Resultado de búsqueda
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows.
We say that uis subharmonic in if u 0 on . This is, of course, equivalent to the condition u 0 in . Note that, in particular, all harmonic functions in are subhar-monic. The set of all subharmonic functions !R is denoted by S(). By de nition, every subharmonic function is twice continuously di erentiable on . Hence, S() ˆC2(). 1. Mean Value ...
Note that a harmonic function is subharmonic. Note that if vis C2 and z 0 is a maximum of v uthen the partials of v uat z 0 vanish and the 2nd derivative is non-positive, so that v= ( v u) 0. In fact this is enough to characterise subharmonic functions whose second partials exist: Lemma 14.6. Let v be a continuous function whose second partial
That is, f is harmonic on every complex line. A function f: U → R ∪ { − ∞} is plurisubharmonic, sometimes plush or psh for short, if it is upper-semicontinuous and for every a, b ∈ Cn, the function of one variable ξ ↦ f(a + bξ) is subharmonic (whenever a + bξ ∈ U ).
23 de ene. de 2021 · Subharmonic function. A function $ u = u ( x): D \rightarrow [ - \infty , \infty ) $ of the points $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, defined in a domain $ D \subset \mathbf R ^ {n} $ and possessing the following properties: 1) $ u ( x) $ is upper semi-continuous in $ D $; 2) for ...
A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset U:f(x)\le\frac{1}{n\alpha(n)r^{n-1}}\int\limits_{\partial\mathbb B_r(x)}f(y)\mathrm \space dS(y)$$
16 de may. de 2024 · Harmonic Functions. Let U subset= C be an open set and f a real-valued continuous function on U. Suppose that for each closed disk D^_ (P,r) subset= U and every real-valued harmonic function h defined on a neighborhood of D^_ (P,r) which satisfies f<=h on partialD (P,r), it holds that f<=h on the open disk D (P,r).
