Subharmonic function: Difference between revisions

Revision as of 18:35, 14 July 2021

Latest revision as of 16:56, 17 June 2025

(10 intermediate revisions by 7 users not shown)

Line 1:

{{Short description|Class of mathematical functions}}{{Moreinline|date=June 2025}}

In [[mathematics]], '''subharmonic''' and '''superharmonic''' functions are important classes of [[function (mathematics)|functions]] used extensively in [[partial differential equations]], [[complex analysis]] and [[potential theory]].

Line 8 ⟶ 10:

Formally, the definition can be stated as follows. Let <math>G</math> be a subset of the [[Euclidean space]] <math>\R^n</math> and let

<math display="block">\varphi \colon G \to \R \cup \{ - \infty \}</math>

be an [[semi-continuity|upper semi-continuous function]]. Then, <math>\varphi </math> is called ''subharmonic'' if for any [[closed ball]] <math>\overline{B(x,r)}</math> of center <math>x</math> and radius <math>r</math> contained in <math>G</math> and every [[real number|real]]-valued [[continuous function]] <math>h</math> on <math>\overline{B(x,r)}</math> that is [[harmonic function|harmonic]] in <math>B(x,r)</math> and satisfies <math>\varphi(y) \leq h(y)</math> for all <math>y</math> on the [[boundary (topology)|boundary]] <math>\partial B(x,r)</math> of <math>B(x,r)</math>, we have <math>\varphi(y) \leq h(y)</math> for all <math>y \in B(x,r).</math>

Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.

Line 17 ⟶ 19:

* A function is [[harmonic function|harmonic]] [[if and only if]] it is both subharmonic and superharmonic.

* If <math>\phi</math> is ''C''² ([[smooth function|twice continuously differentiable]]) on an [[open set]] <math>G</math> in <math>\R^n</math>, then <math>\phi</math> is subharmonic [[if and only if]] one has <math> \Delta \phi \geq 0</math> on <math>G</math>, where <math>\Delta</math> is the [[Laplacian]].

* The [[maxima and minima|maximum]] of a subharmonic function cannot be achieved in the [[interior (topology)|interior]] of its ___domain unless the function is constant, thiswhich is the so-called the [[maximum principle]]. However, the [[minimum]] of a subharmonic function can be achieved in the interior of its ___domain.

* Subharmonic functions make a [[convex cone]], that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.

*The [[pointwise maximum]] of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.

*The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to <math>-\infty</math>).

*Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the [[fine topology (potential theory)|fine topology]] which makes them continuous.

Line 45 ⟶ 47:

If <math>f</math> is a holomorphic function, then

<math display="block">\varphi(z) = \log \left| f(z) \right|</math>

is a subharmonic function if we define the value of <math>\varphi(z)</math> at the zeros of <math>f</math> to be &minus;∞<math>-\infty</math>. It follows that

<math display="block">\psi_\alpha(z) = \left| f(z) \right|^\alpha</math>

is subharmonic for every ''α''&nbsp;> 0. This observation plays a role in the theory of [[Hardy spaces]], especially for the study of ''H<{{i sup>|p</sup>}}'' when 0&nbsp;< ''p''&nbsp;<&nbsp;1.

In the context of the complex plane, the connection to the [[convex function]]s can be realized as well by the fact that a subharmonic function <math>f</math> on a ___domain <math>G \subset \Complex</math> that is constant in the imaginary direction is convex in the real direction and vice versa.

Line 65 ⟶ 67:

<math display="block"> \|M \varphi\|_{L^2(\mathbf{T})}^2 \le C^2 \, \int_0^{2\pi} \varphi(e^{i\theta})^2 \, d\theta.</math>

If ''f'' is a function holomorphic in Ω and 0&nbsp;< ''p''&nbsp;< ∞, then the preceding inequality applies to ''φ''&nbsp;= |''f''&hairsp;{{hair space}}|{{i sup|''p''/2}}. It can be deduced from these facts that any function ''F'' in the classical Hardy space ''H^p'' satisfies

<math display="block"> \int_0^{2\pi} \left( \sup_{0 \le r < 1} \left|F(r e^{i\theta})\right| \right)^p \, d\theta \le C^2 \, \sup_{0 \leq r < 1} \int_0^{2\pi} \left|F(re^{i\theta})\right|^p \, d\theta.</math>

With more work, it can be shown that ''F'' has radial limits ''F''(''e''{{i sup|''iθ''}}) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F_r'', defined by ''F_r''(''e''{{i sup|''iθ''}})&nbsp;= ''F''(''r''&hairsp;{{hair space}}''e''{{i sup|''iθ''}}) tends to ''F'' in ''L''^''p''('''T''').

== Subharmonic functions on Riemannian manifolds ==

Line 74 ⟶ 76:

''Definition:'' Let ''M'' be a Riemannian manifold, and <math>f:\; M \to \R</math> an [[upper semicontinuous]] function. Assume that for any open subset <math>U\subset M</math>, and any [[harmonic function]] ''f''₁ on ''U'', such that <math>f_1 \geq f</math> on the boundary of ''U'', the inequality <math>f_1 \geq f</math> holds on all ''U''. Then ''f'' is called ''subharmonic''.

This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality <math>\Delta f \geq 0</math>, where <math>\Delta</math> is the usual [[Laplace–Beltrami_operator|Laplacian]].<ref>{{Cite journal | author = Greene, R. E. | year = 1974 | title = Integrals of subharmonic functions on manifolds of nonnegative curvature | journal = Inventiones Mathematicae | volume = 27 | pages = 265–298 | doi = 10.1007/BF01425500 | last2 = Wu | first2 = H. | postscriptissue = <!--None-->4| bibcode = 1974InMat..27..265G | issues2cid = 4122233796 }}, {{MathSciNet | id = 0382723}}</ref>

==See also==