Questions tagged [harmonic-functions]
For questions regarding harmonic functions.
211 questions
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Harmonic functions as limits of harmonic functions on graphs?
I have recently learned about Rodin and Sullivan's work that proved a conjecture of Thurston involving giving a construction for the map in the Riemann mapping theorem using circle packings and this ...
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18
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Do Szego Kernel in one variable by fixing another variable in a $C^{\infty}$ bounded domain is Bounded?
Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain. Let $ S(.,.)$ denotes the Szego Kenel of Holomorphic Hardy Space $H^2(\partial\Omega)$. Then for $w\in\Omega$ do $S(.,w)$ is a ...
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$\mathbb Z_k$-harmonic function that distinguishes two vertices of a graph
Let $G$ be a simple, undirected, connected graph on $n$ vertices, and let $A$ be an abelian group. A function $f:V(G)\rightarrow A$, on the vertices of the graph $V(G)$, is said to be $A$-harmonic if ...
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2
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Difference equation satisfied by discrete harmonic functions on square lattice
A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation
$$
\Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~.
$$
...
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1
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1
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Liouville property of hyperbolic spaces
It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...
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8
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1
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Harmonic functions on complete Riemannian manifolds
I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...
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6
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1
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Positive harmonic functions on nilpotent groups & Random walk on groups with a finite number of generators
I want to read the following papers in the English version which I could not find anywhere (the only papers I can get are the Russian versions). Kindly help me out.
Gregory A. Margulis, Positive ...
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0
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128
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Bubble tree convergence: Why is it necessary to consider centre of mass of the energy measure?
In the paper “Bubble Tree Convergence for Harmonic Maps” by Thomas H. Parker, after the picking the energy concentration points, he proceeded by expanding the map around each energy concentration ...
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3
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1
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129
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Component wise convergence of a sequence of complex harmonic functions
It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...
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3
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A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
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Kernel of the Laplacian + a function
It is known that the kernel of the (non-negative) Laplacian operator on a closed manifold consists of constant functions. I would like to ask if some similar phenomena happens for the modified ...
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3
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Differentiability of a weak solution
Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...
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Potential theory as a tool in extrinsic flows
Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
- 217
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545
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Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to
\begin{equation}
\begin{cases}
-\Delta u=0 \quad &\mbox{in $\Omega$}\\
\frac{\partial ...
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2
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131
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Harmonic function over a square with linear Neumann boundary conditions
For a rectangle with height 1 and length 2, here is the unique numerical solution
(showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum)
to ...
- 1,547
8
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2
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773
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Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
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Are Poisson integrals uniquely determined by their radial limits?
Let $\mu$ be a complex Borel measure on the unit circle, and suppose its Poisson integral $u$ satisfies $\lim_{r\to 1-}u(re^{i\theta})=0$ for every $\theta$. Does it follow that $\mu=0$?
This is of ...
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Does the pointwise mean value property imply harmonicity?
Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(...
- 12.9k
3
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1
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473
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Geometric flow by the level sets of a harmonic function
Let $u$ be an harmonic function in a cylindrical domain $B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma_t=\{u=t\}$ are graphs of functions on $B_2^{n-1}$.
Consider a ...
- 7,197
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147
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Dimension of critical set of p-harmonic function
Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...
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Polar growth property for harmonic Maass forms
The definition of a harmonic Maass form consists of three properties; (1) that it is modular, (2) that it is harmonic, and (3) that it has at most polar growth at the cusps (ordered in accordance with ...
- 1,024
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468
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Harmonic functions in infinite domain in Euclidean space
EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...
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3
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1
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About the proof of higher regularity boundary Harnack inequality
I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.
In the paper they used the Hopf lemma to show that $u_\nu>c&...
- 17.2k
3
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1
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273
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Harmonic interpolation with analytic initial condition
Let $n>1$ and $M\subset \mathbb{R}^n$ be a (sufficiently low dimensional) compact analytic submanifold.
Assume that $f:\mathbb{R}^n\to \mathbb{R}$ is an analytic function.
Is there a Harmonic ...
- 6,367
3
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156
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Matrix equation and spherical harmonics
I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...
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6
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Are the two-valued homogeneous harmonic functions classified?
Question. Is there a classification of homogeneous two-valued harmonic functions on $\mathbf{R}^n$, valid in dimensions $n \geq 3$?
For reference, multi-valued functions are familiar objects in ...
- 5,038
4
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243
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How to use blow-up to prove the boundary regularity for a harmonic function
While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem:
Thm. 2.30.
Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
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6
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2
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$\log |f|$ is subharmonic
It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions:
(1) Are there some weaker ...
- 91.8k
8
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2
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622
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Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 17.2k
5
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1
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580
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A question on the monotonicity formula for minimal submanifolds
I'm reading the proof of monotonicity formula from A Course in Minimal Surfaces by Colding-Minicozzi. The theorem says
Suppose $\Sigma^k \subset \mathbb{R}^n$ is a minimal submanifold and $x_0\in\...
- 5,038
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0
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Good (Sidon) Approximation of "Bumps"
Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...
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226
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Spherical harmonic expansion of a power function
Let $f$ be an even continuous function on the sphere $S^{n-1}$.
Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
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Extending harmonic functions defined in the closure of a bounded smooth domain to some larger domain
Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^N$ where $N\geq 2$.
Consider the Laplace equation with a Neumann boundary condition
$$
-\Delta u = 0 \quad\mbox{in } \Omega, \qquad
\frac{\...
- 3,065
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1
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Tangential harmonic $1$-forms are pullbacks of harmonic functions
This question has also been posted on MSE, but maybe here is the right place to obtain an answer.
Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The ...
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117
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Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
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A question about super-harmonic functions
Lets call a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ to be super-harmonic if $\nabla ^2 f = \sum_{i=1}^n \partial_i^2 f \leq 0$.
Now given such a $f$ as above I want to consider the ...
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Mean Value Inequality with Linear Term
I am having trouble proving this modified mean-value inequality.
Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$
Prove that there exists constants $r_0,C>0$ depending only on $c$...
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173
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Function Spaces on the Open Unit Disk defined by Hardy Space norms
I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations....
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Clarification on a potential typo in a theorem from Garnett's Bounded Analytic Functions (2007)
Theorem 6.5 from Garnett's 2007 book Bounded Analytic Functions is as follows. I quote it verbatim, because I am concerned about the possibility of there being a typo:
Theorem 6.5. Let $v\left(z\...
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1
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503
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A harmonic function $\varphi$ with $D\varphi \in L^q(\mathbb R^n)$ is constant
Let $\varphi$ be an harmonic function such that $D\varphi \in L^q(\mathbb R^n)$ for $q \in (1, +\infty)$. I read in Partial Differential Equations of Quin Han and Fanghua Lin that for $q = 2$, $\...
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$W^{1,p}-$regularity on the boundary for solution of Laplace equation with Robin boundary condition
I came across with the attached paper and here is the part that I try to understand.
If the non-tangential maximal function of $\nabla u$, i.e $(\nabla u)^*$, belongs in $L^p(\partial \Omega)$, then ...
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Filled level sets of harmonic functions
Let $f$ be an enitre function. Define the "filled level set of $f$ as follows:
$$A_M(f):=\{z\in{\mathbb C}:\ |f(z)|\le M\}$$
Theorem 1 in Topological Properties of Level-Sets of Entire Functions ...
2
votes
1
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Positive subharmonic functions with constant integral blowing up at boundary
Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying
$\Delta f_n\ge 0$ (subharmonic)
$f_n\ge 0$
$\int_\Omega f_n=I>0$ ...
- 91.8k
7
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1
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Harmonic functions on knot complements
In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} =...
- 217
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1
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Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
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5
votes
1
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324
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Periods of the harmonic conjugate and a Dirichlet problem on a multiply connected domain
Any harmonic function $u$ on a simply connected domain in $\mathbb{R}^2$ is the real part of a holomorphic function. If the domain is multiply connected, then this is no longer true: the harmonic ...
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0
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198
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"Universal" polynomial of bounded norm on the sphere
Consider the space $V_{d,n}=\mathbb{R}[x_1,\ldots,x_n]_d$ of homogeneous polynomials of degree $d$ in $n$ variables. I am interested in the set of bounded polynomials on the sphere $$B_{d,n}=\{f\in V_{...
- 3,031
1
vote
1
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242
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Subharmonic function in unbounded regions
The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$:
$$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
- 91.8k
1
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1
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211
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A question of uniqueness
Let $u$ an harmonique function on $\Omega=(a,b)\times (0,+\infty)$ and boundary conditions :
$\displaystyle u(a,y)=u(b,y)=0,\quad\forall y\geq 0$
$\displaystyle u(x,0)=0,\,\lim_{y\to +\infty} u(x,y)=...
- 91.8k
5
votes
1
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348
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A differential inequality involving gradient and laplacian
Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
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