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Hypersurfaces with prescribed boundary and small Steklov eigenvalues. (English) Zbl 1433.35205

Given a compact connected \(n\)-dimensional submanifold \(M\) in \(\mathbb R^{n+1}\) with boundary \(\Sigma\ne\emptyset\), the authors are able to produce a local perturbation of \(M\) which keeps \(\Sigma\) fixed, such that the \(k\)-th Steklov eigenvalue of the perturbed submanifold is arbitrarily small.
The Steklov eigenvalue problems reads \[ \begin{cases} \Delta u=0\,, & \text{in }M,\\ \partial_{\nu}u=\sigma u\,, & \text{on }\Sigma. \end{cases} \] in the unknowns \(u\) (the eigenfunction) and \(\sigma\) (the eigenvalue). Under suitable assumptions on \(M\) and \(\Sigma\) the spectrum is discrete and consists of an increasing sequence of eigenvalues of finite multiplicity \[ 0=\sigma_0(M)<\sigma_1(M)\leq\cdots\leq\sigma_k(M)\leq\cdots\nearrow+\infty. \] The authors construct, for each \(p\in\Sigma\) and \(k\in\mathbb N\), a sequence \(M_j\) of submanifolds such that \(\partial M_j=\partial M=\Sigma\) for all \(j\) and such that \(M_j=M\) outside a ball centred at \(p\in\Sigma\) of radius \(\frac{1}{j}\), such that \(\lim_{j\rightarrow \infty}\sigma_k(M_j)=0\). This implies that the first \(k\) Steklov eigenvalues of \(M\) can be made arbitrarily small with a local perturbation without perturbing the boundary. Moreover, the volume and the diameter of \(M_j\) converge to those of \(M\) as \(j\rightarrow \infty\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35J25 Boundary value problems for second-order elliptic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58C40 Spectral theory; eigenvalue problems on manifolds
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References:

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