Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 4 1-12; http://www.math.u-szeged.hu/ejqtde/

ISOPERIMETRIC INEQUALITIES FOR SOME NONLINEAR EIGENVALUE PROBLEMS

GABRIELLA BOGNÁR

Abstract. In this paper we intend to review many of the known inequalities for eigenvalues of the Laplacian in Euclidean plane. Our aim is to show that we can generalize some results for the pseudo-Laplacian. We focus on isoperimetric inequalities for the first eigenvalue of the Dirichlet eigenvalue problem.

1. The nonlinear eigenvalue problem

We seek the eigenfunctions u_{j} and corresponding eigenvalues λ_{j} (j = 1, 2, ...) of the following nonlinear eigenvalue problem

(D)  −Q_{p} = λ |u|^{p−1} u in Ω,

where the nonlinear operator Q_{p} is the pseudo-Laplacian defined by

Q_{p} = \frac{∂}{∂x} (|\frac{∂u}{∂x}|^{p−1} \frac{∂u}{∂x}) + \frac{∂}{∂y} (|\frac{∂u}{∂y}|^{p−1} \frac{∂u}{∂y}) for 0 < p < ∞.

The boundary condition corresponding to the problem is

u|_{∂Ω} = 0.

Here Ω denotes a two-dimensional body with boundary ∂Ω. The boundary value problem in which equation (D) is to be solved is called Dirichlet problem.

For p = 1 the problem (D) describes the vibration of an elastic membrane with fixed boundary:

(L)  Δu + λu = 0 in Ω,
u = 0 on ∂Ω.

The function u ∈ W_{0}^{1,p+1} (Ω) is a generalized or weak solution of (D) if for every v ∈ W_{0}^{1,p+1} (Ω)

∫_{Ω} (\frac{∂v}{∂x} \frac{∂u}{∂x} |\frac{∂u}{∂x}|^{p−1} + \frac{∂v}{∂y} \frac{∂u}{∂y} |\frac{∂u}{∂y}|^{p−1}) dx = λ∫_{Ω} vu |u|^{p−1} dx.

It is known (see [3]) that there exist countably many number of distinct normalized eigenfunctions in W_{0}^{1,p+1} (Ω) with associated eigenvalues to the eigenvalue problem (D) . For the eigenvalues λ_{k}(p) of the Dirichlet eigenvalue problem (D)

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 1 the relation λ_{k}(p) → ∞ holds when k → ∞. Every eigenvalue is positive. Here, the first eigenfunction has also many special properties. The first eigenfunction does not change sign and the corresponding first eigenvalue λ_{1} is simple [19].

1991 Mathematics Subject Classification. 35J20, 35P15.

Key words and phrases. eigenvalue, pseudo-Laplace operator, isoperimetric inequality, Faber-Krahn inequality.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The main objective of the paper is to give lower bounds for the first eigenvalue to the eigenvalue problem (D) .

2. The classical isoperimetric inequality

The classical isoperimetric inequality after which all such inequalities are named states that the circle encloses the largest area of all plane curves of given perimeter. Our aim is to show that many Euclidean results remain valid in Minkowski plane

with the norm ∥x∥_{p} := (|x_{1}|^{p+1} + |x_{2}|^{p+1})^{1/(p+1)}. Of course, for p = 1 it is the

usual Euclidean norm on R^{2}.

Let Ω be a simply connected convex domain. We denote the Minkowski arc length of ∂Ω by L_{p}, the usual area of Ω by A, and the radius of the greatest inscribed curve c_{ϱ} of Ω by ϱ, when domain Ω is bounded by curve c_{ϱ}

(c_{ϱ})  |x|^{\frac{1}{p} +1} + |y|^{\frac{1}{p} +1} = ϱ^{\frac{1}{p} +1}, ϱ ∈ R^{+}.

Curve c_{ϱ} is significant because many Euclidean results are preserved if the unitsphere is replaced by the solution of the isoperimetric problem which is called isoperimetrix. The Minkowski length of curve c_{ϱ} is

(2.1) L_{p}(c_{ϱ}) = 4 ∫_{x=0}^{ϱ} \sqrt[p+1]{1 + |y′|^{p+1}}dx = 4 ∫_{x=0}^{ϱ} \sqrt[p+1]{1 + \frac{|x|^{\frac{p+1}{p}}}{ρ^{\frac{p+1}{p}} − |x|^{\frac{p+1}{p}}}} dx = 2P ϱ,

(see [25]) and the area of the domain bounded by this curve is

(2.2)  A(c_{ϱ}) = 4 ∫_{x=0}^{ϱ} [ϱ^{\frac{p+1}{p}} − |x|^{\frac{p+1}{p}}]^{\frac{p}{p+1}} dx = P ϱ^{2},

where

(2.3)  P = 2\frac{p}{p+1} B (\frac{p}{p+1}, \frac{p}{p+1})

and B(\frac{p}{p+1}, \frac{p}{p+1}) is a Beta function (see [13]). If p = 1, then P = π.

G. D. Chakerian [8] proved and applied the Bonnesen inequality

(2.4)  (L_{p}) ϱ ≥ A + P ϱ^{2}

in the Minkowski plane for any convex n-gon (and consequently for any convex curve). This inequality was proved by L. Fejes Tóth [12] for nonconvex curves in the Euclidean plane. This proof can be generalized without difficulty to such Minkowski geometry where the ”circle” is any centrally symmetric convex curve. So the Bonnesen inequality (2.4) is valid for non-convex curves in Minkowski geometry.

In the case p = 1 the inequality (2.4) is reduced to the Bonnesen inequality valid on the Euclidean plane.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 2

From the Bonnesen inequality for a simply connected convex domain G. D. Chakerian [8] showed that the isoperimetric inequality in the Minkowski metric for a simply connected convex domain Ω has the form

(2.5)  (L_{p})^{2} − 4P A ≥ 0.

In (2.5) equality holds if and only if domain Ω is bounded by curve c_{ϱ}.

3. The isoperimetric inequality in ”broader sense”

There are several interesting and important geometrical and physical quantities depending on the shape and size of a curve: e.g., the length of its perimeter, the area included, the moment of inertia, with respect to the centroid, of a homogeneous plate bounded by the curve, the torsional rigidity of an elastic beam the cross-section of which is bounded by the given curve, the principal frequency of a membrane of which the given curve is the rim, the electrostatic capacity of a plate of the same shape and size, and several other quantities.

By the help of the isoperimetric inequalities we can estimate physical quantities on the basis of easily accessible geometrical data.

3.1. Bounds for eigenvalues. Isoperimetric inequalities are also useful in the derivation of explicit a priori inequalities employed in the determination of a priori bounds in various types of initial or boundary value problems.

In the linear case (L) many papers and books were published on the estimation of the first eigenvalue (see [22],.[1], [23] or [18]). Such bounds are based on geometrical data of the domain.

For the case of (L) Lord Rayleigh conjectured that for all membranes with given area A the circle yields the minimum value of λ_{1}. This property can be expressed by the inequality

(3.1)  λ_{1} ≥ \frac{πj_{0}^{2}}{A}

with equality only for the circle and where j_{0} is the first positive zero of the Bessel function of the first kind J_{0}(x). G.Faber [11] and E.Krahn [16] found independently essentially the same proof of Rayleigh’s conjecture. Another proof was given by G.Pólya and G.Szegõ [22] by using Steiner symmetrization.

We know that for the domain Ω with sufficiently smooth boundary ∂Ω the first eigenvalue in the fixed membrane problem (D) admits the following characterization:

λ_{1} = min_{u∈W_{0}^{1,p+1}(Ω)} \frac{∫_{Ω} (|u_{x}|^{p+1} + |u_{y}|^{p+1}) dx}{∫_{Ω} |u|^{p+1} dx}.

This characterization gives us an upper bound for λ_{1}, i.e., for any v ∈ W_{0}^{1,p+1} (Ω)

λ_{1} ≤ \frac{∫_{Ω} (|v_{x}|^{p+1} + |v_{y}|^{p+1}) dx}{∫_{Ω} |v|^{p+1} dx}.

The equality sign will always hold for some choice of v.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 3 3.2. A lower bound for the first eigenvalue. As it was shown in the introduction many papers were published on the estimation of the first eigenvalue for the linear eigenvalue problem.

Let the domain Ω be a simply connected convex domain. We define the curve (c_{ρ_{0}}) by

(3.2)  |x − x_{0}|^{\frac{1}{p} +1} + |y − y_{0}|^{\frac{1}{p} +1} = ρ_{0}^{\frac{1}{p} +1}, ρ_{0} ∈ R^{+}, (x_{0}, y_{0}) ∈ Ω.

In this part let us denote the radius of the greatest inscribed curve (c_{ρ_{0}}) of Ω by ρ. We shall consider (x_{0}, y_{0}) as the origin of the new coordinate system.

Theorem 1. Let λ_{1} be the smallest eigenvalue of the eigenvalue problem (D) such that u_{1} ≥ 0 in the simply connected convex domain Ω∈ R^{2}. Then the inequality

(3.3)  λ_{1} ≥ [\frac{A + σ}{(p + 1)ρA}]^{p+1}

holds, where A is the area of Ω, ρ is the radius of the greatest inscribed isoperimetrix c_{ρ} of Ω, and σ = P ϱ^{2} is the area of the isoperimetrix of radius ρ.

Proof. As in [22] and [15] we shall reconstruct the first eigenfunction u_{1}(x, y) from its level sets Ω(τ) = {(x, y) ∈ Ω| u_{1}(x, y) ≥ c}, where the area of Ω(τ) is τ, so 0 ≤ τ ≤ A and Ω(A) = Ω (see [5]). The domain Ω(τ) may be disconnected but each of its components is simply connected because u_{1} does not have local minima in Ω. Instead of coordinates x, y we introduce the new coordinates τ and s in the domain Ω, where s is the arc length in Minkowski metric of the level line which bounds Ω(τ). Therefore 0 ≤ s ≤ L(τ), where L(τ) is the total length of the level line.

Let the function φ(τ) be defined as

φ(τ) = u_{1}(x, y) on ∂Ω(τ).

Evidently φ(A) = 0 and φ(τ) is monotonically decreasing when 0 ≤ τ ≤ A. We have for the derivatives

\frac{∂u_{1}}{∂τ} = \frac{∂φ}{∂τ} = φ′(τ) = \frac{∂u_{1}}{∂x} \frac{∂x}{∂τ} + \frac{∂u_{1}}{∂y} \frac{∂y}{∂τ} ,

\frac{∂u_{1}}{∂s} = \frac{∂φ}{∂s} = 0 = \frac{∂u_{1}}{∂x} \frac{∂x}{∂s} + \frac{∂u_{1}}{∂y} \frac{∂y}{∂s} .

Since

|\frac{∂x}{∂s}|^{p+1} + |\frac{∂y}{∂s} |^{p+1} = 1.

therefore we get

(3.4)  |\frac{∂u_{1}}{∂x} |^{p+1} + |\frac{∂u_{1}}{∂y} |^{p+1} = \frac{|φ′(τ)|^{p+1}}{|△|^{p+1}} ,

where the Jacobian △ is

△ = | \begin{matrix}\frac{∂x}{∂τ} & \frac{∂y}{∂τ} \\ \frac{∂x}{∂s} & \frac{∂y}{∂s} \end{matrix} |

(similarly as in [17] for the linear case p = 1). The Rayleigh quotient can be expressed by the new coordinates τ and s:

(3.5) λ_{1} = \frac{∫_{Ω} (|\frac{∂u_{1}}{∂x} |^{p+1} + |\frac{∂u_{1}}{∂y} |^{p+1}) dx dy}{∫_{Ω}|u_{1}|^{p+1}dx dy} = \begin{matrix}∫_{τ=0}^{A} & [|φ′(τ)|^{p+1} ∫_{s=0}^{L(τ)} \frac{ds}{|△|^{p}} ] & dτ \\ ∫_{τ=0}^{A} & [|φ(τ)|^{p+1} ∫_{s=0}^{L(τ)}|△|ds] & dτ \end{matrix} .

Since

τ = ∫_{Ω(τ)} dx dy = ∫_{t=0}^{τ} ∫_{s=0}^{L(t)}|△| ds dt

we obtain

(3.6)  ∫_{s=0}^{L(τ)}|△| ds = 1.

Applying the Hölder inequality we get

( ∫_{s=0}^{L(τ)}|△| ds)^{\frac{p}{p+1}} ( ∫_{s=0}^{L(τ)}\frac{ds}{|△|^{p}} )^{\frac{1}{p+1}} ≥ L(τ).

Hence by (3.6)

∫_{s=0}^{L(τ)}\frac{ds}{|△|^{p}} ≥ [L(τ)]^{p+1}.

Now from (3.5) we have

λ_{1} ≥ \frac{∫_{τ=0}^{A} |φ′(τ)L(τ)|^{p+1}dτ}{∫_{τ=0}^{A} |φ(τ)|^{p+1}dτ} .

Let us denote by ρ(τ) the radius of the greatest inscribed curve (c_{ρ_{0}}) of Ω(τ). Now we have the inequality ρ(τ_{1}) ≤ ρ(τ_{2}) when τ_{1} < τ_{2}. We shall use the isoperimetric inequality (2.5)

L(τ) ≥ \sqrt{4P τ}, P = 2 \frac{p}{p + 1} B (\frac{p}{p + 1}, \frac{p}{p + 1})

and the Bonnesen inequality (2.4)

L(τ) ≥ \frac{τ}{ϱ(τ)} + P ϱ(τ).

For σ ≤ τ ≤ A where

σ = P ϱ^{2}

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 5 we show that L(τ) ≥ \frac{τ}{ϱ} + P ϱ. Because the function f(t) = t + \frac{1}{t} is increasing if t > 1 and

\frac{\sqrt{τ}}{\sqrt{P} ϱ(τ)} ≥ \begin{matrix}√ & \overline{τ} \\ \sqrt{P} & ϱ \end{matrix} ≥ \begin{matrix}√ & \overline{σ} \\ \sqrt{P} & ϱ \end{matrix} =1

then

L(τ) ≥ \frac{τ}{ϱ(τ)} + P ϱ(τ) = \sqrt{P τ} [\frac{\sqrt{τ}}{\sqrt{P} ϱ(τ)} + \frac{1}{\frac{\sqrt{τ}}{\sqrt{P} ϱ(τ)}} ] ≥

≥ \sqrt{P τ} [\begin{matrix}√ & \overline{τ} \\ \sqrt{P} & ϱ \end{matrix} + \frac{1}{\begin{matrix}√ & \overline{τ} \\ \sqrt{P} & ϱ \end{matrix}} ] = \frac{τ}{ϱ} + P ϱ.

We define the function X(τ) as follows

X(τ) = \{ \begin{matrix}\sqrt{4P τ}  if 0 ≤ τ < σ, \\ \frac{τ}{ρ} + P ρ  if σ ≤ τ ≤ A, \end{matrix}

and we have the relation

(3.7)  X(τ) ≤ L(τ) (0 ≤ τ ≤ A)

Applying (3.7) we obtain

λ_{1} ≥ \frac{∫_{τ=0}^{A} |φ′(τ)X(τ)|^{p+1}dτ}{∫_{τ=0}^{A} |φ(τ)|^{p+1}dτ} .

For the function X(τ) we find

lim_{τ→σ−0} X(τ) = lim_{τ→σ+0} X(τ) = 2P ϱ,

lim_{τ→σ−0} X′(τ) = lim_{τ→σ+0} X′(τ) = \frac{1}{ϱ}

therefore X(τ) ∈ C^{1}(0, A). Now we introduce the function

Y (τ) = \frac{A + σ}{Aϱ} τ

for which Y (0) = X(0), Y (A) = X(A). For 0 ≤ τ < σ we get

\sqrt{4P τ} ≥ \sqrt{4 P ρ^{2} \frac{τ}{ρ^{2}}} = \sqrt{4 σ \frac{τ}{ρ^{2}}} ≥ \sqrt{4 \begin{matrix}τ & 2 \\ ρ & 2 \end{matrix}} ≥ \sqrt{(1 + \frac{σ}{A})^{2} \begin{matrix}τ & 2 \\ ρ & 2 \end{matrix}} = \frac{A + σ}{Aϱ} τ.

If σ ≤ τ ≤ A then

\frac{τ}{ϱ} +P ϱ ≥ \frac{τ}{ρ} + \frac{P ρ}{A}τ = \frac{τ}{ρ} + \frac{σ}{Aρ}τ = \frac{A + σ}{Aϱ} τ,

therefore Y (τ) ≤ X(τ) when 0 ≤ τ ≤ A.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 6 Using the function Y (τ) we get

λ_{1} ≥ [\frac{A + σ}{Aϱ} ]^{p+1} \frac{∫_{τ=0}^{A} |τφ′(τ)|^{p+1}dτ}{∫_{τ=0}^{A} |φ(τ)|^{p+1}dτ} .

For reals X, Y and p > 0 inequality

(3.8)  |X|^{p+1} + p |Y |^{p+1} − (p + 1)X Y |Y |^{p−1} ≥ 0

holds, with equality if and only if X = Y.

Making use of inequality (3.8) with X = −(p + 1)τφ′ and Y = φ we can write

|(p + 1)τφ′|^{p+1} + p|φ|^{p+1} + (p + 1)^{2}τφ′φ^{{∗ \atop p}} ≥ 0, φ^{{∗ \atop p}} = |φ|^{p}sgn φ,

and therefore

|(p + 1)τφ′|^{p+1} − |φ|^{p+1} = |(p + 1)τφ′|^{p+1} + p|φ|^{p+1} − (p + 1)|φ|^{p+1} ≥
≥ −(p + 1)[(p + 1)τφ′φ^{{∗ \atop p}} + |φ|^{p+1}].

Integrating by parts we have

∫_{τ=0}^{A} [|(p + 1)τφ′|^{p+1} − |φ|^{p+1}]dτ ≥

≥ −(p + 1) ∫_{τ=0}^{A} [(p + 1)τφ′φ^{{∗ \atop p}} + |φ|^{p+1}]dτ =

= −(p + 1) [|φ|^{p+1}τ]_{τ=0}^{A} = 0

since φ(A) = 0. Hence

\frac{∫_{τ=0}^{A} |τφ′(τ)|^{p+1}dτ}{∫_{τ=0}^{A} |φ(τ)|^{p+1}dτ} ≥ \frac{1}{(p + 1)^{p+1}} ,

thus for the first eigenvalue λ_{1} we obtain the inequality (3.3).

In the case p = 1 the lower bound for the smallest eigenvalue is reduced to the lower bound for λ_{1} given by Makai in [17].

3.3. Application of symmetrization. For the nonlinear eigenvalue problem (D) let us consider the bounded domain Ω and the continuous function u(x, y) ∈ W_{0}^{1,p+1} (Ω) on \overline{Ω} satisfying u_{|∂Ω} = 0. We introduce the level set Ω_{c} of u for c ∈ R by

Ω_{c} = \{(x, y) ∈ \overline{Ω} : u(x, y) ≥ c\}.

The most frequently used type of symmetrization is the Schwarz symmetrization which is centrally symmetric. We define the Schwarz symmetrization Ω_{c}^{(0)} of Ω_{c} ⊂ R^{2} by

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 7

Ω_{c}^{(0)} := \{ \begin{matrix}bounded by curve (c_{ρ}) with the same area as Ω_{c} if Ω_{c} ≠ ∅, \\ ∅ if Ω_{c} = ∅. \end{matrix}

For u the level sets Ω_{c} are replaced by concentric curves (c_{ρ}) centered at zero. Therefore the Schwarz symmetrization of u is defined by

u^{(0)} := sup {c ∈ R| (x, y) ∈ Ω_{c}^{(0)}} .

Let us consider the bounded domain Ω and the continuous function u(x, y) on \overline{Ω} satisfying u_{|∂Ω} = 0. We introduce the level set Ω_{c} of u for c ∈ R by

Ω_{c} = {(x, y) ∈ Ω̄ : u(x, y) ≥ c}.

Clearly

Ω_{c′} ⊃ Ω_{c′′} if u_{min} ≤ c′ < c′′ ≤ u_{max},

and

Ω_{c} = Ω if c < u_{min},
Ω_{c} = ∅ if c > u_{max}.

By these properties we can replace function u by a related function u^{(0)} which has some desired properties (see [15] p.21).

Theorem 2. Let λ_{1} be the smallest eigenvalue and u_{1} be the corresponding eigenfunction, respectively, to the eigenvalue problem (D) on the domain Ω and λ_{1}^{(0)} be the smallest eigenvalue with the corresponding eigenfunction u_{1}^{(0)} on the domain Ω^{(0)}. Then λ_{1} > λ_{1}^{(0)} unless ∂Ω is a curve (c_{ϱ}).

Proof. Our proof is based on the proof given by G.Faber [11] and E. Krahn [16] in the linear case (L). First of all we give the differences between the linear and nonlinear cases. We consider the level sets of the first eigenfunction u_{1}

Ω_{c} = {(x, y) ∈ Ω̄ : u_{1}(x, y) ≥ c}, c ∈ R.

Instead of coordinates x, y we introduce the new coordinates w, and s. The intersection of the plane w = c with the surface of w = u_{1} gives the level sets Ω_{c}. Therefore we get 0 ≤ w ≤ a, where a is the maximum value of u_{1}. Let the coordinate s be the arc length of the level line from 0 the total length L (w) of ∂Ω_{c} We have the following connections:

\frac{∂u_{1}}{∂w} = 1, \frac{∂u_{1}}{∂s} = 0

that is

\frac{∂u_{1}}{∂x} \frac{∂x}{∂w} + \frac{∂u_{1}}{∂y} \frac{∂y}{∂w} = 1,
\frac{∂u_{1}}{∂x} \frac{∂x}{∂s} + \frac{∂u_{1}}{∂y} \frac{∂y}{∂s} = 0,

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 8 and

|\frac{∂x}{∂s}|^{p+1} + |\frac{∂y}{∂s} |^{p+1} = 1.

For the new coordinates we have the Jacobian

Δ = | \begin{matrix}\frac{∂x}{∂w} & \frac{∂x}{∂s} \\ \frac{∂y}{∂w} & \frac{∂y}{∂s} \end{matrix} | .

Hence we obtain

|(u_{1})_{x}|^{p+1} + |(u_{1})_{y}|^{p+1} = \frac{1}{|Δ|^{p+1}}

so

∫_{Ω} [|(u_{1})_{x}|^{p+1} + |(u_{1})_{y}|^{p+1}]dx dy = a ; ∫_{w=0}L(w) ; ∫_{s=0} \frac{1}{|Δ|^{p}} ds dw.

We denote by A (w) the area of Ω_{c} that is

A (w) = ∫_{D_{c}}dx = − ∫^{w} ; w̃=a L(w̃) ; ∫_{s=0} |Δ| ds dw̃

and A (0) =mes Ω, A (a) = 0. Therefore we have

(3.9)  A′ (w) = −L(w) ; ∫_{s=0} |Δ| ds,

moreover by the Hölder inequality and by inequality (2.5) we obtain

(3.10)  L(w) ; ∫_{s=0} |Δ| ds [L(w) ; ∫_{s=0} \frac{1}{|Δ|^{p}} ds]^{\frac{1}{p}} ≥ |L(w)|^{\frac{1}{p} +1} ≥ [4P A(w)]^{\frac{p+1}{2p}} .

By equation (3.9) we get

L(w) ; ∫_{s=0} \frac{1}{|Δ|^{p}} ds ≥ |−A′(w)|^{−p−1} [−A′(w)] [4P A(w)]^{\frac{p+1}{2}}

therefore

∫_{Ω} [|(u_{1})_{x}|^{p+1} + |(u_{1})_{y}|^{p+1}]dx dy ≥ (3.11)

(4P)^{\frac{p+1}{2}} a ; ∫_{w=0} |−A′(w)|^{−p−1} [−A′(w)] [A(w)]^{\frac{p+1}{2}} dw.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 9 Since u_{1}^{(0)} is symmetric function we can write

u_{1}^{(0)} (x, y) = v (ρ) .

We obtain

∫_{Ω} [|(u_{1}^{(0)})_{x}|^{p+1} + |(u_{1}^{(0)})_{y}|^{p+1}]dx dy = ∫^{a} ; w=0L(w) ; ∫_{s=0} |\frac{dv}{dρ}|^{p} ds dw =

= a ; ∫_{w=0}2P ρ |\frac{dv}{dρ}|^{p} dw = 2^{p+1}P a ; ∫_{w=0}ρ^{p+1} |\frac{dw}{d (ρ^{2})}|^{p} dw. (3.12)

Since A(w) = P ρ^{2},

ρ = (\frac{A(w)}{P})^{\frac{1}{2}}

thus

|\frac{dw}{d (ρ^{2})}| = −\frac{P}{A′(w)},

and from (3.12)

∫_{Ω} [|(u_{1}^{(0)})_{x}|^{p+1} + |(u_{1}^{(0)})_{y}|^{p+1}]dx dy = (3.13)

= (4P)^{\frac{p+1}{2}} a ; ∫_{w=0} |−A′(w)|^{−p−1} [−A′(w)] [A(w)]^{\frac{p+1}{2}} dw.

Comparing with (3.11) we obtain

∫_{Ω} [|(u_{1})_{x}|^{p+1} + |(u_{1})_{y}|^{p+1}]dx dy ≥ ∫_{Ω} |(u_{1}^{(0)})_{x}|^{p+1} + |(u_{1}^{(0)})_{y}|^{p+1}dx dy.

Using the property ii.) from [15] we get

(3.14)  ∥u_{1}∥_{L^{p+1}}^{p+1} = ∫_{Ω} |(u_{1}^{(0)})|^{p+1} dx dy.

Therefore (3.14) yields the estimate

λ_{1} = \frac{∫_{Ω} [|(u_{1})_{x}|^{p+1} + |(u_{1})_{y}|^{p+1}]dx dy}{∥u_{1}∥_{L^{p+1}}^{p+1}} ≥ \frac{∫_{Ω} [|(u_{1}^{(0)})_{x}|^{p+1} + |(u_{1}^{(0)})_{y}|^{p+1}]dx dy}{∫_{Ω} |(u_{1}^{(0)})|^{p+1}dx dy} ≥

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 10

≥ inf_{v∈F (Ω^{(0)})} \frac{∫_{Ω} (|v_{x}|^{p+1} + |v_{y}|^{p+1}) dx dy}{∫_{Ω} |v|^{p+1} dx dy} = λ_{1}^{(0)}.

From the isoperimetric inequality (2.5) it follows that in (3.11) we have equality (also in (3.10), respectively) if the level lines are curves (c_{ϱ}). Thus λ_{1} > λ_{1}^{(0)} unless ∂Ω is a curve (c_{ϱ}).

Corollary 1. In the case when the domain Ω is bounded by the curve (c_{ϱ}), then the first eigenvalue of (D) can be given by

λ_{1}^{(0)} = (\frac{h_{0}}{ρ} )^{p+1} = \frac{P^{\frac{p+1}{2}} h_{0}^{p+1}}{A^{\frac{p+1}{2}}} ,

where h_{0} is the first positive zero of the generalized nonlinear Bessel function H_{0}(x) satisfying the nonlinear ordinary differential equation

(3.15)  \frac{d}{dx} [|\frac{dH_{0}}{dx} |^{p−1} \frac{dH_{0}}{dx} ] + \frac{1}{x} |\frac{dH_{0}}{dx} |^{p−1} \frac{dH_{0}}{dx} + λ |H_{0}|^{p−1} H_{0} = 0

with conditions

(3.16)  H_{0} (1) = 0

and

\frac{dH_{0}}{dx} (0) = 0,

λ = 1 and P = 2\frac{p}{p + 1} B (\frac{p}{p + 1}, \frac{p}{p + 1}) (see [6]). Hence for any domain on the

plane with area A we get a lower bound for the first eigenvalue:

(3.17)  λ_{1} ≥ \frac{P^{\frac{p+1}{2}} h_{0}^{p+1}}{A^{\frac{p+1}{2}}} .

Inequality (3.17) is the generalization of the Faber-Krahn inequality (3.1) to the nonlinear problem (D).

References

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(Received September 30, 2003)

Mathematical Institute, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

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