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Dirichlet problem for an elliptic operator

consider de Dirichlet problem $Lu=0$ on the unit ball B of $\Bbb C^n$ and $u=f$ on the unit sphere $S^{2n-1}$, we suppose that $L$ is an elliptic operator. My question is there is a formula of the ...
Edward's user avatar
  • 21
2 votes
0 answers
86 views

Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space

Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
SMS's user avatar
  • 1,407
0 votes
1 answer
172 views

Harmonic functions and monotonic decay

I have a general question surrounding certain harmonic functions. I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, ...
HtmlProg's user avatar
1 vote
0 answers
43 views

Behaviour of higher order Laplacian in punctured domain

Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
User 11111's user avatar
8 votes
3 answers
701 views

Regularity of Newtonian potential along smooth boundary

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define $$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$ Is it true that $V(z) \in C^{\infty}(\partial \Omega)$? ...
student's user avatar
  • 1,350
2 votes
1 answer
203 views

Global Hölder regularity

I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...
Sean's user avatar
  • 375
2 votes
0 answers
134 views

Critical points of a strictly subharmonic function

Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant: \begin{equation} \Delta u = A > 0....
Leo Moos's user avatar
  • 5,038
2 votes
1 answer
197 views

Linear elliptic equation

Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
Samir's user avatar
  • 43
3 votes
1 answer
109 views

A harmonic function degenerate in one direction

Question. Let $u: B^3 \to \mathbf{R}$ be a harmonic function with $u(0) = 0$, $Du(0) = 0$, where its homogeneous harmonic blow-up is a polynomial $p = p(x,y)$ in two variables, so independent of $z$; ...
Leo Moos's user avatar
  • 5,038
2 votes
0 answers
53 views

Has the nodal map been studied?

Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary. Define ...
Leo Moos's user avatar
  • 5,038
2 votes
1 answer
106 views

'Dirichlet problem' along axis for harmonic functions

Question. Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the ...
Leo Moos's user avatar
  • 5,038
2 votes
1 answer
203 views

Reference for harmonic functions in cylinders

Question. What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders? I am trying to gain a better understanding of the behavior of ...
Leo Moos's user avatar
  • 5,038
7 votes
1 answer
410 views

Limit of zero sets of harmonic functions

Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...
user492517's user avatar
4 votes
1 answer
346 views

Is there a harmonic function with just one singular point?

Let $D \subset \mathbf{R}^2$ be the unit disc, and $L > 0$. Let $u: D \times (-L,L) \to \mathbf{R}$ satisfy \begin{equation} \begin{cases} \Delta u = 0 \quad \text{ on $D \times (-L,L)$ } \\ \frac{...
Leo Moos's user avatar
  • 5,038
7 votes
1 answer
977 views

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 ...
Llohann's user avatar
  • 369
3 votes
0 answers
85 views

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 ...
sharpe's user avatar
  • 721
5 votes
0 answers
545 views

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 ...
student's user avatar
  • 1,350
8 votes
2 answers
773 views

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 \...
Ali's user avatar
  • 4,135
7 votes
1 answer
339 views

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(...
Guy Fsone's user avatar
  • 1,101
0 votes
1 answer
468 views

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\...
asv's user avatar
  • 21.8k
3 votes
1 answer
206 views

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&...
Holden Lyu's user avatar
4 votes
0 answers
243 views

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 $C^{k, \alpha}$ ...
Holden Lyu's user avatar
6 votes
0 answers
326 views

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 ...
Leo Moos's user avatar
  • 5,038
2 votes
0 answers
147 views

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 ...
cork_twist's user avatar
8 votes
2 answers
622 views

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, ...
Jacob Lu's user avatar
  • 903
3 votes
1 answer
273 views

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 ...
Ali Taghavi's user avatar
0 votes
0 answers
184 views

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{\...
kichr's user avatar
  • 11
1 vote
0 answers
117 views

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 $c$...
dylan's user avatar
  • 11
2 votes
0 answers
113 views

$W^{1,p}-$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 ...
kaithkolesidou's user avatar
3 votes
0 answers
117 views

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(\...
Asaf Shachar's user avatar
  • 6,741
2 votes
1 answer
184 views

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) $f_n\ge 0$ $\int_\Omega f_n=I>0$ ...
Fozz's user avatar
  • 287
4 votes
1 answer
221 views

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 ...
Ali's user avatar
  • 4,135
5 votes
1 answer
348 views

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 ...
tituf's user avatar
  • 311
1 vote
1 answer
200 views

Smooth approximation of a subharmonic function in the barrier sense

Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f_{\epsilon}$ ...
Totoro's user avatar
  • 2,535
5 votes
1 answer
183 views

Can harmonic maps with immersive boundary conditions have singular points?

Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
Asaf Shachar's user avatar
  • 6,741
3 votes
1 answer
540 views

regularity of p-harmonic functions

We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$. Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
user143090's user avatar
2 votes
1 answer
170 views

The positive solutions of the weighted Laplacian equation

Let $u$ be a positive function on $\mathbb R^n$ such that $$ \Delta u-\partial_{x_1}u=0, $$ where $\Delta$ is the Laplacian operator $\partial_{x_1}^2+\partial_{x_2}^2+\cdots+\partial_{x_n}^2$. Can ...
Totoro's user avatar
  • 2,535
4 votes
0 answers
169 views

Can the rank of harmonic maps decrease far from the boundary?

Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be ...
Asaf Shachar's user avatar
  • 6,741
7 votes
2 answers
2k views

Eigenvalues of Laplace-Beltrami on half sphere

Let $ \Delta_\theta$ denote the Laplace-Beltrami operator on $S^{N-1}$. The eigenvalues of this are well known. I assume the same is the case of this operator on the upperhalf sphere; say $ S^{N-...
Math604's user avatar
  • 1,385
5 votes
1 answer
162 views

Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell: Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form? Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...
Asaf Shachar's user avatar
  • 6,741
1 vote
0 answers
105 views

Is every "higher-order" harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ $\newcommand{\TstarM}{...
Asaf Shachar's user avatar
  • 6,741
8 votes
2 answers
471 views

Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist a ...
Asaf Shachar's user avatar
  • 6,741
1 vote
0 answers
145 views

On the solution of Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an annular (bounded and connected) domain with inner and outer boundary $\Gamma_1$ and $\Gamma_2$, respectively. It is known that the PDE system $$ \begin{...
Julienne Franz's user avatar
1 vote
1 answer
126 views

Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field

Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
Ali Taghavi's user avatar
1 vote
1 answer
349 views

Functions orthogonal to harmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$ and $f_1,f_2 \in C^2(\bar{\Omega})$. Suppose $\int_{\Omega}(f_2-f_1)\varphi \, dx=0$ and $\int_{\Omega}(f_2 \Delta^{-1} f_2- f_1 \Delta^{-1} f_1)\...
A random mathematician's user avatar
11 votes
2 answers
2k views

Harmonic function properties on $\mathbb R^3$

Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...
vibe's user avatar
  • 211
3 votes
0 answers
156 views

Wolff's article: Note on counterexamples in strong unique continuation problems

I am reading Wolff's Note on counterexamples in strong unique continuation problems: http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf On Page 3, ...
Thomas Yang's user avatar
2 votes
1 answer
520 views

The flow of Harmonic vector fields

A map or a vector field $g: \mathbb{R}^n \to \mathbb{R}^n $ is called a harmonic map if all its components are harmonic functions. Motivated by conversations on this questions we ask: ...
Ali Taghavi's user avatar
2 votes
1 answer
295 views

Laplace equation in a domain with holes

Suppose $u_r(x_1,x_2)$ in $B_1(0) \setminus B_r(0)$ satisfy \begin{align*} \begin{cases} \Delta u_r &=0\\ u_r(|x|=r)&=-\log r \\ u_r(|x|=1)&=0 \end{cases} \end{align*} with $0<r<1$....
mnmn1993's user avatar
1 vote
1 answer
251 views

Is maximum principle valid in the case of non-smooth boundaries?

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad. Denote $U=U_2 \...
user101829's user avatar