ Θ ϕ | v | p dx. If we take absolute values, rearrange terms and apply H¨older’s inequality, we get ˆ Θ ϕ | v | p dx ≤ p (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ∇ v i| v | p − sgn( v ) dx (cid:12)(cid:12)(cid:12)(cid:12) + τ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ | u τ | p − u τ | v | p dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ p ˆ Θ h Q~g τ , ~g τ i p − h Q ∇ v, ∇ v i | v | p − dx + τ k u τ k p − L ∞ (Θ) k v k pL p (Θ) ≤ p (cid:18) ˆ Θ h Q~g τ , ~g τ i p | v | p dx (cid:19) p ′ (cid:18) ˆ Θ h Q ∇ v, ∇ v i p dx (cid:19) p (6.4) + τ k u τ k p − L ∞ (Θ) k v k pL p (Θ) . To estimate (6.4), define A = ˆ Θ h Q~g τ , ~g τ i p | v | p dx. Let ~h = pu τ | v | p − sgn( v ) ∇ v + | v | p ~g τ ; then by Lemma 6.1 we have ( u τ | v | p , ~h ) ∈ b H ,pQ, (Θ).Moreover, we have that ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ~h i dx = A + p ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ∇ v i u τ sgn( v ) | v | p − dx. LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 21
Since ~ u τ is a weak solution of (4.1) and ( u τ | v | p , ~h ) can be used as a test function, wehave that A ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ~h i dx (cid:12)(cid:12)(cid:12)(cid:12) + p (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ h Q~g τ , ~g τ i p − h Q~g τ , ∇ v i u τ sgn( v ) | v | p − dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ ϕu τ | v | p dx (cid:12)(cid:12)(cid:12)(cid:12) + τ ˆ Θ | u τ | p | v | p dx + p ˆ Θ h Q~g τ , ~g τ i p − h Q ∇ v, ∇ v i | u τ || v | p − dx ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Θ ϕu τ | v | p dx (cid:12)(cid:12)(cid:12)(cid:12) + τ ˆ Θ | u τ | p | v | p dx + C ( p ) ˆ Θ h Q ∇ v, ∇ v i p | u τ | p dx + 12 ˆ Θ h Q~g τ , ~g τ i p | v | p dx ;the last inequality follows from Young’s inequality. The last term equals A . More-over, since k ϕ k L q (Θ) = 1, by Proposition 5.1 there exists a constant K , independentof τ , such that k u τ k L ∞ (Θ) ≤ K . Therefore, the above inequality yields(6.5) A ≤ C ( p ) (cid:20) K ˆ Θ ϕ | v | p dx + τ K p k v k pL p (Θ) + K p ˆ Θ h Q ∇ v, ∇ v i p dx (cid:21) . Irrespective of which term on the right-hand side of (6.5) is the maximum, if wecombine (6.4) with (6.5) we get(6.6) ˆ Θ ϕ | v | p dx ≤ C ( p, K ) (cid:20) ˆ Θ h Q ∇ v, ∇ v i p dx + τ k v k pL p (Θ) (cid:21) . To see that this is true, suppose first that the largest term is τ K p k v k pL p (Θ) . Then byYoung’s inequality we have that ˆ Θ ϕ | v | p dx ≤ Cτ p − p K p − k v k p − L p (Θ) (cid:18) ˆ Θ h Q ∇ v, ∇ v i p dx (cid:19) p + τ K p − k v k pL p (Θ) = CK p − (cid:20) τ p − p k v k p − L p (Θ) (cid:18) ˆ Θ h Q ∇ v, ∇ v i p dx (cid:19) p + τ k v k pL p (Θ) (cid:21) ≤ CK p − (cid:20) ˆ Θ h Q ∇ v, ∇ v i p dx + τ k v k pL p (Θ) (cid:21) . The other estimates are proved similarly.
Given inequality (6.6) it is now straightforward to prove the desired estimate: (cid:18) ˆ Θ ϕ | v | p dx (cid:19) p ≤ C ( p, K ) p (cid:20) ˆ Θ h Q ∇ v, ∇ v i p dx + τ k v k pL p (Θ) (cid:21) p ≤ C ( p, K ) p (cid:20)(cid:18) ˆ Θ h Q ∇ v, ∇ v i p dx (cid:19) p + τ p (cid:18) ˆ Θ | v | pη dx (cid:19) pη (cid:21) . Fix τ < τ C ( p, K ) ≤ . Since this constant is independent of ϕ , if wecombine this inequality with the duality estimate (6.3), we get that (cid:18) ˆ Θ | v | pη dx (cid:19) pη ≤ C ( p, K ) (cid:18) ˆ Θ h Q ∇ v, ∇ v i p dx (cid:19) p + 12 (cid:18) ˆ Θ | v | pη dx (cid:19) pη . If we re-arrange terms we get (6.2) and our proof is complete. (cid:3)
References [CRW] Seng-Kee Chua, S. Rodney and R. L. Wheeden, A compact embedding theorem for gener-alized Sobolev spaces, Pacific J. Math. 265 (2013), 17–57.[CIM] D. Cruz-Uribe, J. Isralowitz, and K. Moen, Two weight bump conditions for matrix weights,preprint, 2017.[CMN] D. Cruz-Uribe, K. Moen, and V. Naibo, Regularity of solutions to degenerate p-Laplacianequations, J. Math. Anal. Appl. 401 no. 1 (2013), 458–478.[CN] D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse H¨older classes, Trans. Amer.Math. Soc., 347(8) (1995), 29412960.[CRR] D. Cruz-Uribe, S. Rodney, and E. Rosta, Poincar´e inequalities and Neumann problems forthe p -Laplacian, Canad. Math. Bull., to appear.[E] L.C. Evans, Partial Differential Equations, graduate studies in mathematics 19, AMS (2010).[FP] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference in honor of A.Zygmund, Wadsworth Math. Series, 1981.[FKS] E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerateelliptic equations, Comm. P. D. E. 7 (1982), 77–116.[GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,Springer Verlag, 1998.[HK] P. Haj lazj and P. Koskela, Sobolev met Poincar´e, Mem. Amer. Math. Soc. 145, no. 688 (2000).[KMR] L. Korobenko, D. Maldonado, and C. Rios, From Sobolev inequality to doubling, Proc.Amer. Math. Soc. 143 (2015), no. 9, 4017–4028.[PL] P. Lindqvist, Notes on the p-Laplace equation, volume 102 of Report. University of Jyv¨askyl¨aa Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Jyv¨askyl¨a, 2006.[M] J. O. MacLellan, undergraduate honors thesis, University of Cape Breton, 2017.[MR] D. D. Monticelli and S. Rodney, Existence and spectral theory for weak solutions of Neu-mann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients, J.Differential Equations, 259(8) (2015), 4009–4044.[MRW1] D. D. Monticelli, S. Rodney and R. L. Wheeden, Boundedness of weak solutions of degen-erate quasilinear equations with rough coefficients, J. Diff. Int. Eq. 25 (2012), 143–200. LOBAL SOBOLEV INEQUALITIES AND DEGENERATE P-LAPLACIAN EQUATIONS 23 [MRW2] D. D. Monticelli, S. Rodney and R. L. Wheeden, Harnack’s inequality and H¨older conti-nuity for weak solutions of degenerate quasilinear equations with rough coefficients, NonlinearAnalysis (2015), http://dx.doi.org/10.1016/j.na.2015.05.029.[R] S. Rodney, A degenerate Sobolev inequality for a large open set in a homogeneous space, Trans.Amer. Math. Soc. 362 (2010), 673–685.[SW1] E. T. Sawyer and R. L. Wheeden, H¨older continuity of weak solutions to subelliptic equationswith rough coefficients, Memoirs Amer. Math. Soc. 847 (2006).[SW2] E. T. Sawyer and R. L. Wheeden, Degenerate Sobolev spaces and regularity of subellipticequations, Trans. Amer. Math. Soc., 362 (2010), 1869–1906.[Sh] R. E. Showalter, Monotone Operators in Banach Spaces and Non-Linear Partial DifferentialEquations, Mathematical Surveys and Monographs Vol. 49, AMS (1996).
David Cruz-Uribe, OFS, Dept. of Mathematics, University of Alabama, Tuscaloosa,AL 35487, USA
E-mail address : [email protected] Scott Rodney, Dept. of Mathematics, Physics and Geology, Cape Breton Univer-sity, Sydney, NS B1Y3V3, CA
E-mail address ::