相关论文: Convexity Properties of Harmonic Functions on Para…
We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the…
We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine…
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…
We obtain effective estimates for the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a…
Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…
We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic…
We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces…
The celebrated Almgren monotonicity formula for harmonic functions $u:\mathbb{R}^n \rightarrow \mathbb{R}$ says that its $L^2-$energy concentrated on a sphere of radius $r$, when measured in a suitable sense, is non-decreasing: if $u$…
We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds…
We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…
In this paper we discuss convexity, its average principle, an extrinsic average variational method in the Calculus of Variations, an average method in Partial Differential Equations, a link of convexity to $p$-subharmonicity, subsolutions…
We consider functions which are subfunctions with respect to the differential operator $$L_\rho = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2\rho \frac{\partial}{\partial x} + \rho^2 $$ and are doubly periodic in…
In this article, we investigate the regularity for certain elliptic systems without a $L^2$-antisymmetric structure. As applications, we prove some $\epsilon$-regularity theorems for weakly harmonic maps from the unit ball $B= B(m) \subset…
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…
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower…
We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $\lambda_1$-eigenfunctions. We extend such inequality to…
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 bound the $L^2$-norm of an $L^2$ harmonic $1$-form in an orientable cusped hyperbolic $3$-manifold $M$ by its topological complexity, measured by the Thurston norm, up to a constant depending on $M$. It generalizes two inequalities of…
A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…