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Subharmonic function

In , particularly in and partial differential equations, a subharmonic is a real-valued function u defined on an \Omega \subset \mathbb{R}^n that satisfies the sub-mean value property: for every B_r(x) \subset \Omega with center x \in \Omega and radius r > 0, the value u(x) is less than or equal to the of u over the or its . For twice continuously differentiable functions, this is equivalent to the condition that the Laplacian \Delta u \geq 0 in \Omega. In the context of , where \Omega \subset \mathbb{C}, subharmonic functions are upper semicontinuous functions that are bounded above by functions agreeing with them on circle boundaries, generalizing the notion to non-smooth cases. Subharmonic functions form a convex cone under addition and positive scaling, and they are closed under taking maxima; notably, the maximum of two subharmonic functions is again subharmonic. A fundamental consequence is the maximum principle: if a subharmonic function attains its maximum value at an interior point of a connected , it must be constant throughout the domain. This principle underscores their role in bounding solutions to elliptic equations, as subharmonic functions provide upper envelopes for functions and appear in applications like , where they model potentials below equilibrium. In higher dimensions, they connect to plurisubharmonic functions in several complex variables, extending the theory to Kähler geometry and holomorphic mappings.

Definition and Basic Concepts

Formal Definition

A subharmonic function on an open domain \Omega \subset \mathbb{R}^n is a function u: \Omega \to [-\infty, \infty) that is upper semicontinuous and satisfies the sub-mean value property. Upper semicontinuity means that for every x \in \Omega, \limsup_{y \to x} u(y) \leq u(x). The sub-mean value property requires that for every x \in \Omega and every r > 0 such that the closed ball \overline{B_r(x)} \subset \Omega, u(x) \leq \frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)} u \, d\sigma, where |\partial B_r(x)| denotes the surface area of the sphere \partial B_r(x) and d\sigma is the surface measure. For functions that are twice continuously differentiable, subharmonicity is equivalent to the condition that the Laplacian satisfies \Delta u \geq 0 in the distributional sense, meaning \int_\Omega u \Delta \phi \, dx \geq 0 for all nonnegative test functions \phi \in C_c^\infty(\Omega). Harmonic functions represent the boundary case where equality holds in the mean value property.

Relation to Harmonic and Superharmonic Functions

Subharmonic functions are closely related to functions, which represent the equality case within the broader class of subharmonic functions. Specifically, a u that is subharmonic in a \Omega \subset \mathbb{R}^n is if and only if it satisfies the mean value equality u(x) = \frac{1}{|\partial B(x,r)|} \int_{\partial B(x,r)} u(y) \, d\sigma(y) for every ball B(x,r) \subset \Omega, where \partial B(x,r) denotes of radius r centered at x, and d\sigma is the surface measure. This equality distinguishes functions from the general subharmonic case, where the u(x) \leq average holds. Superharmonic functions form the dual class to subharmonic functions. A function v is defined to be superharmonic in \Omega if -v is subharmonic there, which equivalently means that v satisfies the super-mean value property v(x) \geq \frac{1}{|\partial B(x,r)|} \int_{\partial B(x,r)} v(y) \, d\sigma(y) for every ball B(x,r) \subset \Omega. This duality implies that if u is subharmonic, then -u is superharmonic, and conversely, the negation of a superharmonic function is subharmonic. Harmonic functions are precisely those that are both subharmonic and superharmonic. The concept of subharmonic functions was introduced by in the early to extend the of potentials beyond the strict equality of the mean value property. In his seminal work, Riesz developed the foundational ideas linking subharmonic functions to , emphasizing their role in generalizing classical .

Key Properties

Mean Value Inequality

A subharmonic function u on an \Omega \subseteq \mathbb{R}^n satisfies the mean value inequality: for every x \in \Omega and every r > 0 such that \overline{B(x, r)} \subseteq \Omega, u(x) \leq \frac{1}{\sigma_{n-1}(\partial B(x, r))} \int_{\partial B(x, r)} u(y) \, d\sigma_{n-1}(y), where \sigma_{n-1} denotes the (n-1)-dimensional surface measure on the sphere \partial B(x, r). An equivalent form holds for the ball interior: u(x) \leq \frac{1}{m(B(x, r))} \int_{B(x, r)} u(y) \, dm(y), with m the Lebesgue measure. For C^2 subharmonic functions (those with \Delta u \geq 0), the inequality follows from applying the to the \nabla (u(y) |y - x|^{2 - n}) for n \geq 3 (or an analogous identity for n=2), yielding that the spherical mean M_u(x, r) = \frac{1}{\sigma_{n-1}(\partial B(x, r))} \int_{\partial B(x, r)} u(y) \, d\sigma_{n-1}(y) satisfies M_u'(x, r) = \frac{1}{n \omega_n r^{n-1}} \int_{B(x, r)} \Delta u(y) \, dm(y) \geq 0, so M_u(x, r) is nondecreasing in r and thus u(x) = M_u(x, 0+) \leq M_u(x, r). The ball version then arises by integrating the spherical means: the volume average equals \frac{n}{r} \int_0^r t^{n-1} M_u(x, t) \, dt / \int_0^r t^{n-1} dt \geq M_u(x, 0+) = u(x), since M_u(x, t) \geq u(x) for all t > 0. In the general case, subharmonic functions are upper semicontinuous functions satisfying the above inequality locally (or equivalently, \Delta u \geq 0 in the distributional sense). The inequality extends from the smooth case via approximation: any subharmonic u can be approximated from above on compact subsets by subharmonic functions u_k \uparrow u (using with mollifiers and the upper of u to control the limit), preserving the inequality in the limit k \to \infty. This relation to appears in one dimension, where subharmonic functions coincide with functions (as \Delta u = u'' \geq 0); there, directly implies the mean value inequality, since for u, u(x) \leq \frac{1}{2r} \int_{x-r}^{x+r} u(y) \, dy as the x is the average of the endpoints. For non-circular domains, the inequality extends via the : if \Omega is a bounded with G(x, y), then for x \in \Omega, u(x) \leq \int_{\partial \Omega} u(y) \, d\mu_x(y), where \mu_x is the harmonic measure (the of the at x onto \partial \Omega), obtained by approximating balls inscribed in \Omega or solving the .

Maximum Principle and Harnack's Inequality

The strong for subharmonic functions states that if u is a non-constant subharmonic function on a connected \Omega \subseteq \mathbb{R}^n, then u attains no maximum value in \Omega; if it does attain a maximum at some interior point, then u must be throughout \Omega. This follows from the mean value inequality: if u attains a maximum M at an interior point x_0, then u(x_0) \leq average of u over any B \subset \Omega centered at x_0, implying the average equals M and thus u = M on B; iterating over smaller balls and using connectedness of \Omega, u is constant. A related result is the weak maximum principle, which asserts that for a subharmonic function u on a bounded \Omega, \sup_{\Omega} u = \sup_{\partial \Omega} u. The proof again relies on the mean value inequality, showing that the supremum cannot exceed the boundary values without violating subharmonicity. Harnack's inequality provides bounds on positive s, which are a special case of subharmonic functions. For a positive harmonic function u on the B_r(0) \subset \mathbb{R}^n with n=2, the inequality states that \frac{r - |x|}{r + |x|} u(0) \leq u(x) \leq \frac{r + |x|}{r - |x|} u(0) for all x \in B_r(0). In dimension n=3, it takes the form \left( \frac{r - |x|}{r + |x|} \right)^2 u(0) \leq u(x) \leq \left( \frac{r + |x|}{r - |x|} \right)^2 u(0). In general dimension n \geq 2, the inequality is \left( \frac{r - |x|}{r + |x|} \right)^{n-1} u(0) \leq u(x) \leq \left( \frac{r + |x|}{r - |x|} \right)^{n-1} u(0) for x \in B_r(0). These estimates derive from the Poisson integral and the applied to auxiliary functions.

Examples and Applications

Classical Examples in Euclidean Space

In one dimension, convex functions u: I \to \mathbb{R} defined on an open interval I \subset \mathbb{R} provide fundamental examples of subharmonic functions, as their second derivative satisfies u'' \geq 0, which coincides with the condition \Delta u \geq 0 for the one-dimensional Laplacian. This equivalence holds because convexity implies the sub-mean value property over intervals, aligning with the definition of subharmonicity. In \mathbb{R}^2, the logarithmic potential u(x) = \log |x| serves as a classical example of a subharmonic function outside the , where it is since \Delta u = 0 , but in the distributional sense over \mathbb{R}^2, \Delta u = 2\pi \delta_0 \geq 0. Similarly, in \mathbb{R}^n for n \geq 3, the function u(x) = -|x|^{2-n} (up to a positive normalization constant) is subharmonic outside the , as it satisfies \Delta u = 0 away from the and \Delta u = c \delta_0 \geq 0 distributionally, with c > 0. Another family of examples in \mathbb{R}^n consists of power functions u(x) = |x|^p for $0 < p \leq 2. These are subharmonic because their Laplacian is nonnegative in the distributional sense; for radial functions, the Laplacian computes as \Delta u = p(p + n - 2) |x|^{p-2} for |x| > 0, and since p > 0 and p + n - 2 \geq 0 under the given range (with equality at p = 2 yielding a constant positive Laplacian equal to $2n), the part is nonnegative, supplemented by a nonnegative at the for p < 2. For p = 2, u(x) = |x|^2 is strictly subharmonic with \Delta u = 2n > 0. Trivial yet illustrative examples include constant functions, which have \Delta u = 0 \geq 0 and thus satisfy subharmonicity everywhere, and any harmonic function, which also has \Delta u = 0 and inherits the sub-mean value property with equality. These cases demonstrate that the class of subharmonic functions properly contains the harmonic functions.

Examples from Complex Analysis

In complex analysis, a fundamental example of a subharmonic function arises from the modulus of a holomorphic function. If f is holomorphic and nowhere zero on a domain \Omega \subset \mathbb{C}, then u(z) = \log |f(z)| is subharmonic on \Omega. This property stems from the sub-mean value inequality satisfied by |f(z)| on circles centered at any point in \Omega, combined with the fact that the logarithm is an increasing concave function, preserving the subharmonicity. A key construction for generating further examples involves compositions. If u is subharmonic on \Omega and \phi: \mathbb{R} \to \mathbb{R} is and non-decreasing, then \phi \circ u is also subharmonic on \Omega. This classical result allows the creation of new subharmonic functions from existing ones, such as applying \phi(t) = e^t to \log |f(z)| to obtain |f(z)| itself as subharmonic. Specific instances illustrate these principles in the . The function u(z) = |z|^2 is subharmonic on \mathbb{C}, as its Laplacian is \Delta u = 4 \geq 0. Similarly, u(z) = \operatorname{Re}(z^2) = x^2 - y^2 (where z = x + iy) is harmonic, hence subharmonic, since \Delta u = 0. For entire functions, if f is entire and non-constant, \log |f(z)| provides a non-harmonic subharmonic example, reflecting the growth of f via the . Blaschke products offer another prominent example. A finite or infinite B(z) is holomorphic in the unit disk \mathbb{D}, bounded by 1 in modulus, and \log |B(z)| is subharmonic in \mathbb{D}, with singularities only at the zeros of B. This subharmonicity aids in studying the distribution of zeros and boundary behavior in the disk.

Representation and Construction

The provides a canonical representation for subharmonic functions in . For a subharmonic function u defined on a \Omega \subset \mathbb{R}^n (n \geq 2) admitting a Green function G(x,y), there exists a unique positive \mu_u on \Omega and a unique h on \Omega such that u(x) = \int_{\Omega} G(x,y) \, d\mu_u(y) + h(x) for all x \in \Omega. In this framework, the measure \mu_u is termed the Riesz measure associated with u, and it equals the Laplacian of u in the distributional sense: \Delta u = c_n \mu_u, where c_n > 0 is a constant depending on the dimension n (specifically, c_n = (n-2) \omega_n for the Newtonian kernel in n \geq 3, with \omega_n the surface area of sphere). This ensures \mu_u is nonnegative, reflecting the subharmonicity condition \Delta u \geq 0. The proof of the theorem proceeds by approximating u with smooth subharmonic functions and applying to extract the measure component, or alternatively by constructing h as the greatest harmonic minorant of u via Perron's and showing the remainder is a potential. The (sweeping) offers another approach, redistributing mass from u onto sets while preserving the subharmonic property to isolate the part. follows from the fact that if two such representations exist, their difference would be both and superharmonic (hence constant), and adjusting for behavior yields equality; moreover, \mu_u = 0 if and only if u is .

Perron's Method for Constructing Subharmonic Functions

Perron's method provides a constructive approach to solving the Dirichlet problem for Laplace's equation by utilizing families of subharmonic and superharmonic functions to build generalized solutions that are subharmonic in nature. For a bounded open domain \Omega \subset \mathbb{R}^n and continuous boundary data \phi on \partial \Omega, the upper Perron solution is defined as H_\phi(x) = \inf \{ v(x) \mid v \text{ is superharmonic on } \Omega, \, v \geq \phi \text{ on } \partial \Omega \}. This infimum yields the smallest superharmonic majorant of the boundary data, and under the assumption of continuous \phi, H_\phi is harmonic in \Omega. Since harmonic functions satisfy the sub-mean value property, H_\phi is subharmonic on \Omega. Complementing this, the subharmonic envelope, or lower Perron solution, is constructed as the supremum of all subharmonic functions on \Omega that are bounded above by \phi on \partial \Omega: \underline{H}_\phi(x) = \sup \{ u(x) \mid u \text{ is subharmonic on } \Omega, \, u \leq \phi \text{ on } \partial \Omega \}. This yields the largest subharmonic minorant below the boundary data, which is itself subharmonic by the closure properties of subharmonic functions under pointwise suprema. When the boundary \partial \Omega is regular (e.g., satisfies the Wiener criterion at every point), \underline{H}_\phi = H_\phi, and the common value provides the unique harmonic solution to the Dirichlet problem that continuously extends to \phi on the boundary. The method ensures through sequences: decreasing sequences of superharmonic majorants converge to H_\phi, while increasing sequences of subharmonic minorants converge to \underline{H}_\phi, with regularity results guaranteeing harmonicity in the interior via the and . For irregular boundaries, H_\phi and \underline{H}_\phi remain subharmonic but may fail to attain the boundary values at singular points, highlighting the role of barrier functions in assessing boundary regularity. Historically, Perron's method was developed by Oskar Perron in 1923 as a generalization of earlier potential-theoretic ideas, initially using superharmonic majorants to address existence for the in \mathbb{R}^2. It was independently discovered by in the same year for \mathbb{R}^3 and refined by Marcel Riesz in 1926 through his axiomatic definition of subharmonicity, which solidified the sub-mean value property as central to the construction.

Subharmonic Functions in Complex Analysis

Subharmonic Functions in the Complex Plane

In the complex plane \mathbb{C}, identified with \mathbb{R}^2, subharmonic functions are defined as upper semicontinuous functions u: \Omega \to [-\infty, \infty) on an open domain \Omega \subset \mathbb{C} that satisfy the sub-mean value property over disks. Due to the conformal invariance of the Laplacian operator under holomorphic transformations, subharmonicity in \mathbb{C} is equivalently characterized by the sub-mean value inequality over circles centered at any point in the domain. Specifically, for a subharmonic function u on \Omega and for every a \in \Omega and r > 0 such that the closed disk \overline{D(a, r)} \subset \Omega, u(a) \leq \frac{1}{2\pi} \int_0^{2\pi} u(a + r e^{i\theta}) \, d\theta. This circular mean value inequality follows from the general sub-mean property in \mathbb{R}^2 and the rotational symmetry inherent to the complex structure. In one complex variable, subharmonic functions coincide precisely with plurisubharmonic functions, as the complex Hessian reduces to the in this setting, and the sub-mean property over complex lines aligns with the circular averages. Subharmonic functions bounded above on a in \mathbb{C} admit a harmonic majorant, meaning there exists a function h such that u \leq h throughout the domain; this provides key growth estimates and ensures the existence of least harmonic majorants via the Perron method.

Harmonic Majorants and Radial Maximal Functions

A harmonic majorant of a subharmonic function u defined on a domain \Omega \subset \mathbb{C} is a function v: \Omega \to \mathbb{R} such that v \geq u throughout \Omega. If such a majorant exists, the least harmonic majorant \tilde{u} is the infimum over all harmonic majorants of u, and \tilde{u} coincides with u outside the set where u = -\infty. For a subharmonic function u that is bounded above on the unit disc D = \{ z \in \mathbb{C} : |z| < 1 \}, a harmonic majorant exists and can be constructed via the Poisson integral formula applied to the boundary values derived from the upper semicontinuous regularization of u. In the unit disc D, the radial maximal function of a subharmonic function u at radius r < 1 is defined as M_r(u)(\theta) = \sup_{0 < \rho < r} u(\rho e^{i\theta}), for \theta \in [0, 2\pi). Since subharmonic functions satisfy the sub-mean value property over circles, the function \theta \mapsto M_r(u)(\theta) is subharmonic on the circle of radius r, and the family \{M_r(u)\}_{0 < r < 1} provides a tool to analyze boundary behavior. A fundamental characterization states that a subharmonic function u on D admits a harmonic majorant if and only if \limsup_{r \to 1^-} M_r(u)(\theta) < \infty for almost every \theta \in [0, 2\pi) with respect to Lebesgue measure. In this case, the least harmonic majorant is the Poisson integral of the boundary function given by these limsup values. Moreover, if u has a harmonic majorant, then by a variant of Fatou's lemma adapted to subharmonic functions, the radial limits \lim_{\rho \to 1^-} u(\rho e^{i\theta}) exist and are finite for almost every \theta. This ensures that the boundary function is integrable in the sense required for the Poisson representation.

Generalizations and Extensions

Subharmonic Functions on Riemannian Manifolds

In the setting of a Riemannian manifold (M, g), a function u: M \to \mathbb{R} is subharmonic if it is upper semicontinuous and satisfies \Delta_g u \geq 0 in the distributional sense, where \Delta_g denotes the Laplace-Beltrami operator acting on distributions. For smooth functions, this reduces to the pointwise condition \Delta_g u \geq 0. Equivalently, subharmonicity can be characterized by the sub-mean value property over geodesic balls: for every x \in M and sufficiently small r > 0 such that the geodesic ball B_r(x) is well-defined and the is a onto it, u(x) \leq \frac{1}{\mathrm{Vol}_g(B_r(x))} \int_{B_r(x)} u \, d\mathrm{Vol}_g. This generalizes the definition, where the Euclidean Laplacian is replaced by \Delta_g and by the Riemannian . Key properties of subharmonic functions adapt from the case but depend on the manifold's geometry. Notably, the holds on complete Riemannian manifolds without : a subharmonic function u that is bounded above attains its supremum only if u is constant. This result follows from the Omori-Yau maximum principle, which guarantees the existence of points where u nearly achieves its maximum with controlled gradient and Laplacian, under conditions such as bounded below. On non-complete manifolds or those with , additional assumptions like volume growth controls are needed to ensure the principle applies. A canonical example of a subharmonic function on M is the squared geodesic distance to a fixed point p \in M, given by f(x) = \frac{1}{2} d_g(p, x)^2, where d_g is the Riemannian distance. This function is convex along geodesics away from the cut locus of p and satisfies \Delta_g f \geq \dim M in the distributional sense, rendering it strictly subharmonic. It plays a role in volume comparisons and exhaustion functions for non-compact manifolds. Studying subharmonic functions on Riemannian manifolds presents challenges due to the absence of global coordinates, which hinders explicit computations of the Laplace-Beltrami or integrals. To address this, representations and estimates often employ the associated with \Delta_g, which facilitates probabilistic interpretations of mean value properties via and provides bounds for averages over balls without relying on coordinate charts.

Subharmonic Functions in Several Complex Variables

In several complex variables, the classical notion of subharmonic functions from one variable extends to plurisubharmonic functions, which form the cornerstone of pluripotential theory and are essential for understanding geometric properties like pseudoconvexity. These functions generalize subharmonicity by requiring it along lines rather than real balls, reflecting the CR structure of \mathbb{C}^n for n \geq 2. Introduced independently by Kiyoshi Oka and Pierre Lelong in 1942, plurisubharmonic functions arose in the study of domains of holomorphy and have since underpinned major advances in . A function u: \Omega \to [-\infty, \infty) defined on an open set \Omega \subset \mathbb{C}^n is plurisubharmonic if it is upper semicontinuous, not identically -\infty on any component of \Omega, and its restriction to every affine complex line L \subset \mathbb{C}^n with L \cap \Omega \neq \emptyset is subharmonic on L \cap \Omega. For C^2-smooth functions, this condition is equivalent to the complex Hessian matrix \left( \frac{\partial^2 u}{\partial z_j \partial \overline{z_k}} \right)_{1 \leq j,k \leq n} being at every point in \Omega. Examples include the logarithm of the modulus of a holomorphic function, \log |f(z)| for f holomorphic and nonconstant, as well as quadratic forms like |z|^2 and any on \mathbb{R}^{2n} \cong \mathbb{C}^n. The class is stable under addition, maxima, and certain compositions with holomorphic maps; specifically, if u is plurisubharmonic and f is holomorphic, then u \circ f is plurisubharmonic provided it is upper semicontinuous. Plurisubharmonic functions inherit key properties from their subharmonic counterparts but exhibit distinct behaviors in higher dimensions. They satisfy the : on a connected , if u attains its supremum at an interior point, then u is constant. Unlike subharmonic functions, which relate directly to the real Laplacian \Delta u \geq 0, plurisubharmonic functions are characterized by the positivity of the Levi form, the Hermitian form associated with the complex Hessian. Regularization theorems ensure that every plurisubharmonic function can be approximated uniformly on compact subsets by plurisubharmonic functions, often via with approximate identities. In pluripotential theory, developed prominently by Eric Bedford and starting in the 1970s, plurisubharmonic functions enable the study of the complex Monge-Ampère operator (dd^c u)^n, whose non-pluripolar part defines measures on pseudoconvex s. A defining application is in the characterization of pseudoconvex domains: an open set \Omega \subset \mathbb{C}^n is pseudoconvex if and only if it admits a plurisubharmonic exhaustion function, i.e., a plurisubharmonic u: \Omega \to \mathbb{R} such that u(z) \to \sup u as |z| \to \infty and the superlevel sets \{u < c\} are relatively compact for c < \sup u. This extends Oka's work on domains of holomorphy, where pseudoconvexity ensures the domain is a domain of existence for holomorphic functions. Further, plurisubharmonic functions facilitate approximation results, such as the solution to the for the complex Monge-Ampère equation on pseudoconvex domains with continuous boundary data, as established by and . These tools have profound implications for envelope theorems, capacity theory, and the of holomorphic functions across analytic sets.