On the Hardy-Sobolev-Maz'ya inequality and its generalizations
aa r X i v : . [ m a t h . A P ] M a r On the Hardy-Sobolev-Maz’ya inequalityand its generalizations
Yehuda Pinchover
Department of MathematicsTechnion - Israel Institute of TechnologyHaifa 32000, [email protected]
Kyril Tintarev
Department of MathematicsUppsala UniversitySE-751 06 Uppsala, [email protected]
June 7, 2018
Abstract
The paper deals with natural generalizations of the Hardy-Sobolev-Maz’ya inequality and some related questions, such as the optimalityand stability of such inequalities, the existence of minimizers of the as-sociated variational problem, and the natural energy space associatedwith the given functional. . . Key words . quasilinear elliptic operator, p -Laplacian, ground state, positive solutions, Hardy-Sobolev-Maz’ya inequality. The term “inequalities of Hardy-Sobolev type” refers, somewhat vaguely, tofamilies of inequalities that in some way interpolate the Hardy inequality Z Ω |∇ u ( x ) | p d x ≥ C ( N, p, K, Ω) Z Ω | u ( x ) | p dist( x, K ) p d x u ∈ C ∞ (Ω \ K ) , (1.1)1here Ω ⊂ R N is an open domain and K ⊂ ¯Ω is a nonempty closed set, andthe Sobolev inequality Z Ω |∇ u ( x ) | p d x ≥ C (cid:18)Z Ω | u ( x ) | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) , (1.2)where C >
0, 1 < p < N , and p ∗ def = pN/ ( N − p ) is the corresponding Sobolevexponent. Throughout the paper we repeatedly consider the following par-ticular case. Example 1.1.
Let Ω = R N = R n × R m , where 1 ≤ m < N , and let K = R n ×{ } . We denote the variables of R n and R m as z and y respectively,and set R N = R n × ( R m \ { } ). It is well known that the Hardy inequality(1.1) holds with the best constant C ( N, p, R n × { } , R N ) = (cid:12)(cid:12)(cid:12)(cid:12) m − pp (cid:12)(cid:12)(cid:12)(cid:12) p . (1.3)An elementary family of Hardy-Sobolev inequalities can be obtained byH¨older interpolation between the Hardy and the Sobolev inequalities. Moresignificant inequalities of Hardy-Sobolev type with the best constant in theHardy term can be derived as consequences of Caffarelli-Kohn-Nirenberg in-equality ([7, 15]) that provides estimates in terms of the weighted gradientnorm R | ξ | α |∇ u | p d ξ . The substitution u = | y | β v into the Caffarelli-Kohn-Nirenberg inequality can be used to produce inequalities that combine termswith the critical exponent and with the Hardy potential. Such inequalitiesare known as Hardy-Sobolev-Maz’ya (or HSM for brevity) inequalities. Inparticular, in [18, Section 2.1.6, Corollary 3] Maz’ya proved the HSM in-equality Z R N |∇ u | d y d z − (cid:18) m − (cid:19) Z R N | u | | y | d y d z ≥ C Z R N | u | ∗ d y d z ! / ∗ u ∈ C ∞ ( R N ) , (1.4)where C > N >
2, and 1 ≤ m < N . This HSM inequality is false for m = N and reduces to the Sobolev inequality for m = 2. Since the left-hand2ide of (1.4) induces a Hilbert norm, the inequality holds on D , ( R N ), thecompletion of C ∞ ( R N ) in the gradient norm, which coincides with D , ( R N )for all m >
1, in particular, C ∞ ( R N ) may be replaced by C ∞ ( R N ) unless m = 1.A joint paper of Filippas, Maz’ya and Tertikas [10] gives the followinggeneralization of the HSM inequality (1.4). Example 1.2.
Let 2 ≤ p < N , p = m < N , and let Ω ⊂ R N be a boundeddomain. Let K be a compact C -manifold without boundary embedded in R N , of codimension m such that K ⋐ Ω for 1 < m < N (i.e., K is compactin Ω), or K = ∂ Ω for m = 1. Assume further that − ∆ p (cid:2) dist ( · , K ) ( p − m ) / ( p − (cid:3) ≥ \ K, (1.5)where ∆ p ( u ) def = ∇ · ( |∇ u | p − ∇ u ) is the p -Laplacian. Then for all u ∈ C ∞ (Ω \ K ) we have Z Ω |∇ u ( x ) | p d x − (cid:12)(cid:12)(cid:12)(cid:12) m − pp (cid:12)(cid:12)(cid:12)(cid:12) p Z Ω | u ( x ) | p dist ( x, K ) p d x ≥ C (cid:18)Z Ω | u ( x ) | p ∗ d x (cid:19) p/p ∗ . (1.6)For N = 3 Benguria, Frank and Loss [4] have shown recently that thebest constant C in (1.4) is the Sobolev constant S . Mancini and Sandeep[16] have studied the analog of HSM on the hyperbolic space and its closeconnection to the original HSM inequality.In the present paper we consider a nonnegative functional Q of the form Q ( u ) def = Z Ω ( |∇ u | p + V | u | p ) d x u ∈ C ∞ (Ω) , (1.7)where Ω ⊆ R N is a domain, V ∈ L ∞ loc (Ω), and 1 < p < ∞ . We study severalquestions related to extensions of inequalities (1.4) and (1.6). In Section 2,we deal with generalizations of these HSM inequalities for the functional Q .It turns out, that in the subcritical case a weighted HSM inequality holdstrue, where the weight appears in the Sobolev term. In the critical case,one needs to add a Poincar´e-type term (a one-dimensional p -homogeneousfunctional), and we call it Hardy-Sobolev-Maz’ya-Poincar´e (or HSMP forbrevity) inequality. We show that under “small” perturbations such HSM-type inequalities are preserved (with the original Sobolev weight). We also3ddress the question concerning the optimal weight in the generalized HSMinequality.In Section 3, we study a natural energy space D , V (Ω) for nonnegative sin-gular Schr¨odinger operators, and discuss the existence of minimizers for theHSM inequality in this space, that is, minimizers of the equivalent Caffarelli-Kohn-Nirenberg inequality. Finally, in Section 4 we prove that a relatedfunctional ˆ Q which satisfies C − Q ≤ ˆ Q ≤ CQ for some C > C ∞ (Ω)-functions. For p = 2, this normcoincides (on the above cone) with the D , V (Ω)-norm defined in [20]. It isour hope that this approach paves the way to circumvent the general lack ofconvexity of the nonnegative functional Q for p = 2. We need the following definition.
Definition 2.1.
Let Ω ⊆ R N be a domain, V ∈ L ∞ loc (Ω), and 1 < p < ∞ .Assume that the functional Q ( u ) = Z Ω ( |∇ u | p + V | u | p ) d x (2.1)is nonnegative on C ∞ (Ω). A function ϕ ∈ C (Ω) is a ground state for thefunctional Q if ϕ is an L p loc -limit of a nonnegative sequence { ϕ k } ⊂ C ∞ (Ω)satisfying Q ( ϕ k ) → , and Z B | ϕ k | p d x = 1 , for some fixed B ⋐ Ω (such a sequence { ϕ k } is called a null sequence ). Thefunctional (1.7) is called critical if Q admits a ground state and subcritical or weakly coercive if it does not.The following statement (see [22]) is a generalization of HSM inequality.Inequality (2.4) might be called Hardy-Sobolev-Maz’ya-Poincar´e (HSMP)-type inequality. Theorem 2.2.
Let Q be a nonnegative functional on C ∞ (Ω) of the form (1.7) , and let < p < N . i) The functional Q does not admit a ground state if and only if thereexists a positive continuous function W such that Q ( u ) ≥ (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) . (2.2) (ii) If Q admits a ground state ϕ , then ϕ is the unique global positive(super)-solution of the Euler-Lagrange equation Q ′ ( u ) def = − ∆ p ( u ) + V | u | p − u = 0 in Ω . (2.3) Moreover, there exists a positive continuous function W such that for everyfunction ψ ∈ C ∞ (Ω) with R Ω ψϕ d x = 0 , the following inequality holds Q ( u ) + C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψu d x (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:18)Z Ω W | u | p ∗ d x (cid:19) p/p ∗ u ∈ C ∞ (Ω) (2.4) with some suitable constant C > . Remark 2.3.
For the relationships between the criticality of Q in Ω andthe p -capacity (with respect to the functional Q ) of closed balls see [22,Theorem 4.5] and [26, 27].Theorem 2.2 applies to the case of Ω = R N and the Hardy potential (seeExample 1.1, and in particular (1.4)), but it does not specify that the weight W in the Sobolev term is the constant function. We note that Example 1.2provides another Hardy-type functional satisfying the HSM inequality withthe weight W = constant.On the other hand, let Ω = R N , with m = N , then the correspondingHardy functional admits a ground state ϕ ( x ) = | x | ( p − N ) /p , and thereforethe HSM inequality does not hold with any weight. Moreover, the HSMPinequality (2.4), which by Theorem 2.2 holds with some weight W , is falsewith the weight W = constant ([11] and Example 2.5).Let us present few other examples which illustrate further the questionof the admissible weights in the HSM and HSMP inequalities. The first twoexamples are elementary but general. In the first one the HSM inequality(2.2) holds with the constant weight function, while in the second example(Example 2.5) such an inequality is false.5 xample 2.4. Consider a nonnegative functional Q of the form (1.7), where V ∈ L ∞ loc (Ω) is nonzero function, and 1 < p < N . For λ ∈ R we denote Q λ ( u ) def = Z Ω ( |∇ u | p + λV | u | p ) d x. Then for every λ ∈ (0 ,
1) there exists
C > Q λ ( u ) ≥ C k u k pp ∗ u ∈ C ∞ (Ω) , (2.5)where C = C ( N, p, λ ) >
0. This HSM inequality follows from Q λ ( u ) = (1 − λ ) Z Ω |∇ u | p d x + λQ ( u ) ≥ (1 − λ ) Z Ω |∇ u | p d x, and the Sobolev inequality. Example 2.5.
Let Q ≥ < p < N . Suppose that Q admits ground state ϕ / ∈ L p ∗ (Ω), and let { ϕ k } be a null sequence (seeDefinition 2.1) such that ϕ k → ϕ locally uniformly in Ω (for the existenceof a locally uniform convergence null sequence, see [22, Theorem 4.2]). Let V ∈ L ∞ (Ω) be a nonzero nonnegative function with a compact support.Then Q ( ϕ k ) + Z Ω V | ϕ k | p d x → Z Ω V | ϕ | p d x < ∞ , while Fatou’s lemma implies that k ϕ k k p ∗ → ∞ . Therefore, the subcriticalfunctional Q V ( u ) def = Q ( u ) + Z Ω V | u | p d x does not satisfy the HSM inequality (2.2) with the constant weight. Similarargument shows that the critical functional Q does not satisfy the HSMPinequality with the constant weight. Remark 2.6.
Example 2.5 can be slightly generalized by replacing the as-sumption ϕ / ∈ L p ∗ (Ω) with ϕ / ∈ L p ∗ (Ω , W d x ), where W is a continuous pos-itive weight function. Under this assumption it follows that the functional Q V and Q do not satisfy HSM and respectively HSMP inequality with theweight W . 6 xample 2.7. In [12, Theorem C], Filippas, Tertikas and Tidblom provedthat a nonnegative functional Q of the form (1.7) with p = 2 satisfies theHSM inequality in a smooth domain Ω with W = constant if the equation Q ′ ( u ) = 0 has a positive C -solution ϕ such that the following L -Hardy-typeinequality Z Ω ϕ N − / ( N − |∇ u | d x ≥ C Z Ω ϕ N/ ( N − |∇ ϕ | | u | d x u ∈ C ∞ (Ω) . holds true. Example 2.8.
Consider the function X ( r ) def = ( | log r | ) − r > . Let Ω ⊂ R N , N >
2, be a bounded domain and let
D > sup x ∈ Ω | x | . Thefollowing inequality is due to Filippas and Tertikas [11, Theorem A, and thecorresponding Corrigendum], see also [1]. Z Ω |∇ u | d x − (cid:18) N − (cid:19) Z Ω | u | | x | d x ≥ C (cid:18)Z Ω | u | ∗ X ( | x | /D ) N/ ( N − d x (cid:19) / ∗ u ∈ C ∞ (Ω) . (2.6)In this case the HSM inequality does not hold with W = constant (cf. Ex-ample 2.5 and Remark 2.6).We now consider the question whether the weight W in the HSM inequal-ity (2.2) is preserved (up to a constant multiple) under small perturbations. Theorem 2.9.
Let Ω be a domain in R N , N > , and let V ∈ L ∞ loc (Ω) .Assume that the following functional Q satisfies the HSM inequality Q ( u ) def = Z Ω (cid:0) |∇ u | + V | u | (cid:1) d x ≥ (cid:18)Z Ω W | u | ∗ d x (cid:19) / ∗ u ∈ C ∞ (Ω)(2.7) with some positive continuous function W . Let ˜ V ∈ L ∞ loc (Ω) be a nonzeropotential satisfying | ˜ V | N/ W (2 − N ) / ∈ L (Ω) , (2.8)7 nd consider the one-parameter family of functionals ˜ Q λ defined by ˜ Q λ ( u ) def = Q ( u ) + λ Z Ω ˜ V | u | d x, where λ ∈ R .(i) If ˜ Q λ is nonnegative on C ∞ (Ω) and does not admit a ground state,then ˜ Q λ ( u ) ≥ C (cid:18)Z Ω W | u | ∗ d x (cid:19) / ∗ u ∈ C ∞ (Ω) , (2.9) where C is a positive constant.(ii) If ˜ Q λ is nonnegative on C ∞ (Ω) and admits a ground state v , then forevery ψ ∈ C ∞ (Ω) such that R Ω ψv d x = 0 we have ˜ Q λ ( u ) + C (cid:18)Z Ω ψu d x (cid:19) ≥ C (cid:18)Z Ω W | u | ∗ d x (cid:19) / ∗ u ∈ C ∞ (Ω) (2.10) with suitable positive constants C, C > .(iii) The set S def = { λ ∈ R | ˜ Q λ ≥ on C ∞ (Ω) } is a closed interval with a nonempty interior which is bounded if and only if ˜ V changes its sign on a set of a positive measure in Ω . Moreover, λ ∈ ∂S ifand only if ˜ Q λ is critical in Ω .Proof. (i)–(ii) Let D , λ ˜ V (Ω) denote the completion of C ∞ (Ω) with respect tothe norm defined by the square root of the left-hand side of (2.9) if ˜ Q λ doesnot admit a ground state, and by the square root of the left-hand side of(2.10) if ˜ Q λ admits a ground state (see [20]). Similarly, we denote by D , V (Ω)the completion of C ∞ (Ω) with respect to the norm defined by the square rootof the left-hand side of (2.7). We denote the norms on D , λ ˜ V (Ω) and D , V (Ω)by k · k D , λ ˜ V and k · k D , V respectively.Assume that (2.9) (respect. (2.10)) does not hold. Then there exists asequence { u k } ⊂ C ∞ (Ω) such that k u k k D , λ ˜ V → , and Z Ω W | u k | ∗ d x = 1 . (2.11)By [20, Proposition 3.1], the space D , λ ˜ V (Ω) is continuously imbedded into W , (Ω) and therefore, u k → W , (Ω). Consequently, for any K ⋐ Ω we8ave lim k →∞ Z K | ˜ V || u k | d x = 0 . (2.12)On the other hand, (2.8) and H¨older inequality imply that for any ε > K ε ⋐ Ω such that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω \ K ε ˜ V | u k | d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z Ω \ K ε | ˜ V | N/ W (2 − N ) / d x (cid:19) /N (cid:18)Z Ω W | u k | ∗ d x (cid:19) / ∗ < ε. (2.13)Since k u k k D , V ≤ k u k k D , λ ˜ V + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω λ ˜ V | u k | d x (cid:12)(cid:12)(cid:12)(cid:12) / , it follows from (2.11)–(2.13) that the sequence u k → D , V (Ω). There-fore, (2.7) implies that R Ω W | u k | ∗ d x → R Ω W | u k | ∗ d x = 1. Consequently, (2.9) (resp. (2.10)) holds true.(iii) It follows from [21, Proposition 4.3] that S is an interval, and that λ ∈ int S implies that Q λ is subcritical in Ω. The claim on the boundednessof S is trivial and left to the reader.On the other hand, suppose that for some λ ∈ R the functional ˜ Q λ issubcritical. By part (i), ˜ Q λ satisfies the HSM inequality with weight W .Therefore, (2.13) (with K ε = ∅ ) implies that˜ Q λ ( u ) ≥ C (cid:18)Z Ω W | u | ∗ d x (cid:19) / ∗ ≥ C (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ˜ V | u | d x (cid:12)(cid:12)(cid:12)(cid:12) u ∈ C ∞ (Ω) . (2.14)Therefore, λ ∈ int S . Consequently, λ ∈ ∂S implies that ˜ Q λ is critical in Ω.In particular, 0 ∈ int S . Example 2.10.
Let Ω = R N , where N ≥
3, and let V ∈ L N/ ( R N ) suchthat V (cid:3) V is a short range potential). Fix µ < ( N − /
4. Thenthe classical Hardy inequality together with Example 2.4 and Theorem 2.9imply that there exists λ ∗ > λ < λ ∗ , we have the followingHSM inequality Z R N |∇ u | d x − µ Z R N | u | | x | d x + λ Z R N V ( x ) | u | d x ≥ C λ (cid:18)Z R N | u | ∗ d x (cid:19) / ∗ u ∈ C ∞ ( R N ) . (2.15)9n the other hand, if λ = λ ∗ , then the associated functional is critical andsatisfies the corresponding HSMP inequality with the weight function W =constant. Recall that the HSM and HSMP inequalities for µ = ( N − / W = constant (see, Example 2.5 and [11]). Example 2.11.
Consider again Example 1.2 with p = 2 < N , and 2 = m
We note that even under the less restricted assumptions of[17, Theorem 1.2], with p = 2 and λ = λ ∗ , one can show that the positivesolution u ∗ of Equation (1.14) in [17] is actually a ground state. Therefore, u ∗ is the unique (up to a multiplicative constant) global positive supersolutionof that equation, and the corresponding functional is critical.Indeed, Lemma 5.1 of [17] implies that any positive supersolution of [17,Equation (1.14)] satisfies Cu ( x ) ≥ dist ( x, ∂ Ω) / x ∈ Ω . (2.21)On the other hand, [17, Theorem 1.2] implies that the positive solution u ∗ satisfies u ∗ ( x ) ≍ dist ( x, ∂ Ω) / x ∈ Ω , (2.22)where f ≍ g means that there exists a positive constant C such that C − ≤ f /g ≤ C in Ω. Now, take a positive supersolution u , and let ε be the maximal11ositive number such that u − εu ∗ ≥ ε is well defined. By the strong maximum principle it follows that either u = εu ∗ , or u − εu ∗ >
0. Consequently, (2.21) and (2.22) imply that thereexists a positive constant C such that u − εu ∗ ≥ C dist ( x, ∂ Ω) / ≥ C u ∗ in Ω , which is a contradiction to the definition of ε . D , V (Ω) and minimizers for theHSM inequality Consider again the HSM inequality (1.4). This inequality clearly extends to D , ( R N ) for m > D , ( R N ) for m = 1, but since the quadratic form Q ( u ) in the left-hand side of (1.4) induces a scalar product on C ∞ ( R N ), thenatural domain of Q is the completion of C ∞ ( R N ) with respect to the norm Q ( · ) / . Recall [20] that given a general subcritical functional Q of the form(1.7) (with p = 2), we denote such a completion by D , V (Ω). Similarly to thestandard definition of D , ( R N ) for N = 1 ,
2, when Q admits a ground state,one appends to Q ( u ) a correction term of the form (cid:0)R Ω ψu d x (cid:1) . Hence, by(2.2) and (2.4) the space D , V (Ω) is continuously imbedded into a weighted L ∗ -space.In the particular case (1.4), V is the Hardy potential [( m − / | y | − .By (1.4), the space D , V ( R N ) is continuously imbedded into L ∗ ( R N ), thusits elements can be identified as measurable functions. The substitution u = | y | (2 − m ) / v transforms HSM inequality (1.4) into an inequality of Caffarelli-Kohn-Nirenberg type: Z R N | y | − m |∇ v | d y d z ≥ C (cid:18)Z R N | y | (2 − m )2 ∗ / | v | ∗ d y d z (cid:19) / ∗ v ∈ D , ( R N , | y | − m d y d z ) . (3.1)The left-hand side of (3.1) defines a Hilbert space isometric to D , V ( R N ).However, the Lagrange density |∇ u | − (cid:18) m − (cid:19) | u | | y | (3.2)12s no longer integrable for an arbitrary u ∈ D , V ( R N ). The integrable La-grange density of (3.1), | y | − m |∇ ( u | y | ( m − / ) | can be equated to (3.2) bypartial integration when u ∈ C ∞ ( R N ), but this connection does not extendto the whole of D , V ( R N ) as the terms that mutually cancel in the partialintegration on C ∞ ( R N ) might become infinite. In particular, it should not beexpected a priori that the minimizer for HSM inequality in D , V ( R N ) wouldhave a finite gradient in L ( R N , d x ).Existence of minimizers for the variational problem associated with (3.1)is proved in [25] for all codimensions 0 < m < N , where N >
3. The ex-istence proof is based on concentration compactness argument that utilizesinvariance properties of the problem. Similarly to other problems where lackof compactness stems from a noncompact equivariant group of transforma-tions, some general domains and potentials admit minimizers and some donot, and analogy with similar elliptic problems in D , ( R N ) provides usefulinsights (see for example [23]). Q for p > The definition of D , V (Ω) cannot be applied to other values of p , since for p = 2 the positivity of the functional Q on C ∞ (Ω) does not necessarilyimply its convexity, and thus it does not give rise to a norm. For the lackof convexity when p >
2, see an elementary one-dimensional counterexampleat the end of [8], and also the proof of Theorem 7 in [14]. For p <
2, see [13,Example 2].On the other hand, by [21, Theorem 2.3], the functional Q is nonnegativeon C ∞ (Ω) if and only if the equation Q ′ ( u ) = 0 in Ω admits a positive globalsolution v . With the help of such a solution v , one has the identity [9, 2, 3]: Q ( u ) = Z Ω L v ( w ) d x u ∈ C ∞ (Ω) , where w def = u/v , the Lagrangian L v ( w ) is defined by L v ( w ) def = | v ∇ w + w ∇ v | p − w p |∇ v | p − pw p − v |∇ v | p − ∇ v ·∇ w ≥ w ∈ C ∞ (Ω) , (4.1)and C ∞ (Ω) denotes the cone of all nonnegative functions in C ∞ (Ω).The following proposition claims that the nonnegative Lagrangian L v ( w ),which contains indefinite terms, is bounded from above and from below bymultiples of a simpler Lagrangian. 13 roposition 4.1 ([19, Lemma 2.2]) . Let v be a positive solution of the equa-tion Q ′ ( u ) = 0 in Ω . Then L v ( w ) ≍ v |∇ w | ( w |∇ v | + v |∇ w | ) p − ∀ w ∈ C ∞ (Ω) . (4.2) In particular, for p ≥ , we have L v ( w ) ≍ ˆ L v ( w ) def = v p |∇ w | p + v |∇ v | p − w p − |∇ w | ∀ w ∈ C ∞ (Ω) . (4.3)Define the simplified energy ˆ Q byˆ Q ( u ) def = Z Ω ˆ L v ( w ) d x w = u/v ∈ C ∞ (Ω) . (4.4)It is shown in [19] that for p > Q is dominated by the other.It follows from Proposition 4.1 that Q ( u ) = Q ( | u | ) ≍ ˆ Q ( | u | ) u ∈ C ∞ (Ω) . In [24], the solvability of equation Q ′ ( u ) = f is proved in the class offunctions u satisfying Q ∗∗ ( u ) < ∞ , where Q ∗∗ ≤ Q is the second convexconjugate (in the sense of Legendre transformation) of Q . If the inequality Q ≤ CQ ∗∗ is true, then Q ∗∗ /p ( u ) would define a norm, and Q would extendto a Banach space, which should be regarded as the natural energy space forthe functional Q .On the other hand, if p >
2, it is not clear whether the functional ˆ Q is convex due to the second term in (4.3). It has, however, the followingconvexity property. Proposition 4.2.
Assume that p ≥ , and let v ∈ C (Ω) be a fixed positivefunction. Consider the functional Q ( ψ ) def = ˆ Q ( vψ /p ) ψ ∈ C ∞ (Ω) , where ˆ Q is defined by (4.3) and (4.4) . Then the functional Q is convex on C ∞ (Ω) . roof. We first split each of the functionals ˆ Q and Q into a sum of twofunctionals:ˆ Q ( u ) def = Z Ω v p |∇ w | p d x, ˆ Q ( u ) def = Z Ω v |∇ v | p − w p − |∇ w | d x w = u/v ∈ C ∞ (Ω) , Q ( ψ ) def = ˆ Q ( vψ /p ) = Z Ω v p |∇ ( ψ /p ) | p d x ψ ∈ C ∞ (Ω) , Q ( ψ ) def = ˆ Q ( vψ /p ) = Z Ω v |∇ v | p − ψ p − /p |∇ ( ψ /p ) | d x ψ ∈ C ∞ (Ω) . Thus, ˆ Q = ˆ Q + ˆ Q , and Q = Q + Q .For t ∈ [0 ,
1] and w , w ∈ C ∞ (Ω), let w t def = h (1 − t ) w p/ + tw p/ i /p . Then ∇ w t = (1 − t ) w p/ − ∇ w + tw p/ − ∇ w h (1 − t ) w p/ + tw p/ i − /p . Therefore, |∇ w t | ≤ [(1 − t ) /p w ] p/ − (1 − t ) /p |∇ w | + ( t /p w ) p/ − t /p |∇ w | h (1 − t ) w p/ + tw p/ i − /p . (4.5)Applying H¨older inequality to the sum in the numerator of (4.5) (with theterms (1 − t ) /p |∇ w | and t /p |∇ w | raised to the power p/
2) and taking intoaccount that the conjugate of p/ − /p , we have |∇ w t | p/ ≤ (1 − t ) |∇ w | p/ + t |∇ w | p/ . (4.6)From (4.6) it follows easily that |∇ w t | p ≤ (1 − t ) |∇ w | p + t |∇ w | p . Setting ψ t def = w p/ t , t ∈ [0 , Q is convex asa function of ψ . The same conclusion extends to Q once we note that w p − |∇ w | = (2 /p ) |∇ w p/ | , and use (4.6) for p = 4. 15et N ( ψ ) def = [ Q ( ψ )] / = h ˆ Q ( vψ /p ) i / ψ ∈ C ∞ (Ω) . (4.7)It is immediate that N ( ψ ) > ψ ∈ C ∞ (Ω), unless ψ = 0, and that N ( λψ ) = λN ( ψ ) for λ ≥
0. Due to Proposition 4.2, the functional N ( · )satisfies the triangle inequality N ( ψ + ψ ) ≤ N ( ψ ) + N ( ψ ) ψ , ψ ∈ C ∞ (Ω) . Thus, we have equipped the cone C ∞ (Ω) with a norm. For p = 2 the func-tional Q = ˆ Q is a positive quadratic form, and thus convex. Consequently,in the subcritical case, Q / extends the functional N to a norm on the whole C ∞ (Ω), and then by completion, to the Hilbert space D , V (Ω). It would beinteresting to introduce D ,pV (Ω) for p > N to C ∞ (Ω). Acknowledgments
Part of this research was done while K. T. was visiting the Technion. K. T.would like to thank the Technion for the kind hospitality. Y. P. acknowledgesthe support of the Israel Science Foundation (grant No. 587/07) founded bythe Israeli Academy of Sciences and Humanities, and the B. and G. GreenbergResearch Fund (Ottawa).
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