Related papers: Results on gradients of harmonic functions on Lips…
We show that gradient descent can converge to any local minimum of a smooth semi-algebraic function. This holds if the step sizes are nonsummable and sufficiently small. The same results hold for the subgradient method on locally Lipschitz…
We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian…
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet,…
We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…
We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude…
Classical results show that gradient descent converges linearly to minimizers of smooth strongly convex functions. A natural question is whether there exists a locally nearly linearly convergent method for nonsmooth functions with quadratic…
We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions…
Level proximal subdifferential was introduced by Rockafellar recently for studying proximal mappings of possibly nonconvex functions. In this paper a systematic study of level proximal subdifferential is given. We characterize variational…
We study Lipschitz continuity for solutions of the $\bar{\alpha}$-Poisson equation in planar cases. We also review some recently obtained results. As corolary we can restate results for harmonic and gradient harmonic functions.
We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the…
For a locally Lipschitz continuous function $f:X\to\mathbb{R}$ the generalized gradient $\partial f(x)$ of Clarke is used to develop some (set-valued) gradient on a set $A\subset X$. Existence, uniqueness and some approximation are…
We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the…
In this article, by applying the well known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and…
Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $\Omega$. The gradient estimates…
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this…
We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions…
We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…
Numerical experiments with smooth surface extension and image inpainting using harmonic and biharmonic functions are carried out. The boundary data used for constructing biharmonic functions are the values of the Laplacian and normal…