Related papers: Deformations of harmonic function germs
We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance…
Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial…
Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\"uller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is…
Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$…
We prove that bounded harmonic functions of anisotropic fractional Laplacians are H\"older continuous under mild regularity assumptions on the corresponding L\'evy measure. Under some stronger assumptions the Green function, Poisson kernel…
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant…
We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a…
We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf…
We study some harmonic properties of slice regular functions in one and several Clifford variables and give explicit formulas of the iterated Laplacian applied to slice regular functions and to their spherical derivative, which are new also…
We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and…
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value…
We study the relation between the Laplacian associated to an odd metric on a supermanifold and harmonic superfunctions, through the application of the calculus of variations to a supersymmetric sigma model.
A problem list in singularity theory. Most of these problems are related with the algorithmic enumeration of possible topological types of non-discriminant Morsifications of real function singularities, and/or with the Picard--Lefschetz…
We consider germs of holomorphic vector fields at the origin of $\mathbb{C}^3$, with non-isolated singularities that are tangent to a holomorphic foliation of codimension one. This configuration is known as a $2$-flag of foliations. The…
M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing…
We provide a complete system of invariants for the formal classification of complex analytic unipotent germs of diffeomorphism at $\cn{n}$ fixing the orbits of a regular vector field. We reduce the formal classification problem to solve a…
We study the Laplacian on a metrized graph, and its eigenfunctions.
Symmetries in the Lagrangian formalism of arbitrary order are analysed with the help of the so-called Anderson-Duchamp-Krupka equations. For the case of second order equations and a scalar field we establish a polynomial structure in the…
We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…