Abstract:
We construct a multiply connected domain in $\mathbb{R}^2$ for which the second eigenfunction of the Laplacian with Robin boundary conditions has an interior nodal line. In the process, we adapt a bound of Donnelly-Fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains, which does not depend directly on any boundary behaviour. This also gives a new proof of the nodal line property of the example in the Dirichlet case.

Abstract:
In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2\pi\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2\pi\mathbb{Z})\times(\mathbb{R}/2\rho\pi\mathbb{Z})\,$, with $\rho=\frac{1}{\sqrt{3}}\,$, and construct on it an eigenfunction with three nodal domains.

Abstract:
We show that for any convex surface S in a contact 3-manifold, there exists a metric on S and a neighbourhood contact isotopic to $S \times I$ with contact structure given as $\ker(ud - \star du)$ where u is an eigenfunction of the Laplacian on S, and $\star$ is the Hodge star from the metric on $S$. This answers a question posed by Komendarczyk.

Abstract:
We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number of S^{1}-invariant eigenvalues can be made to be arbitrarily small, while the non-invariant spectrum becomes arbitrarily large.

Abstract:
Let $\Omega$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $\Omega$, under suitable conditions on $\frac{\partial}{\partial \theta}$ .

Abstract:
We consider the problem of prescribing the nodal set of the first nontrivial eigenfunction of the Laplacian in a conformal class. Our main result is that, given a separating closed hypersurface $\Sigma$ in a compact Riemannian manifold $(M,g_0)$ of dimension $d \geq 3$, there is a metric $g$ on $M$ conformally equivalent to $g_0$ and with the same volume such that the nodal set of its first nontrivial eigenfunction is a $C^0$-small deformation of $\Sigma$ (i.e., $\Phi(\Sigma)$ with $\Phi : M \to M$ a diffeomorphism arbitrarily close to the identity in the $C^0$ norm).

Abstract:
We prove the existence of one positive, one negative, and one sign-changing solution of a $p$-Laplacian equation on $\mathbb{R}^N$, with a $p$-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of $\mathbb{R}^N$ have only been scarcely investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

Abstract:
We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

Abstract:
In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface \Sigma in a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is \Sigma. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.

Abstract:
We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a "zero"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros, where the "nodal surplus" $s$ is an integer between 0 and the number of cycles on the graph. We then perturb the Laplacian by a weak magnetic field and view the $n$-th eigenvalue as a function of the perturbation. It is shown that this function has a critical point at the zero field and that the Morse index of the critical point is equal to the nodal surplus $s$ of the $n$-th eigenfunction of the unperturbed graph.