We prove radial symmetry of singular solutions to an overdetermined boundary
value problem for a class of degenerate quasilinear elliptic equations.Comment: 8 pages, to appear on Rendiconti dell'Istituto di Matematica
  dell'Universita' di Triest

Alessandrini, Giovanni

Rosset, Edi

English

arXiv

We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations

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Rend. Istit. Mat. Univ. TriesteVol. XXXIX, 1–8 (2007)Symmetry of Singular Solutions ofDegenerate QuasilinearElliptic EquationsGiovanni Alessandrini and Edi Rosset (∗)In ricordo di FabioSummary. - We prove radial symmetry of singular solutions to anoverdetermined boundary value problem for a class of degeneratequasilinear elliptic equations.1. IntroductionWe consider solutions u todiv(a(|∇u|)∇u) = 0, in Ω \ {O}, (1)which vanish on ∂Ωu = 0, on ∂Ω, (2)and have a positive singularity at the origin Olimx→Ou(x) = M ∈ (0,+∞]. (3)We prove that if u satisfies the overdetermined boundary condition∂u∂ν= −c, on ∂Ω, (4)(∗) Work supported in part by MiUR, PRIN no. 2006014115.Authors’ addresses: Giovanni Alessandrini and Edi Rosset, DMI, Uni-versità di Trieste, via A. Valerio 12/1, 34127 Trieste, Italy; E-mail:alessang@univ.trieste.it, rossedi@univ.trieste.itKeywords: Symmetry, singular solutions, moving planes, p-laplacian.AMS Subject Classification: 35J60, 35J70, 35R35.2 G. ALESSANDRINI AND E. ROSSETwith c > 0 constant, then Ω is a ball centered at O and u is radiallysymmetric.To be more specific, we shall assume Ω to be a bounded connectedopen set in Rn, n ≥ 2, containing the origin O, and with C2,α-smoothboundary ∂Ω. The nonlinearity a is assumed to be a C1 functionfrom (0,+∞) to (0,+∞) and satisfying the degenerate ellipticitycondition0 < λ ≤ 1 +sa′(s)a(s)≤ Λ, for every s > 0, (5)for some positive constants λ, Λ.Such a class of quasilinear degenerate elliptic equations, whichstrictly contains the one of p-Laplacian type equations, was intro-duced by Lieberman [5] and independently, in the two dimensionalcase, by Alessandrini, Lupo and Rosset [3].With such assumptions the main result of this note is the follow-ing.Theorem 1.1. Let u ∈ C1,α(Ω \ {O}) be a weak solution to (1),satisfying the conditions (2), (3). If, in addition, u satisfies (4) thenΩ is a ball centered at the origin O and u is radially symmetric.Let us observe that in view of the regularity results by Lieberman[5] it is reasonable to treat solutions in the C1,α-class.Note also that for singular solutions satisfying (1)–(3) the limitM in (3) may be finite or infinite depending on the nonlinearity a.This fact is particularly evident in the special case when a(t) = tp−2,p > 1. One readily sees that when p ≤ n then we have M = +∞,whereas for p > n one must have M < +∞. See in this respectKichenassamy and Veron [4] and, for a detailed study of singularsolutions in two variables we refer to Rosset [6].Our proof is based on an adaptation of the well-known methodof moving planes by Alexandrov and Serrin. The adaptation of themethod to degenerate equations was initiated in [2]. Here we refermostly to arguments introduced in [1], however, the presence of thesingularity adds a little further difficulty, since it may appear, at afirst glance, that the method of moving planes cannot be used afterthe moving plane has crossed the singularity. We shall show thatSYMMETRY OF SINGULAR SOLUTIONS 3if this is the case for a certain direction ξ, then for the oppositedirection −ξ the problem of “hitting the singularity” cannot occur.Remark 1.2. We take this opportunity to point out that an erratumis in order in [1]. In fact, it is improperly stated there that thenonlinearity a may depend on |∇u| and also on u. This is not correct,in fact one should assume a = a(|∇u|) and with this proviso all thestatements there are correct.2. Proof of Theorem 1.1We recall some basic properties of solutions to equation (1) thatwe shall use repeatedly in our arguments. Local solutions to theequationdiv(a(|∇u|)∇u) = 0, (6)are obtained as limits in C1,α of solutions uǫ to regularized equationsdiv(aǫ(|∇uǫ|)∇uǫ) = 0, (7)where aǫ satisfies the same conditions as a and in addition is C∞and aǫ ≥ ǫ > 0 everywhere. Consequently uǫ can also be seen as astrong solution to the non-divergence uniformly elliptic equation∑i,j(δij +a′ǫ(|∇uǫ|)|∇uǫ|2aǫ(|∇uǫ|)∂uǫ∂xi∂uǫ∂xj)∂2uǫ∂xixj= 0. (8)Consequently solutions to (6) inherit some properties of strong solu-tions to uniformly elliptic equations.We quote, in particular, the Harnack inequality, that is, thereexists C = C(λ,Λ) such that: if u solves (6) in BR(x0) and u ≥ 0thenmaxBR/2(x0)u ≤ C minBR/2(x0)u. (9)A further consequence of the use of the regularized solutions is thatsolutions to (6) satisfy the comparison principle in the weak form.That is, if v, u solve (6) in a domain G and v ≤ u on ∂G then v ≤ ualso inside. In addition, if |∇u| + |∇v| > 0 in G then the strongversion of the comparison principle holds, that is if v ≤ u on ∂Gthen either v ≡ u or v < u in G.4 G. ALESSANDRINI AND E. ROSSETLet us also observe that, if u is a solution to (6), then for ev-ery constant C, also C − u is a solution. Hence, by the Harnackinequality, one readily obtains that the solution to (1)–(3) satisfies0 < u(x) < M, for every x ∈ Ω \ {O}. (10)Let us now introduce the moving plane apparatus. For any di-rection ξ ∈ Rn, |ξ| = 1, and for any t ∈ R, we define the hyperplaneΠξt = {x ∈ Rn | x · ξ = t}.We denote by Rξt the reflection in Πξt , that isRξtx = 2(t − x · ξ)ξ + x.We shall denote(Rξtu)(x) = u(Rξt x).If we agree to say that if x · ξ < t, x is on the left hand side of Πξt ,and conversely x is on the right hand side of Πξt if x ·ξ > t, we denoteby RξtΩ the reflection of the part of Ω which is on the left hand sideof Πξt , that isRξtΩ = {x ∈ Rn | x · ξ > t,Rξt x ∈ Ω}.Given ξ, we fix t such that RξtΩ = ∅. Letting t > t increase, wedenote by t(ξ) the largest number such thatRξt Ω ⊂ Ω, for every t ∈ (t, t(ξ)).As is well-known since Serrin [7], when t = t(ξ) one of the followingtwo cases is satisfied(i) ∂(Rξt Ω) is tangent to ∂Ω at a point P 6∈ Πξt ,(ii) ∂(Rξt Ω) is tangent to ∂Ω at a point P ∈ Πξt .Let us consider the family of moving planes associated to the oppositedirection −ξ and the corresponding reflections. One can easily verifythatΠξt = Π−ξ−t , for every t,SYMMETRY OF SINGULAR SOLUTIONS 5and alsoRξt = R−ξ−t , for every t.Now we observe that for every s < t(ξ) we also have −s > t(−ξ). InfactRξsΩ & Ω ∩ {x · ξ > s}and therefore, applying R−ξ−s to both sides,Ω ∩ {x · ξ < s} & R−ξ−sΩ,that is R−ξ−sΩ is not contained in Ω and hence −s > t(−ξ).Hence, letting s increase to t(ξ), we obtaint(ξ) + t(−ξ) ≤ 0 (11)Consequently either t(ξ) = t(−ξ) = 0 or one of the two numberst(ξ), t(−ξ) is strictly negative.If t(ξ) = t(−ξ) = 0 then, obviously, Ω is symmetric in Πξ0 = Π−ξ0 .Assume now t(ξ) < 0 (the other case t(−ξ) < 0 being equivalent).We simplify our notation by posingΠ = Πξt(ξ), R = Rξt(ξ), G = Rξt(ξ)Ω, v = Rξt(ξ)u.Since the origin O is on the right hand side of Π, by (10), we havethat there exists N , 0 < N < M such that the level setE = {x ∈ Ω \ {O} | u(x) ≥ N}is strictly on the right hand side of Π.Consequently, on G, v = Rξt(ξ)u < N . Now we observe that on∂(G \ E) we have v ≤ u, in fact ∂(G \ E) can be decomposed as∂(G \ E) = (∂G ∩ Π) ∪ (∂G \ (E ∪ Π)) ∪ (∂E ∩ G)and we havev = u, on ∂G ∩ Π,v = 0 ≤ u, on ∂G \ (E ∪ Π),v < N ≤ u, on ∂E ∩ G.6 G. ALESSANDRINI AND E. ROSSETHence, by the weak comparison principle,v ≤ u, in G \ E,and also, trivially,v < N ≤ u, in G ∩ E.Consequentlyv ≤ u in G. (12)From now on we rephrase arguments in [1] to prove that Π is a planeof symmetry for Ω.Let U be an ǫ-neighborhood of ∂Ω in Ω, with ǫ small enough tohave |∇u| > 0 in U and O 6∈ U . Let A be the connected componentof (RU) ∩ U such that P ∈ ∂A. In view of the boundary conditions(2), (4), we have(u − v)(P ) = 0, ∇(u − v)(P ) = 0.Moreover, when case II) occurs, by applying the arguments in [7] toequation (6) we also have that∂2∂η2(u − v)(P ) = 0,for every direction η. Since A ⊂ G ∩ U , u − v is a non-negativesolution of a uniformly elliptic equation in A. By using the Hopflemma when case I) occurs and its variant due to Serrin [7, Lemma2] when case II) occurs and by the strong comparison principle, itfollows thatu = v, in A. (13)Now, let us prove that R(∂Ω) ⊂ ∂Ω, obtaining that Π is a plane ofsymmetry of Ω. Assume, by contradiction, that there exists a pointQ ∈ R(∂Ω) \ ∂Ω. Let γ be an arc in R(∂Ω) joining Q with P . Onecan find a subarc γ′ in γ ∩ (U ∪ ∂Ω) having as endpoints P and apoint R ∈ U . Since in any neighborhood of γ′ one can find points ofRU ∩ U , which can be joined to P through paths inside RU ∩ U , itfollows that γ′ ⊂ ∂A. Therefore the set R(∂Ω)∩U ∩∂A is non-emptyand on such a set v = 0 < u, contradicting (13).SYMMETRY OF SINGULAR SOLUTIONS 7We have therefore proved that for any direction ξ there exists aplane Π orthogonal to ξ which is a plane of symmetry for Ω. HenceΩ is a ball BR(x0). It remains to prove that x0 = O and that u isradially symmetric.Let b = b(s) be the inverse function to ta(t), let K = Rn−1c a(c)and setv(r) =∫ Rrb(Kρ1−n)dρ, 0 < r ≤ R.One can easily verify that v(|x − x0|) solvesdiv(a(|∇v|)∇v) = 0, in BR(x0) \ {x0},v = 0, on ∂BR(x0),∂v∂ν= −c, on ∂BR(x0).Now, in BR(x0)\{O,x0}, u(x) and v(|x−x0|) satisfy the same ellipticequation and have the same Cauchy data on ∂BR(x0).We recall that for such an equation the unique continuation prop-erty holds as long as one of the two solutions has non-vanishing gra-dient. It follows from a standard continuity argument that u(x) =v(|x − x0|) for every x 6= O,x0.Consequently u and v must have the same singular point, that isx0 = O and finally u(x) = v(|x|). We conclude the proof observingthat the singular value M can be computed as followsM =∫ R0b((Rρ)n−1c a(c))dρ.References[1] G. Alessandrini, A symmetry theorem for condensers, Math. Meth-ods Appl. Sci. 15, no. 5 (1992) 315–320.[2] G. Alessandrini and N. Garofalo, Symmetry for degenerateparabolic equations, Arch. Rational Mech. Anal. 108, no. 2 (1989)161–174.[3] G. Alessandrini, D. Lupo and E. Rosset, Local behavior and ge-ometric properties of solutions to degenerate quasilinear elliptic equa-tions in the plane, Appl. Anal. 50, no. 3-4 (1993) 191–215.8 G. ALESSANDRINI AND E. ROSSET[4] S. Kichenassamy and L. Véron, Singular solutions of the p-Laplace equation, Math. Ann. 275, no. 4 (1986) 599–615.[5] G.M. Lieberman, The natural generalization of the natural condi-tions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm.Partial Differential Equations 16, no. 2-3 (1991) 311–361.[6] E. Rosset, Isolated singularities of solutions to the equationdiv(a(|Du|)Du) = 0 in the plane, Complex Variables Theory Appl.25, no. 1 (1994) 69–96.[7] J. Serrin, A symmetry problem in potential theory, Arch. RationalMech. Anal. 43 (1971), 304–318.Received November 14, 2007.

Symmetry of Singular Solutions of Degenerate Quasilinear Elliptic Equations

We prove radial symmetry of singular solutions to an
overdetermined boundary value problem for a class of degenerate
quasilinear elliptic equations

ALESSANDRINI, GIOVANNI

ROSSET, EDI

Archivio istituzionale della ricerca - Università di Trieste

Symmetry of singular solutions of degenerate quasilinear elliptic equations

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Symmetry of singular solutions of degenerate quasilinear elliptic equations

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