Immanuel Ben Porat
We investigate variants of a Three Circles type Theorem in the context of \mathcal{Q}-valued functions. We prove some convexity inequalities related to the L^{2} growth function in the \mathcal{Q}-valued settings. Optimality of these inequalities and comparsion to the case of real valued harmonic functions is also discussed.
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In this note we give a quantitative estimate on the number of singular points of multiplicity $Q$ of a $2$-dimensional $Q$-valued energy minimizing map, in terms of the value of its frequency function.
We develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity and compactness techniques to investigate existence of solution of the multivalued equation $\displaystyle - Δ_Φ u \in \partial j(.,u) + λh \mbox{in} Ω$, where $Ω\subset {\bf R}^{N}$ is a bounded smooth domain, $Φ: {\r} \longrightarrow [0,\infty)$ is a suitable N-function, $Δ_Φ$ is the corresponding $Φ$-Laplacian, $λ> 0$ is a parameter, $h:Ω\rightarrow{\r}$ is integrable and $\partial j(., u)$ is the subdifferential of a function $j$ associated with critical growth…
Characterisation of some multivalued harmonic functions on $\mathbb{R}^{n}$
针对Rⁿ上的Z₂-调和函数，本文结合Donaldson与Dashen Yan的构造结果，证明其无穷远二次增长、光滑余维2分支集附近O(r^(3/2))增长的性质可唯一刻画该类函数，仅差Rⁿ的刚体运动。
For the $p$-harmonic function with strictly convex level sets, we find a test function which comes from the combination of the norm of gradient of the $p$-harmonic function and the smallest principal curvature of the level sets of $p$-harmonic function. We prove that this curvature function is convex with respect to the height of the $p$-harmonic function. This test function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on the ball. For the minimal graph, we obtain a similar results.
It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $\mathbf{R}^3.$ Moreover, we give a new proof of the convexity resul…
It is shown that, given a point $x\in\mathbbm{R}^d$, $d\ge 2$, and open sets $U_1,...,U_k$ containing $x$, any convex combination of the harmonic measures for $x$ with respect to $U_n$, $1\le n\le k$, is the limit of a sequence of harmonic measures for $x$ with respect to open subsets $W_m$ of $U_1\cup... \cup U_k$ containing $x$. This answers a question raised in connection with Jensen measures. More generally, we prove that, for arbitrary measures on an open set $W$, the set of extremal representing measures, with respect to the cone of continuous potentials on $W$ or with respect to the con…
Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of coercivity for the intrinsic bi-energy. In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit $4$-ball $B_1 \subset \mathbf{R}^4$ into the sphere $\mathbf{S}^n$. This is a higher-order analogue of the energy convexity a…
We establish a maximum principle for a two-point function in order to analyze the convexity of level sets of harmonic functions. We show that this can be used to prove a strict convexity result involving the smallest principal curvature of the level sets.
We discuss various representations of planar $p$-harmonic systems of equations and their solutions. For coordinate functions of $p$-harmonic maps we analyze signs of their Hessians, the Gauss curvature of $p$-harmonic surfaces, the length of level curves as well as we discuss curves of steepest descent. The isoperimetric inequality for the level curves of coordinate functions of planar $p$-harmonic maps is proven. Our main techniques involve relations between quasiregular maps and planar PDEs. We generalize some results due to P. Lindqvist, G. Alessandrini, G. Talenti and P. Laurence.
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 which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.
Strong convexity for harmonic functions on compact symmetric spaces
本文证明了紧对称空间及正规齐性紧黎曼流形上调和函数的 |h|² 在 Laplace-Beltrami 算子任意次作用下保持非负，从而推广了欧氏空间中的相应强凸性结果。
We consider functions which are subfunctions with respect to the differential operator $$L_ρ= \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + 2ρ\frac{\partial}{\partial x} + ρ^2 $$ and are doubly periodic in the plane. These functions play an important role in describing the asymptotic behavior of entire and subharmonic functions of finite order. In studying their properties we are led to problems concerning the uniqueness of Martin functions and the critical value for the parameter $ρ$ in the homogeneous boundary problem for the operator $L_ρ$ in a domain on the torus.
We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology, as time goes to infinity, to the unique limiting harmonic map.
Let $Ω$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $Ω$ that vanishes on the boundary of $Ω$ is called a Martin function. In this note, we show that, when $Ω$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $Ω$ is symmetric, then the maximum of any Martin function along a slice $Ω\cap (\{t\}\times\mathbb{R}^d)$ is attained at $(t,0).$
In this work we establish a connection between two classical notions, unrelated so far: Harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity type and propagation of smallness results for harmonic functions on the lattice.
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 minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes r…
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 (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature $K$. Such geodesic curvature f…
We consider the identification of heterogeneous linear elastic moduli in the context of time-harmonic elastodynamics. This inverse problem is formulated as the minimization of the modified error in constitutive equation (MECE), an energy-based cost functional defined as an weighted additive combination $\mathcal{E}+κ\mathcal{D}$ of the error in constitutive equation (ECE) $\mathcal{E}$, expressed using an energy seminorm, and a quadratic error term $\mathcal{D}$ incorporating the kinematical measurements. MECE-based identification are known from existing computational evidence to enjoy attract…
Convex Functions are $p$-Subharmonic Functions, $p >1$ On $\mathbb{R}^n$ with Applications
本文探讨了凸性与 $p$-次调和性的联系，证明了 $\mathbb{R}^n$ 和 Riemannian 流形上凸函数在特定条件下为 $p$-次调和函数，并应用于完备非紧流形上常函数的判定。
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