Anisotropic Picone identities and anisotropic Hardy inequalities

Journal of Inequalities and Applications, Jan 2017

In this paper, we derive an anisotropic Picone identity for the anisotropic Laplacian, which contains some known Picone identities. As applications, a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy type inequality are shown. MSC: 26D10, 26D15.

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Anisotropic Picone identities and anisotropic Hardy inequalities

Feng and Cui Journal of Inequalities and Applications Anisotropic Picone identities and anisotropic Hardy inequalities Tingfu Feng 0 Xuewei Cui 0 0 Department of Applied Mathematics, Northwestern Polytechnical University , Xi'an, Shaanxi 710072 , P.R. China In this paper, we derive an anisotropic Picone identity for the anisotropic Laplacian, which contains some known Picone identities. As applications, a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy type inequality are shown. MSC: Primary 26D10; secondary 26D15 - ∂u pi– ∂u ∂xi ∂xi ∂u p– ∂u ∂xi ∂xi Allegretto and Huang [], Dunninger [] independently extended (.) to a p-Laplacian, for differentiable functions v >  and u ≥ , u u = |∇u| + v |∇v| –  v ∇v · ∇u = |∇u| – ∇ ∇v. = |∇u|p – ∇ (a(x)u ) + b(x)u = , (a(x)v ) + b(x)v = , where u and v are differentiable functions in x, and proved the identity that, for the differentiable function v(x) = , Theorem . (Anisotropic Picone identity) Let v >  and u ≥  be two differentiable functions in the set ⊂ Rn, and denote R(u, v) = L(u, v) = ∂u pi ∂xi ∂u pi ∂xi upi– ∂v pi– ∂v ∂u pi vpi– ∂xi ∂xi ∂xi ∂v pi– ∂v ∂xi ∂xi ∂u pi dx ≥ 2 Proof of Theorem 1.1 Proof of Theorem . One derives easily that R(u, v) = L(u, v). Moreover, we have L(u, v) ≥ ; ∂u pi ∂xi ∂u pi ∂xi ∂u pi ∂xi = L(u, v), ∂u pi ∂xi R(u, v) = L(u, v) = n ∂ upi ∂v pi– ∂v i= ∂xi vpi– ∂xi ∂xi n piupi– ∂∂xui vpi– – upi (pi – )vpi– ∂∂xvi ∂v pi– ∂v ∂xi ∂xi upi– ∂v pi– ∂v ∂u pi vpi– ∂xi ∂xi ∂xi upi– ∂v pi– ∂u pi vpi– ∂xi ∂xi I = II = i= n  ∂u pi + pi –  pi pi ∂xi pi upi– ∂v pi– ∂u pi vpi– ∂xi ∂xi ∂∂xvi ∂∂xui – ∂∂xvi ∂∂xui .  ∂u pi + pi –  ≤ pi pi ∂xi pi ∂∂xui = uv ∂∂xvi . Using II = , it implies ∂∂xui = c ∂∂xvi . Putting (.) into (.) yields u = cv. (b) If u(x) = , then we denote S = {x ∈ |u(x) = } and ∂∂xui =  a.e. in S. Thus with the norms u W,(pi)( ) = |u| dx + ∂u pi ∂xi u W,(pi)( ) = ∂u pi ∂xi in= f(x)upi–, x ∈ , upi ∂ i= vpi– ∂xi ∂v pi– ∂v ∂xi ∂xi f(x)upi , x ∈ , Proof Suppose that v to (.) does not change sign, without loss of generality, let v >  in . By (.), (.), and (.), we observe must change sign.  ≤ L(u, v) dx = R(u, v) dx ∂u pi ∂xi ∂v pi– ∂v ∂xi ∂xi which is a contradiction. This completes the proof. 3 Proof of Theorem 1.3 To prove Theorem ., we need a lemma from Theorem .. v = ∂ v pi– ∂ v ∂ xi ∂ xi ≥ kihi(x)vpi–,  then, for any  ≤ u ∈ C( ), we have ∂ u pi ∂ xi dx ≥ hi(x)upi dx. Proof By (.) and (.), we see  ≤ L(u, v) dx = R(u, v) dx where βj = ppj– j and vi = ∂ u pi ∂ xi ∂ u pi ∂ xi ∂ u pi ∂ xi ∂ v pi– ∂ v ∂ xi ∂ xi ∂ v pi– ∂ v ∂ xi ∂ xi hi(x)upi dx, which implies (.). Proof of Theorem . Without loss of generality, we let  ≤ u ∈ C∞. To use Lemma ., we introduce the auxiliary function ∂ xi ∂ xi ∂ v pi– ∂ v pi– ∂ v ∂ xi = βipi– v¯ipi–|xi|βipi–βi–pi+, = βipi–vipi–|xi|βipi–βi–pi xi, ∂ v pi– ∂ v ∂ xi ∂ xi i= ai we have by taking ai = |xi|, |∇u| dx = for ai ≥ , i = , . . . , n, ∂ u  A ∂ xi i= |xi| |∇u|p dx ≥ which gives ai ≤ aip ≥ n– p– for ai ≥ , i = , . . . , n, i= |xi| ≥ n– p– Putting (.) into the right-hand side of (.), On the other hand, A i= ∂ u p ∂ xi dx ≤ ∂ xi dx = |∇u|p dx. 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Tingfu Feng, Xuewei Cui. Anisotropic Picone identities and anisotropic Hardy inequalities, Journal of Inequalities and Applications, 2017, 16, DOI: 10.1186/s13660-017-1292-4