## Abstract

A correct proof is given for the following assertions about the two-dimensional sloshing problem. The fundamental eigenvalue is simple and the corresponding stream function may be chosen to be non-negative in the closure of the water domain. New proof is based on stricter assumptions about the water domain; namely, it must satisfy John’s condition.

In the paper Kozlov *et al.* (2004), which deals with the two-dimensional sloshing problem, the proof of assertion (i) of theorem 3.1 (the simplicity of the fundamental eigenvalue) is based on proposition 3.2. The key point in the proof of the latter proposition is the inequality (the definitions of the operators and are given below). However, this inequality is incorrect because
The minus sign was lost on the right-hand side of this chain of equalities in Kozlov *et al.* (2004, p. 2597), and so the proof of proposition 3.2 is incomplete. Nevertheless, assertion (i) of theorem 3.1 is true, provided the water domain satisfies an additional condition; namely, we have the following.

### Theorem 3.1

*Let the water domain W satisfy John’s condition. Then the fundamental eigenvalue ν*_{1} *of the two-dimensional sloshing problem is simple.*

### Proof

In terms of a stream function the two-dimensional sloshing problem is as follows:
3.1
3.2and
3.3
where the boundary condition (3.2) on the free surface *F* contains the spectral parameter *ν*. Problem (3.1)–(3.3) is equivalent to the following operator equation (for the proof see §3*a* in Kozlov *et al.* 2004):
3.4
The particular choice of *F* in equation (3.4) does not restrict the generality; the operators and are defined as follows:
and
maps *ϕ* given on *F* into , where *Φ* satisfies the Dirichlet problem:
It is clear that is a symmetric, positive operator in *L*^{2}(*F*), whereas —the so-called Dirichlet–Neumann operator—is a positive, self-adjoint operator in *L*^{2}(*F*).

Let us show that the operator
is a bounded integral operator. For this purpose we introduce *K*^{(h)}(*x*,*ξ*,*η*) with the following properties. It is a harmonic function of (*ξ*,*η*)∈*W*, which depends on the parameter *x*∈[−1,1] so that
Using this definition and problem (3.1)–(3.3) in the second Green’s identity, we find that
Hence is the integral operator and its kernel is equal to .

Since *K*(*x*,*ξ*) is smooth for *x*≠*ξ*, the same is true for . It is also clear that *K*_{ξ}(*x*,*ξ*) is bounded and belongs to *C*([−1,1];*L*^{2}(*F*)). Then we have , which follows from the results in the book Kenig (1994, ch. 2, §1).

Now, we prove that the kernel is positive on *F* provided *W* satisfies John’s condition (this means that *W* is contained within the semi-strip bounded by and two vertical rays going downwards from the endpoints of ). First, we fix *x*∈(−1,1) and consider
It is clear that *w*(*ξ*,*η*) is harmonic in *W* and vanishes on *F*∩{*ξ*<*x*}. In view of John’s condition, the definition of *K*^{(h)}(*x*,*ξ*,*η*) yields that *w*≤0 on the rest part of ∂*W*. Then Hopf’s lemma implies that *w*_{η}(*ξ*,0)>0 on *F*∩{*ξ*<*x*}, that is, the kernel is positive on this part of *F*. Applying the same considerations to
we obtain that the kernel of is also positive on *F*∩{*x*<*ξ*}. Now, we are in a position to apply Jentzsch’s theorem in the form given, for example, by Vladimirov (1971, ch. 4, §18.7). (In the original paper, Jentzsch 1912 imposed the superfluous continuity condition on the kernel of an integral operator.) According to this theorem, the smallest in absolute value characteristic number of equation (3.4), that is, *ν*_{1} is simple. This yields assertion of the theorem because equation (3.4) is equivalent to the sloshing problem. ■

The assertion of proposition 3.2 in Kozlov *et al.* (2004) is also a consequence of Jentzsch’s theorem, for which purpose it must be combined with the maximum principle. Namely, we have the following.

### Corollary 3.2

*Let the water domain W satisfy John’s condition. Then the fundamental eigenvalue of problem* (3.1)–(3.3) *is simple and the corresponding eigenfunction may be chosen to be positive in W*∪*F*.

### Proof

The first assertion is established in the proof of theorem 3.1.

According to Jentzsch’s theorem, the fundamental eigenfunction of equation (3.4) may be chosen to be positive on *F*. On the other hand, this function is the trace on *F* of the fundamental eigenfunction *v*_{1} of problem (3.1)–(3.3). Thus, *v*_{1} vanishes on one part of ∂*W* and is positive on the complementary part of ∂*W*. Since *v*_{1} satisfies the Laplace equation (3.1), the maximum principle yields the second corollary’s assertion. ■

It is worth mentioning that the fundamental eigenvalue of the sloshing problem is also simple in the following two cases. The water domain is
whereas the free surface *F* consists either of a single interval of the *x* axis or of two such intervals of equal length (see Kuznetsov & Motygin 2008). Note that John’s condition is violated for these water domains.

Theorem 3.1 and proposition 3.2 from Kozlov *et al.* (2004) were used in Kulczycki & Kuznetsov (2009, 2010), but in both papers these results are applied for domains satisfying John’s condition. Therefore, theorem 3.1 and corollary 3.2 must be referred to, where needed in the latter papers, instead of theorem 3.1 and proposition 3.2 from Kozlov *et al.* (2004).

## Acknowledgements

The authors are grateful to Dr M. Kwaśnicki (Wrocław Polytechnic Institute) for drawing their attention to a gap in the proof of proposition 3.2 in Kozlov *et al.* (2004). V.K. was supported by the Swedish Research Council (VR). N.K. acknowledges the financial support from G.S. Magnuson’s Foundation of the Royal Swedish Academy of Sciences and from the Linköping University.

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