On the eigenvalues of the polyharmonic operator

Antonio Boccuto & Roberta Filippucci Department of Mathematics, via Vanvitelli, 1 - 06123 Perugia, Italy

                             boccuto@dipmat.unipg.it

Let B be the unit ball in IR^{n}. We study the eigenvalue problem associated to the polyharmonic operator in B.

We say that μ ∈ IR is an eigenvalue and u ∈ L^{2}(B) is a corresponding eigenfunction of the problem

\{ \begin{matrix}(−Δ)^{m}u(x) = μ σ(x) u(x) & in B, \\ u = \frac{∂^{i}u}{∂ν^{i}} = 0 (i = 1, …, m − 1) & on ∂B, \end{matrix}  (1)

where (−Δ)^{m} is the polyharmonic operator, m ≥ 1, σ ∈ L^{∞}(B), σ > 0 almost everywhere in B, if u is a non-trivial weak solution of (1), namely a function u ∈ H_{0}^{m}(B), u ≠ 0, such that

∫_{B} D_{m}u D_{m}v dx = μ ∫_{B} σ u v dx for all v ∈ H_{0}^{m}(B),  (2)

where

D_{m}ϕ = \{ \begin{matrix}DΔ^{j}ϕ & if m = 2j + 1 \\ Δ^{j}ϕ & if m = 2j. \end{matrix}

The following result holds.

Theorem 1. The first eigenvalue μ_{1} of problem (1) is positive and simple, and is the unique eigenvalue which admits eigenfunctions of constant sign.

Moreover, for each eigenvalue μ of (1), the corresponding eigenfunctions are of class C^{2m−1,β}(\overline{B}) for all β ∈]0, 1[.

From now on, we always suppose that

the function σ in problem (1) is radial.

The following results hold.

Theorem 2. If μ_{1} is as in Theorem 1, then the eigenfunctions of (1) associated with μ_{1} are radial.

The following result is a sufficient condition under which the first eigenfunctions are not only radial and of one sign, but also strictly monotone functions of the radius (measured from the center of B). This result generalizes the analogous one of Pucci and Serrin ([5]), given in the case in which σ ≡ 1 and m = 2 or m = 3. Moreover the proof is somewhat simpler and more transparent, even when given in all detail, than the argument sketched by Lions ([6, Corollary I.2, p. 165]) in the canonical case (σ ≡ 1 and m ≥ 1).

Theorem 3. If μ_{1} is as in Theorem 1, and there exists η ∈]0, 1[ such that σ ∈ C^{0,η}(B), then the eigenfunctions of (1) associated with μ_{1} are radial and strictly monotone along the radius r = |x| ∈ [0, 1].

References

[1] A. BOCCUTO & R. FILIPPUCCI, On the Eigenvalues of the Polyharmonic Operator, to appear on Atti Sem. Mat. Fis. Univ. Modena.

[2] G. CARISTI, E. MITIDIERI & R. SORANZO, Isolated Singularities of Polyharmonic Equations, to appear on Atti Sem. Mat. Fis. Univ. Modena.

[3] H. Ch. GRUNAU, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order, Analysis, 11 (1991), 83-90.

[4] P. L. LIONS, The concentration-compactness principle in the calculus of variations - The limit case - Part I, Revista Matematica Iberoamericana 1 (1985), 145-201.

[5] P. PUCCI & J. SERRIN, Remark on the first eigenspace for polyharmonic operators, Atti Sem. Mat. Fis. Univ. Modena 36 (1988), 107-117.

[6] R. SORANZO, A priori estimates and existence of positive solutions of a superlinear polyharmonic equation, Dyn. Syst. Appl. 3 (1994), 465-487.