Stagnation zones for $mathcal{A}$-harmonic functions on canonical domains

a r X i v :0801.4678v 1 [m a t h .A P ] 30 J a n 2008STAGNATION ZONES FOR A -HARMONIC FUNCTIONS ONCANONICAL DOMAINSVLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINEN Abstract.We study stagnation zones of A -harmonic functions on canonical do-mains in the Euclidean n -dimensional space.Phragmen-Lindel¨o f type theorems are proved.1.IntroductionIn this article we investigate solutions of the A -Laplace equation on canonical domains in the n -dimensional Euclidean space.Suppose that D is a domain in R n ,and let f :D →R be a function.For s >0,a subset ∆⊂D is called s -zone (stagnation zone with the deviation s )of f ,if there exists a constant C such that the difference between C and the function f is smaller than s on ∆.We may,for example,consider difference in the sense of the Euclidean distance,f (x )−C C (∆)=sup x ∈∆|f (x )−C |<s ,the L p-normf (x )−C L p (∆)=∆|f (x )−C |p d H n1/p<s ,or the Sobolev normf (x ) W 1p(∆)=∆|∇f (x )|p d H n1/p<s ,where H d is the d -dimensional Hausdorffmeasure in R n .For discussion about the history of the question,recent results and applications,see [SS06,SS07].Some estimates of stagnation zone sizes for solutions of the A -Laplace equation on locally Lipschitz surfaces and behaviour of solutions in stagnation zones,were given in [Mik07].In this paper we consider solutions of the A -Laplace equation in subdomains of R n of a special form.In two-dimensional case,such domains are2VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENsectors and strips.In higher dimensions,they are conical and cylindrical regions. The special form of domains allows us to obtain more precise results.Below we study stagnation zones of generalized solutions of the A-Laplace equationdiv A(x,∇f)=0(see[HKM93])with boundary conditions of types(see definitions1.7and1.11below):A(x,∇f),n =0,x∈∂D\Gon canonical domains in the Euclidean n-dimensional space,where G is a closed subset of∂D.We will prove Phragm´e n-Lindel¨o f type theorems for solutions of the A-Laplace equation with such boundary conditions.Canonical domains.Let n≥2.Fix an integer k,1≤k≤n,and setd k(x)= k i=1x2i 1/2.We call the setB k(t)={x∈R n:d k(x)<t}a k-ball andΣk(t)={x∈R n:d k(x)=t}a k-sphere in R n.In particular,the symbolΣk(0)denotes the k-sphere with the radius0,i.e.the setΣk(0)= x=(x1,...,x k,...,x n):x k+1=...=x n=0 .For every0<k<n we setp k(x)= n j=k+1x2j 1/2.andΣ∗k(t)={x∈R n:p k(x)=t},t≥0.Let0<α<β<∞befixed,and letD kα,β={x∈R n:α<p k(x)<β}.For k=n−1we also assume that x n>0.Then for k=n−1the D n−1α,βis the a layerbetween two parallel hyperplanes,and for1≤k<n−1the boundary of the domain D kα,βconsists of two coaxial cylindrical surfaces.The intersectionsΣk(t)∩D kα,βare precompact for all t>0.Thus,the functions d k(x)are exhaustion functions for D kα,β.STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS3Figure:D11,2(left)and D21,2in R3.Structure conditions.Let D be a subdomain of R n and letA(x,ξ):D the functionA(x,ξ):R n→R nis defined and continuous with respect toξ.We assume that the functionx→A(x,ξ)is measurable in the Lebesgue sense for allξ∈R n and(1.1)A(x,λξ)=λ|λ|p−2A(x,ξ),λ∈R\{0},p≥1.Suppose that for a.e.x∈D and for allξ∈R n the following properties hold:(1.2)ν1|ξ|p≤ ξ,A(x,ξ) ,|A(x,ξ)|≤ν2|ξ|p−1with p≥1and some constantsν1,ν2>0.We consider the equation(1.3)div A(x,∇f)=0.An important special case of(1.3)is the Laplace equation∆f=ni=1∂2f4VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENFrequencies.Fix t ≥0and p ≥1.Let O be an open subset of Σ∗k (t )(with respect to the relative topology of Σ∗k (t )),and let P be a nonempty closed subset of ∂O .We let(1.4)λp,P (O )=infuO|∇u |p d H n −1O )with(1.5)u |P =0,andn is the unit normal vector to ∂O .If P =∂O we call λp (O )≡λp,P (O )the first frequency of the order p ≥1of the set O .If P =∂O the quantity λp,P (O )is the third frequency .The second frequency is the following quantity:(1.6)µp (O )=sup CinfuO|∇u |p d H n −1O ).Seealso Pˆo lya and Szeg¨o [PS51],Lax[Lax57].Generalized boundary conditions.Let ϕ:D ⊂R n →R be a locally Lipschitz function.We denote by D b (ϕ)the set of all points x ∈D at which ϕdoes not have the differential.Let U ⊂D be a subset and let ∂′U =∂U \∂D be its boundary with respect to D .If ∂′U is (H n −1,n −1)-rectifiable,then it has locally finite perimeter in the sense of De Giorgi and H n −1-almost everywhere on ∂U ,an unit normal vectorU ∩n =0,x ∈∂D \G ,STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS5 if for every subdomain U∈(G,D),(1.9)H n−1 ∂′U∩D b(f) =0,and for every locally Lipschitz functionϕ:n d H n−1= U A(x,∇f),∇ϕ d H n.Heren =0,x∈∂D\G,if for every subdomain U∈(G,D)with(1.9),and for every locally Lipschitz function ϕ:n d H n−1= U A(x,∇f),∇(ϕf) d H n.In the case of a smooth boundary∂D,and f∈C2(D),the relation(1.10)implies (1.3)with(1.8)everywhere on∂D\G.This requirement(1.13)implies(1.3)with (1.12)on∂D\G.See[Mik06,Section7.2.1].The surface integrals exist by(1.9).Indeed,this assumption guarantees that∇f(x) exists H n−1-a.e.on∂′U.The assumption that U∈(G,U)implies existence of a normal vectorn is defined andfinite a.e.on∂′U.2.Saint-Venant principleIn this section,we will prove the Saint-Venant principle for solutions of the A-Laplace equation on canonical domains.Let0<k<n.Fix a domain D=D kα,β∩B k(t0)with t0>0and0<α<β<∞. We write P={x∈∂D:p k(x)=α}and Q={x∈∂D:p k(x)=β}and G=P∪Q. Let t,τ∈(α,β),t<τ,and∆k(t,τ)={x∈D:t<p k(x)<τ}.For s≥0we setσk(s)= x∈∆k(0,∞):p k(x)=s .2.1.Theorem.Letα<τ′<τ′′<β.If f:D→R is a generalized solution of(1.3) with the generalized boundary condition(1.8)on∂D\G,then the inequality(2.2)I1(t,τ′)+C1(t)/ν1≤ I1(t,τ′′)+C1(t)/ν1 exp −ν16VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENholds for all t∈(α,τ′].If f:D→R is a generalized solution of(1.3)with the generalized boundary con-dition(1.12)then(2.3)I1(t,τ′)+C2(t)/ν1≤ I1(t,τ′′)+C2(t)/ν1 exp −ν1U\G→R be a locally Lipschitz function.By(1.13)we haveϕf A(x,∇f),∂′∆k(t,τ)n=∇p k(x)for x∈σk(τ)andSTAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS7 whereC2(t)= σk(t)f A(x,∇f),∇p k(x) d H n−1.Note that we may also choose(2.6)˜C2(τ)=− σk(τ)f A(x,∇f),∇p k(x) d H n−1,to obtain an inequality similar to(2.5).Next we will estimate the right side of(2.5).By(1.2)and the H¨o lder inequalityσk(τ)f A(x,∇f),∇p k(x) d H n−1≤ σk(τ)|f||A(x,∇f)|d H n−1≤ν2 σk(τ)|f||∇f|p−1d H n−1≤ν2 σk(τ)|f|p d H n−1 1/p σk(τ)|∇f|p d H n−1 (p−1)/p. We note that the surfacesσk(τ)are parallel to∂D kα,β.By using(1.4)we may write (2.7) σk(τ)|f|p d H n−1≤λ−1p,Z f(τ)(σk(τ)) σk(τ)|∇f|p d H n−1andσk(τ)f A(x,∇f),∇p k(x) d H n−1 ≤ν2λ−1/p p,Z f(τ)(σk(τ)) σk(τ)|∇f|p d H n−1. By(2.5)and the Fubini theoremν1I1(t,τ)+C2(t)≤ν2λ−1/pp,Z f(τ)(σk(τ))dI1ν2λ1/pp,Z f(τ)(σk(τ))≤dI1ν1 .By integrating this differential inequality we haveexp ν1I1(t,τ′)+C2(t)/ν18VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINEN for arbitraryτ′,τ′′∈(α,β)withτ′<τ′′.We have shown that (2.8)I1(t,τ′)+C2(t)/ν1≤ I1(t,τ′′)+C2(t)/ν1 exp −ν1n d H n−1=0.For an arbitrary constant C,we get from this and(1.10) (2.9) σk(t)∪σk(τ)(f−C) A(x,∇f),STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS9 and by(2.10)we haveν1I1(t,τ)+C1(t)≤ν2µ−1p(σk(τ)) σk(τ)|∇f|p d H n−1orν1I1(t,τ)+C1(t)≤ν2µ−1p(σk(τ))dI1ν2τ′′τ′µp(σk(τ))dτ .3.Stagnation zonesNext we apply the Saint-Venant principle to obtain information about stagnation zones of generalized solutions of the equation(1.3).Wefirst consider zones with respect to the Sobolev norm.Other results of this type follow immediately from well-known imbedding theorems.Stagnation zones with respect to the W1p-norm.We rewrite(2.2)and(2.3)in another form.Fix D kα,β.Letp∗k(x)=p k(x)−α+β2we have−β∗<p∗k(x)<β∗,and we denoteD∗,kβ∗={x∈R n:−β∗<p∗k(x)<β∗}.Let t0>0and−β∗<τ′≤τ′′<β∗.We write(3.2)∆∗,k(τ′,τ′′)={x∈B k(t0):τ′<p∗k(x)<τ′′}andI2(τ′,τ′′)= ∆∗,k(τ′,τ′′)|∇f|p d H n.10VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENLet 0<τ′<τ′′<β∗.By (2.3)we have for t ∈(−τ,τ)I 2(t,τ′)+C 4(t )/ν1≤ I 2(t,τ′′)+C 4(t )/ν1 exp−ν1ν2−τ′−τ′′λ1/pp,Z ∗f(τ)(σ∗,k (τ))dτ,where (3.4)σ∗,k (s )={x ∈∆∗,k (−∞,∞):p ∗k (x )=s }.By adding these inequalities and noting that C 4(t )+˜C4(t )=0we obtain I 2(−τ′,t )+I 2(t,τ′)≤ I 2(−τ′′,t )+I 2(t,τ′′)×max exp −ν1ν2τ′′τ′λ1/pp,Z ∗f (τ)(σ∗,k (τ))dτ.Thus we have the estimate (3.5)I 2(−τ′,τ′)≤I 2(−τ′′,τ′′)×max exp−ν1ν2τ′′τ′λ1/pp,Z ∗f(τ)(σ∗,k (τ))dτ.Similarly,from (2.2)we obtain (3.6)I 2(−τ′,τ′)≤I 2(−τ′′,τ′′)×max exp−ν1ν2τ′′τ′µp (σ∗,k (τ))dτ.From this we obtain the following theorem on stagnation W 1p -zones:3.7.Theorem.Let t 0>0,β>α>0,and let −β∗<τ′≤τ′′<β∗where β∗is as in(3.1).If f is a solution of (1.3)on D =D ∗,kβ∗∩B k (t 0)with the generalized boundarySTAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS11 condition(1.8)on∂D\G,where G={x∈∂D:p∗k(x)=±β∗}and max exp −ν1ν2τ′′ τ′µp(σ∗,k(τ))dτ <s1/p,or a solution of(1.3)on D with the generalized boundary condition(1.12)on∂D\G andmax exp −ν1ν2τ′′ τ′λ1/p p,Z∗f(τ)(σ∗,k(τ))dτ <s1/p,then the subdomain∆∗,k(−τ′,τ′)is a s-zone with respect to the W1p-norm i.e.,∆∗,k(−τ′,τ′)|∇f|p d H n<s,where∆∗,k is as in(3.2).Stagnation zones with respect to the L p-norm.Let t0>0,β>α>0,and let −β∗<τ′≤τ′′<β∗whereβ∗is as in(3.1).Denote by C5(k,p,α,β,t0)the best constant of the imbedding theorem from W1p(D∗,kβ∗)to L p(D∗,kβ∗),i.e.in the inequalityg−CL p(D∗,kβ∗)≤C5(k,p,α,β,t0) gW1p(D∗,kβ∗).The domains D∗,kβ∗∩B k(t0)are convex and such a constant C5exists(see Maz’ya[Maz85]or[AF03,p.85]).In this case we obtain from(3.5),(3.6)(3.8) f−C pL p(∆∗,k(−τ′,τ′))≤C p5(k,p,β∗−τ′′,β∗+τ′′,t0)I2(−τ′′,τ′′)×max exp −ν1ν2τ′′ τ′λ1/p p,Z∗f(τ)(σ∗,k(τ))dτ , and(3.9) f−C pL p(∆∗,kτ′)≤C p5(k,p,β∗−τ′′,β∗+τ′′,t0)I2(−τ′′,τ′′)×max exp −ν1ν2τ′′ τ′µp(σ∗,k(τ))dτ .These relations can be used to obtain information about stagnation zones with respect to the L ly,we have:12VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINEN3.10.Theorem.If f is a solution of(1.3)on D=D∗,kβ∗∩B k(t0)with the generalizedboundary condition(1.8)(or(1.12))on∂D\G,where G={x∈∂D:p∗k(x)=±β∗},and the right side of(3.8)(or(3.9))is smaller than s>0,then the domain ∆∗,k(−τ′,τ′)is a stagnation zone with the deviation s p in the sense of the L p-norm on D.Stagnation zones for bounded,uniformly continuous functions.Let t0>0,β>α>0,and let−β∗<τ′≤τ′′<β∗whereβ∗is as in(3.1).As above,denote by C6(k,p,α,β,t0)the best constant of the imbedding theorem from W1p(D∗,kβ∗)to C(D∗,kβ∗),i.e.in the inequality(3.11) g−CC(D∗,kβ∗)≤C6(k,p,α,β,t0) gW1p(D∗,kβ∗).A domain D∗,kβ∗is convex and hence(3.11)holds for p>n(see Maz’ya[Maz85]or[AF03,p.85]).In this case from(3.5),(3.6)we obtain(3.12) f−C C(∆∗,k(−τ′,τ′))≤C p6(k,p,β∗−τ′,β∗+τ′,t0)I2(−τ′′,τ′′)×max exp −ν1ν2τ′′ τ′λ1/p p,Z∗f(τ)(σ∗,k(τ))dτ , and(3.13) f−C C(∆∗,k(−τ′,τ′))≤C p6(k,p,β∗−τ′,β∗+τ′,t0)I2(−τ′′,τ′′)×max exp −ν1ν2τ′′ τ′µp(σ∗,k(τ))dτ .These relations can be used to obtain theorems about stagnation zones for bounded uniformly continuous functions.3.14.Theorem.If f is a solution of(1.3),p>n,on D=D∗,kβ∗∩B k(t0)with thegeneralized boundary condition(1.8)(or(1.12))on∂D\G where G={x∈∂D: p∗k(x)=±β∗},and the right side of(3.12)(or(3.13))is smaller than s>0,then the domain∆∗,k(−τ′,τ′)is a stagnation zone with the deviation s in the sense of the norm · C0(∆∗,k(−τ′,τ′)).4.Other applicationsNext we prove Phragm´e n-Lindel¨o f type theorems for the solutions of the A-Laplace equation with boundary conditions(1.8)and(1.12).STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS13Estimates for W1p -norms.Let t0>0,β>α>0,and let beβ∗is as in(3.1).First we will prove some estimates of the W1p-norm of a solution.Let f be a solutionof(1.3)on D∗,kβ∗with the generalized boundary condition(1.8)on∂D\G.Fix0<τ′<τ′′<β∗and estimate f W1p (∆∗,k(−τ′,τ′)).Letψ:[τ′,τ′′]→(0,∞)be a Lipschitz function such that (4.1)ψ(τ′)=1,ψ(τ′′)=0.We choose(4.2)φ(t)= 1for|t|<τ′,ψ(|t|)forτ′≤|t|≤τ′′. The functionϕ(x)=φ(p∗k(x))is admissible in Definition1.7forU=∆∗,k(−τ′′,τ′′).As in(2.9)we may by(1.10)writeσ∗,k(−τ′′)∪σ∗,k(τ′′)φp(p∗k(x))(f−C) A(x,∇f),14VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENand by the H¨o lder inequality∆∗,k(−τ′′,τ′′)φp−1(p∗k(x))|f−C||∇f|p−1|∇φ(p∗k(x))|d H n= ∆∗,k(−τ′′,τ′′)φp−1(p∗k(x))|∇f|p−1|φ′(p∗k(x))||f−C|d H n≤ ∆∗,k(−τ′′,τ′′)φp(p∗k(x))|∇f|p d H n (p−1)/p ∆∗,k(−τ′′,τ′′)|φ′(p∗k(x))|p|f−C|p d H n 1/p From this inequality and(4.3)we obtainνp1 ∆∗,k(−τ′,τ′)φp(p∗k(x))|∇f|p d H n≤p pνp2 ∆∗,k(−τ′′,τ′′)|φ′(p∗k(x))|p|f−C|p d H n. Becauseφ(p∗k(x))≡1on∆∗,k(−τ′,τ′)we have the following inequality: (4.4)νp1 ∆∗,k(−τ′,τ′)|∇f|p d H n≤p pνp2∆∗,k(−τ′′,τ′′)\∆∗,k(−τ′,τ′)|ψ′(p∗k(x))|p|f−C|p d H n. Next we willfindminψ∆∗,k(−τ′′,τ′′)\∆∗,k(−τ′,τ′)|ψ′(p∗k(x))|p|f−C|p d H n,where the minimum is taken over allψin(4.2).We have∆∗,k(−τ′′,τ′′)\∆∗,k(−τ′,τ′)|ψ′(p∗k(x))|p|f−C|p d H n=−τ′−τ′′|ψ′(τ)|p dτ σ∗,k(τ)|f(x)−C|p d H n−1+τ′′τ′|ψ′(τ)|p dτ σ∗,k(τ)|f(x)−C|p d H n−1STAGNATION ZONES FOR A -HARMONIC FUNCTIONS ON CANONICAL DOMAINS 15and (4.5)minψ∆∗,k (−τ′′,τ′′)\∆∗,k (−τ′,τ′)|ψ′(p ∗k (x ))|p |f (x )−C |p d Hn≤minψ−τ′−τ′′|ψ′(τ)|p dτσ∗,k (τ)|f (x )−C |p d H n −1+minψτ′′τ′|ψ′(τ)|p dτσ∗,k (τ)|f (x )−C |p d H n −1≡A 1+A 2.Because by the H¨o lder inequality 1≤τ′′ τ′|ψ′(τ)|dτp≤τ′′ τ′|ψ′(τ)|p dτσ∗,k (τ)|f (x )−C |p d H n −1×τ′′τ′dτσ∗,k (τ)|f (x )−C |p d H n −11/(1−p ) p −1,we have τ′′τ′dτσ∗,k (τ)|f (x )−C |p d H n −11/(1−p ) 1−p≤τ′′τ′|ψ′(τ)|p dτσ∗,k (τ)|f (x )−C |p d H n −1,and henceA 2≥τ′′τ′dτσ∗,k (τ)|f (x )−C |p d H n −11/(1−p ) 1−p .It is easy to see that here the equality holds for a special choice of ψ.ThusA 2=τ′′τ′dτσ∗,k (τ)|f (x )−C |p d H n −11/(1−p ) 1−p.16VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINEN Similarly,A1= −τ′ −τ′′dτ σ∗,k(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p.From(4.5)we obtain|ψ′(p∗k(x))|p|f−C|p d H nminψ∆∗,k(−τ′′,τ′′)\∆∗,k(−τ′,τ′)≤ −τ′ −τ′′dτ σ∗,k(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p+ τ′′ τ′dτ σ∗,k(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p. By using(4.4)we obtain the inequality:p−p ν1STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS17 Similarly,for solutions of the A-Laplace equation with the boundary condition (1.12)we may prove thatp−p ν118VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENBy using(3.6)we obtain from this the inequality:I2(−τ′,τ′)≤C7max −τ′′ −τ′′−1dτ σ∗,n−1(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p,τ′′+1dτ σ∗,n−1(τ)|f(x)−C|p d H n−1 1/(1−p) 1−pτ′′×max exp −ν1ν2τ′′ τ′µp(σ∗,n−1(τ))dτ . We observe that in this case(4.8)µp σ∗,n−1(τ) ≡µp σn−1(0) ,and henceτ′′µp σ∗,n−1(τ) dτ=µp σn−1(0) (τ′′−τ′).τ′It follows that(4.9)I2(−τ′,τ′)≤C7max −τ′′ −τ′′−1dτ σ∗,n−1(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p,τ′′+1dτ σ∗,n−1(τ)|f(x)−C|p d H n−1 1/(1−p) 1−pτ′′×exp −ν1STAGNATION ZONES FOR A-HARMONIC FUNCTIONS ON CANONICAL DOMAINS19 Similarly for a solution f of(1.3)with(1.1)and(1.2),satisfying the boundary condition(1.12)we may write(4.11)I2(−τ′,τ′)≤C7max −τ′′ −τ′′−1dτ σ∗,n−1(τ)|f(x)|p d H n−1 1/(1−p) 1−p,τ′′+1dτ σ∗,n−a(τ)|f(x)|p d H n−1 1/(1−p) 1−pτ′′×max exp −ν1ν2τ′′ τ′λ1/p p,Z∗f(τ)(σ∗,n−1(τ))dτ . However here we do not have any identity similar to(4.8).We have:4.12.Theorem.Let t0>0,and let f:B n−1(t0)→R be a generalized solution of (1.3)with(1.1)and(1.2)satisfying the boundary condition(1.12)onΣn−1(t0).If the right side of(4.11)tends to0asτ′′→∞,then f≡0on the cylinder B n−1(t0)If f(x)=0everywhere on the boundaryΣn−1(t0)of the cylinder,then an identity similar to(4.8)holds in the following form:(4.13)λ1/p p(σ∗,n−1(τ))≡λ1/p p(σn−1(0))(λ).As above,wefind(4.14)I2(−τ′,τ′)≤C7max −τ′′ −τ′′−1dτ σ∗,n−1(τ)|f(x)|p d H n−1 1/(1−p) 1−p,τ′′+1dτ σ∗,n−1(τ)|f(x)|p d H n−1 1/(1−p) 1−pτ′′×exp −ν120VLADIMIR M.MIKLYUKOV,ANTTI RASILA,AND MATTI VUORINENPhragm´e n-Lindel¨o f type theorems II.We prove Phragm´e n-Lindel¨o f type theo-rems for canonical domains of an arbitrary form.Let1≤k<n−1and let t0>0 befixed.We consider a domainD=B k(t0)= x=(x1,...,x k,x k+1,...,x n):d k(x)<t0 .Let f be a generalized solution of(1.3)with(1.1)and(1.2)satisfying the boundary condition(1.8)onΣk(t0)= x∈R n:d k(x)<t0 .Fixτ0>0.Letτ0<τ′<τ′′<∞.By(4.6)we may writeD k0,τ′|∇f|p d H n≤C8 τ′′ τ′dτ σk(τ)|f(x)−C|p d H n−1 1/(1−p) 1−p,where C8=C7/2.As in(3.6)we obtain from(2.2)the estimateD k0,τ|∇f|p d H n≤ D k0,τ′|∇f|p d H n exp −ν1ν2τ′τ0µp(σk(τ))dτ .The inequality(4.16)holds for arbitrary constant C and everyτ′′>τ′.Thus the following statement holds:4.17.Theorem.Let t0>0,and let f:B k(t0)→R be a generalized solution of(1.3) with(1.1)and(1.2)satisfying the boundary condition(1.8)onΣk(t0),1≤k<n−1. 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