Questions tagged [potential-theory]

Potential theory concerns solutions of elliptic partial differential equations (especially Laplace's equation) that are represented by integration against a measure or a more general distribution.

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Average behavior of Green's function near the boundary

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for ...
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$u\in C_0^3(\Omega)$ satisfis $-\Delta u=f(x), x\in\Omega.$Prove that $\sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} dx\leq n\int_{\Omega}f^2 dx.$

Suppose $u\in C_0^3(\Omega)$ satisfis $$-\Delta u=f(x), x\in\Omega.$$ Prove that $$\sum_{i,j=1}^{n}\int_\Omega u^2_{x_i x_j} \mathrm{d}x\leq n\int_{\Omega}f^2\mathrm{d}x.$$ I have tried as ...
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How to prove potencies of matrices by induction

a) The matrix has the property ajk = 0 for 1≤j≤k≤N. Proof: A^N = 0. Note: use the induction over N. b) With the help of a), calculate the expression B^20 for the real matrix enter image description ...
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Potential flow around ellipse

I stuck at the following problem: Consider a planar incompressible potential flow (i.e., an irrotational flow) around an elliptic profile \begin{equation*} P = \lbrace (x,y) \in \mathbb{R}^2 | \frac{...
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Sum of subharmonic functions attaining a maximum in an open, connected set.

I want to prove the following statement: If $p_1,..,p_n$ are subharmonic functions on an open, connected subset $V$ of the complex plane and if $p_1+...+p_n$ attains a maximum on $V$, then each $p_i$ ...
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Riesz potentials of $\mathscr{L}^p_\beta\left(\mathbb{R}^d\right)$ functions

In the Chapter V of Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30. Princeton, N.J.: Princeton University Press. XIV, 287 p. (1970)....
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Reverse Hardy-Litllewood-Sobolev inequality

Some time ago, I asked here: How to use Hardy-Littlewood-Sobolev inequality to estimate an integral involving two fuctions and Riesz Potential. about estimates involving the Hardy-Littlewood-Sobolev ...
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Taking the Vector Gradient of a Function in $(r)$ where $r = |\mathbf{\vec{r}}|$

I have tried looking for the answer to this online, but probably due to my personal lack of knowledge have been unable to ask the question effectively. Essentially I would like to understand how to ...
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How can I solve this PDE $h(\phi_{xx}+\phi_{yy})+h_{x}\phi_{x}+h_{y}\phi_{y}=0, h=5-e^{-x^4-y^2}$

$\phi(x,y)$ is the velocity potential ($\frac{\partial \phi}{\partial x}=-u,\frac{\partial \phi}{\partial y}=-v$) for a steady flow in a 2D ocean. $h(x,y)=5-e^{-x^4-y^2}$ is the water depth at $(x,y)$....
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Can "abstract potential theory" be used to establish maximum principles for more general PDOs?

As I understand it, this paper uses ideas from "abstract potential theory" (harmonic spaces, sheaves etc.) to find a maximum principle applying to unbounded domains. But its assumptions on ...
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Extension of superharmonic functions

In the "Classical Potential Theory" of Armitage and Gardiner, the following result is stated as an exercise (pg 195). Let $K$ be a compact of $\mathbb{R}^m$ ($m\geq2$) and $\Omega$ an open ...
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Helmholtz equation Inequality (potential theory)

I was reading the chapter about surface potential. I came across an inequality $$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|,$$ $k$ is a complex number here. Since $k$ is a complex number here, I am ...
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Let $u$ be a negative subharmonic function on $D(0,1)$. Then $\limsup_{r\to 1^-} \frac{u(rz_0)}{1-r}< 0$ for each $z_0\in \partial D(0,1)$.

This exercise has a hints to follow which is apply maximum principle $f(z)=u(z)+c\log|z|$ on the domain $\frac{1}{2}<|z|<1$. Now if I choose $c>0$ then since $u<0$ in the whole domain so ...