4 citations · 5 across the 9 of their papers we have counts for
15 papers
Optimisation of Steklov transmission eigenvalues and minimal surfaces
Mikhail Karpukhin, Alain Didier Noutchegueme
In the present paper, we study the variational properties of Steklov transmission eigenvalues, which can be seen as eigenvalues of the sum of two Dirichlet-to-Neumann operators on…
Second Variation Formula for Eigenvalue Functionals on Surfaces
Mikhail Karpukhin
Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a rema…
Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces
Mikhail Karpukhin, Antoine Métras, Iosif Polterovich
Consider a Dirac operator on an oriented compact surface endowed with a Riemannian metric and spin structure. Provided the area and the conformal class are fixed, how small can the…
Flexibility of Steklov eigenvalues via boundary homogenisation
Mikhail Karpukhin, Jean Lagacé
Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their resul…
Weyl's law for the Steklov problem on surfaces with rough boundary
Mikhail Karpukhin, Jean Lagacé, Iosif Polterovich
The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundaries is a well-known open question in spectral geometry. We answer this question in two dimension…
Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces
Mikhail Karpukhin, Mickaël Nahon, Iosif Polterovich +1
We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class.…
Laplace and Steklov extremal metrics via -harmonic maps
Mikhail Karpukhin, Antoine Métras
We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of -harmonic maps. Our approach…
The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
Alexandre Girouard, Mikhail Karpukhin, Michael Levitin +1
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat s…
Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems
Alexandre Girouard, Mikhail Karpukhin, Jean Lagacé
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and ot…
Conformally maximal metrics for Laplace eigenvalues on surfaces
Mikhail Karpukhin, Nikolai Nadirashvili, Alexei V. Penskoi +1
The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this…
Maximal metrics for the first Steklov eigenvalue on surfaces
Mikhail Karpukhin
In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper…
An isoperimetric inequality for Laplace eigenvalues on the sphere
Mikhail Karpukhin, Nikolai Nadirashvili, Alexei V. Penskoi +1
We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, i…
Steklov problem on differential forms
Mikhail Karpukhin
In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov.…
Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds
Mikhail A. Karpukhin
Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eig…
Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces
Mikhail Karpukhin, Gerasim Kokarev, Iosif Polterovich
We prove two explicit bounds for the multiplicities of Steklov eigenvalues on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the…