Related papers: Topological regularity for solutions to the genera…

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over the complex numbers. Then the primitive cohomology $H^{1,1}_{prim}(X)$ has, up to a sign, a natural orthonormal basis $(\eta_i)_{i\in [1, N]}$ given by certain meromorphic…

We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and…

A normal form is derived for Hamiltonian-Hopf bifurcations of solitary waves in generalized nonlinear Schr\"odinger equations. This normal form is a simple second-order nonlinear ordinary differential equation that is asymptotically…

This paper is concerned with a class of nonlocal dispersive models -- the $\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$…

Considered is the equation $$ (T(a)+H(b))\phi=f, $$ where $T(a)$ and $H(b)$, $a,b\in L^\infty(\mathbb{T})$ are, respectively, Toeplitz and Hankel operators acting on the classical Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the…

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

The tension equation for a mapping $f:{\mathbb C}\to {\mathbb C}$ is the nonlinear second order equation \[ \Delta f +\varphi(f) f_z f_{\bar z} = 0\] Solutions are "harmonic" mappings. Here we give a complete description of the solution…

In this paper has been considered probability-one global convergence of NFPH (Newton-Fixed Point Homotopy) algorithm for system of nonlinear equations and has been proposed a probability-one homotopy algorithm to solve a regularized…

Here is the simplest particular case of our main result: let $f:{\bf R}\to {\bf R}$ be a function of class $C^1$, with $\sup_{\bf R}f'>0$, such that $$\lim_{|\xi|\to +\infty}{{f(\xi)}\over {\xi}}=0\ .$$ Then, for each $\lambda>{{\pi^2}\over…

We prove two H\"older regularity results for solutions of generated Jacobian equations. First, that under the A3 condition and the assumption of nonnegative $L^p$ valued data solutions are $C^{1,\alpha}$ for an $\alpha$ that is sharp. Then,…

If H is a Hopf algebra whose square of the antipode is the identity, $v\in\l (V)\otimes H$ is a corepresentation, and $\pi :H\to\l (W)$ is a representation, then $u=(id\otimes\pi)v$ satisfies the equation $(t\otimes id)u^{-1}=((t\otimes…

Using infinite compositions, we solve the general equations $P(\lambda w) = p(w)f(P(w))$ for holomorphic functions $p$ and $f$. We describe the situations in which this equation is palpable; and their effectiveness at describing dynamical…

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…

We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear…

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential…

In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain $\Omega$ satisfies the exterior Reifenberg…

We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with H$\ddot{\text{o}}$lder continuous top-order coefficients and lower-order coefficients in a Morrey space.

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of…

We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on…