Eigenvalue problems for the Laplacian

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Problem 2.3.10 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.3 "Integral Calculus"

Let D\mathcal{D} denote the open unit disc in R2\mathbb {R}^{2}. Let uu be an eigenfunction for the Laplacian in D\mathcal{D}; that is, a real valued function of class C2C^2 defined in D‾\overline{\mathcal{D}}, zero on the boundary of D\mathcal{D} but not identically zero, and satisfying the differential equation

∂2u∂x2+∂2u∂y2=λu, \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = \lambda u,

where λ\lambda is a constant. Prove that

∬D∣grad⁡u∣2 dx dy+λ∬Du2 dx dy=0,(∗) \iint _{\mathcal{D}} \left|\operatorname {grad} u\right|^2 \, dx \, dy + \lambda \iint _{\mathcal{D}} u^2 \, dx \, dy = 0, \tag {$*$}

and hence that λ<0\lambda < 0.

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Problem 2.3.11 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.3 "Integral Calculus"

Let λ,a∈R\lambda , a \in \mathbb {R}, with a<0a < 0. Let u(x,y)u(x,y) be an infinitely differentiable function defined on an open neighborhood of closed unit disc D\mathcal{D} such that

∂2u∂x2+∂2u∂y2=λuin int⁡(D) \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = \lambda u \qquad \text{in } \operatorname {int}(\mathcal{D}) Dnu=auin ∂D. D_n u = au \qquad \text{in } \partial \mathcal{D}.

Here DnuD_n u denotes the directional derivative of uu in the direction of the outward unit normal. Prove that if uu is not identically zero in the interior of D\mathcal{D} then λ<0\lambda < 0.

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References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.