Analysis

Potential theory studies harmonic functions, subharmonic functions, and the Laplace operator, including harmonic measure, regular boundary points, and the Dirichlet problem. The Brownian representation of the Dirichlet problem connects these analytic objects with exit probability distributions.

An integral equation asks for an unknown function that occurs under an integral. Integral equations with a variable upper limit have a triangular dependence structure that often permits successive approximation.

A Volterra integral equation has a variable integration limit, so its value at a point depends only on earlier values. Repeated integration produces factorial decay, which often makes its Neumann series converge without a smallness assumption.

For time-dependent integral kernels on a Riemannian manifold, composition also integrates over the intermediate point: $(A\ast B)(t,x,y)={\int }_0^t{\int }_{M}A(t-s,x,z)B(s,z,y)d{V}_{z}ds$. It is associative under the usual absolute-integrability hypotheses. The integration simplex gives factorial bounds for iterated convolutions of bounded kernels.

Starting from $\Vert \tau g(t){\Vert }^2\le C^{2}t{\int }_0^t\Vert g(s){\Vert }^{2}ds$, induction integrates $s^{2n-2}$ at the $n$th step. Its denominator accumulates the odd integers $1,3,\dots ,2n-1$. The factorial bound for a Volterra iterate is stronger because $(n!{)}^2\ge (2n-1)!!$.

If $\Vert \tau f(t)\Vert \le C{\int }_0^t\Vert f(s)\Vert ds$, repeated integration produces the time-ordered simplex of volume $t^n/n!$. Thus the Neumann series ${\sum }_{n\ge 0}{\tau }^n$ converges on every finite interval, without requiring the single-step operator norm to be less than one.

A spherical harmonic is an eigenfunction of the Laplacian on the unit sphere:

The integral of three spherical harmonics on the sphere. In the conventional complex basis it is real and equals a product of two Wigner 3j symbols and $\sqrt{(2{\ell }_1+1)(2{\ell }_2+1)(2{\ell }_3+1)/(4\pi )}$. It vanishes unless the multipoles satisfy a triangle condition, ${\ell }_1+{\ell }_2+{\ell }_3$ is even, and $m_1+m_2+m_3=0$. It carries the angular geometry of the reduced CMB bispectrum.

A regular solid harmonic is $r^l{Y}_l^m(\theta ,\varphi )$: it is homogeneous of degree $l$ and solves the Laplace equation. Thus $\mathbf{r}\cdot \nabla F=lF$ and ${\nabla }^{2}F=0$. These identities turn angular spherical harmonic expansions of displacement, density and Newtonian gravitational potential into radial equations.

For orthonormal complex spherical harmonics, $P_{\ell }(\hat{\mathbf{n}}\cdot \hat{\mathbf{m}})=4\pi {\sum }_{m=-\ell }^{\ell }{Y}_{\ell m}(\hat{\mathbf{n}})Y_{\ell m}^{\ast }(\hat{\mathbf{m}})/(2\ell +1)$. This converts an angle between two directions into a separable sum. Together with Orthogonality of Legendre polynomials, it projects rotationally invariant scattering kernels onto angular multipoles.

A spin-weight-$s$ field on an oriented sphere acquires a phase under rotation of its local tangent-frame basis. Spin-weighted spherical harmonics form the angular basis for such fields. The polarization combinations $Q\pm iU$ carry opposite spin weights of magnitude two; ordinary scalar functions have spin weight zero.

The angular basis ${}_s{Y}_{\ell m}$ for a spin-weight-$s$ field generalizes ordinary spherical harmonics. It has $\ell \ge |s|$. For scalar-source CMB polarization about a wavevector, only $m=0$ survives and ${}_{\pm 2}{Y}_{20}\propto {\sin }^2\theta$.

On a spin-$s$ field $f$, the spin-raising and lowering operations are $ðf=-{\sin }^s\theta ({\partial }_{\theta }+i\csc \theta {\partial }_{\phi })({\sin }^{-s}\theta f)$ and $\overline{ð}f=-{\sin }^{-s}\theta ({\partial }_{\theta }-i\csc \theta {\partial }_{\phi })({\sin }^s\theta f)$. The intermediate spin changes after each application. For an axisymmetric scalar $f(\mu )$, either operation applied twice yields $(1-{\mu }^2)f^{\prime \prime }(\mu )$.

Here $\mu =\cos \theta$ and $a_{\ell ,m}=\sqrt{({\ell }^2-m^2)/[(2\ell +1)(2\ell -1)]}$. This follows by combining the associated Legendre function recurrence $(2\ell +1)\mu P_{\ell }^m=(\ell -m+1)P_{\ell +1}^m+(\ell +m)P_{\ell -1}^m$ with the normalized spherical harmonic constants. It gives the adjacent-multipole couplings in a photon Boltzmann hierarchy.

The Rayleigh plane-wave expansion iswhere $j_{\ell }$ is a spherical Bessel function.

A power law is a relation of the form $f(x)=C{x}^p$ for constants $C$ and $p$. Under rescaling of its argument it satisfies $f(\lambda x)={\lambda }^{p}f(x)$.

A cored power-law density profile is finite at its centre and approaches a power law outside a core radius. A simple continuous example is $n(r)=n_0$ for $r\le r_c$ and $n(r)=n_0(r/r_c{)}^{-\beta }$ for $r>r_c$. The normalization $n_0$ is the core density. Smoothing the transition changes integrated coefficients but not convergence thresholds fixed by the outer slope.

A function has polynomial growth when $|f(x)|\le C(1+|x|{)}^N$ for some constants $C,N$. Such growth is slow enough for a locally integrable function to define a tempered distribution.

Complex dynamics studies iteration of holomorphic and rational maps, especially the division of the Riemann sphere into stable Fatou behavior and chaotic Julia behavior.

A family of holomorphic or meromorphic maps is normal when every sequence has a subsequence converging locally uniformly in the spherical metric, possibly to infinity.

A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.

The Fatou set of a rational map is the largest open subset of the Riemann sphere on which its iterates form a normal family. Its complement is the Julia set.

A periodic Fatou component of a rational map is an immediate attracting basin, an immediate parabolic basin, a Siegel disc, or a Herman ring. Attracting and parabolic basins capture critical orbits, while the boundaries of rotation domains lie in the closure of the postcritical set.

Every immediate basin of an attracting fixed point of a rational map contains a critical point. Otherwise the map on the basin would be an unbranched covering; lifting it to the unit disc would give an automorphism fixing a point, contradicting strict contraction at the attracting fixed point.

The postcritical set is the closure of the union of the forward orbits of all critical values.

A parabolic basin consists of points whose iterates converge to a parabolic cycle along a specified attracting direction. Each connected basin component lies in the Fatou set, and its boundary lies in the Julia set.

Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.

The Julia set is the complement of the Fatou set. It is nonempty, closed, completely invariant, and perfect for every rational map of degree at least two.

If a closed set $E$ satisfies $f^{-1}(E)=E$ for a rational map of degree at least two, then either $E$ has at most two points and lies in the Fatou set, or $J(f)\subseteq E$. The complement omits every point of $E$, so Montel theorem proves normality when $|E|\ge 3$.

The accumulation-point set of $J(f)$ is closed and completely invariant. Applying the classification of completely invariant closed sets shows that it equals $J(f)$; hence the Julia set is perfect.

For every $d\ge 2$, choose a $d$th root of unity $r\ne 1$ with $d|1-r|>1$, put $a=1-r$, and setIts only critical points are $0$ and infinity, and both land on the repelling fixed point $r$. The postcritically finite map has no possible Fatou component, so its Julia set is the whole sphere.

The degree-$d$ Chebyshev polynomial has Julia set the interval $[-1,1]$ after the standard normalization. Its Fatou set is the connected complement of that interval in the Riemann sphere, so it has exactly one Fatou component.

Near a superattracting fixed point, a holomorphic map is conformally conjugate to its leading monomial.

For a degree-$d$ polynomial $f(z)=a_d{z}^d+\cdots$, the Böttcher coordinate near infinity satisfiesIts modulus extends naturally throughout the basin of infinity.

The escape-rate Green function isIt vanishes on the filled Julia set, is positive and harmonic on the basin of infinity, satisfies $G_f\circ f=d{G}_f$, and equals $\log |{\varphi }_f|$ near infinity and by continuation throughout the basin.

The Julia set of a polynomial is connected exactly when every finite critical point belongs to the filled Julia set. In this case the Böttcher coordinate extends conformally over the entire basin of infinity; an escaping critical point is precisely an obstruction to that continuation.

For $f_c(z)=z^2+c$ outside the Mandelbrot set, evaluate the dynamical Böttcher coordinate at the critical value:This parameter Böttcher map is a conformal isomorphism from the complement of the Mandelbrot set to the exterior unit disc, proving that the Mandelbrot set is full and connected.

The holomorphic fixed-point index isAt a simple fixed point of multiplier $\lambda \ne 1$, it equals $1/(1-\lambda )$. The indices of all fixed points of a rational map, counted appropriately, sum to one.

If a quadratic rational map has three distinct fixed points with multipliers ${\lambda }_0,{\lambda }_1,{\lambda }_2$, then ${\lambda }_i{\lambda }_j\ne 1$ for $i\ne j$. Otherwise the two corresponding indices already sum to one, contradicting the nonzero third index.

A parabolic cycle is a periodic orbit whose multiplier is a root of unity; an iterate then has multiplier one and attracting petals.

A hyperbolic component in a parameter space consists of maps with a specified attracting cycle. Its multiplier gives a holomorphic coordinate on the component in the quadratic family, and root-of-unity boundary multipliers give parabolic parameters.

For every prime $p$, the polynomial $f_c^p(0)$ has a nonzero root, giving a superattracting cycle of exact period $p$. Moving in its hyperbolic component to a boundary point with cycle multiplier $-1$ gives a parabolic cycle of exact period $p$. Thus the set of parabolic periods in the quadratic family is infinite.

A functional equation specifies a function through an identity involving values of that function at several arguments.

An additive function on an additive semigroup satisfies $f(x+y)=f(x)+f(y)$ whenever $x$, $y$, and $x+y$ belong to its domain.

Every Lebesgue-measurable additive function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is continuous. Small-value preimages are measurable; one of their integer dilates has positive measure, so the Steinhaus theorem puts a neighbourhood of zero in their difference set. Additivity then makes $f$ small on that neighbourhood.

If $f:[0,\infty )\to \mathbb{R}$ is continuous and is an additive function, then $f(t)=tf(1)$. Additivity proves this first for nonnegative rational numbers; because $f$ is continuous, the identity extends to every nonnegative real number.

Nonlinear analysis studies equations and variational problems in which superposition fails. Its tools include nonlinear functional inequalities, conserved energies, compactness, phase-plane methods, and concentration arguments.

For $0<\lambda <d$ and suitable conjugate exponents, the Hardy–Littlewood–Sobolev inequality bounds convolution with the Riesz kernel $|x{|}^{-\lambda }$. One useful form is

The generalized Korteweg–De Vries equation is the nonlinear partial differential equationThe cases below the exponent $p=5$ are energy-subcritical in $H^1(\mathbb{R})$.

Sufficiently regular decaying solutions conserveBoth identities follow by integration by parts from the divergence form of the equation.

For $p<5$, the one-dimensional Gagliardo-Nirenberg interpolation inequality givesThe exponent of $\Vert u_x{\Vert }_2$ is below two, so the conserved mass and energy control the $H^1$ norm. The blowup alternative then extends every local $H^1$ solution globally.

If $Q^{\prime \prime }-Q+Q^p=0$, thensatisfies $Q_c^{\prime \prime }-c{Q}_c+Q_c^p=0$, and $u(t,x)=Q_c(x-ct)$ is a traveling-wave solution of the Generalized Korteweg–De Vries equation.

For $p<5$, a generalized KdV solitary wave is orbitally stable in $H^1(\mathbb{R})$ modulo spatial translations. The variational slope has the stable sign becauseis strictly increasing in $c$. Conserved mass and energy then furnish a Lyapunov functional transverse to the translation orbit.

The gravitational Hartree equation couples a wave function to its attractive Newtonian potential:

The gravitational Hartree flow conservesThe self-adjointness of the Newtonian convolution turns the time derivative of the potential energy into $\mathrm{Re}\int \varphi u\overline{u_t}$, which cancels the derivative of the kinetic energy by the equation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality and Gagliardo-Nirenberg interpolation inequality giveConserved mass and energy therefore bound the $H^1$ norm by Young inequality, and the blowup alternative gives global existence.

For a finite-variance solution in four dimensions,satisfies $I^{\prime \prime }(t)=16E(u)$. The identity uses the degree $-2$ homogeneity of the Newtonian kernel. Negative-energy finite-variance data consequently blow up in finite time by the convexity argument.

The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

A Lane-Emden equation is a semilinear elliptic equation of the formFor a radial function it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.

A Pohozaev identity is an integral identity for a nonlinear partial differential equation, obtained by multiplying the equation by a scaling vector field such as $x\cdot \nabla u$ and applying integration by parts. It relates kinetic, potential, and nonlinear energies.

Testing the ground-state equation against $Q$ and applying the Pohozaev identity givesConsequently $E(Q)=0$ and $\inf J=\frac{d}{d+2}\Vert Q{\Vert }_2^{4/d}$.

The symmetric decreasing rearrangement $u^{\ast }$ of a measurable function is the radial, radially nonincreasing function whose superlevel sets are centered balls with the same measures as those of $|u|$. It preserves $L^p$ norms, and the Pólya-Szegő inequality gives $\Vert \nabla u^{\ast }{\Vert }_2\le \Vert \nabla u{\Vert }_2$.

For $u\in W^{1,p}({\mathbb{R}}^d)$, the symmetric decreasing rearrangement satisfies

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

The mass-critical focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$ isIts scaling $u(t,x)\mapsto {\lambda }^{d/2}u({\lambda }^{2}t,\lambda x)$ preserves the Lebesgue space norm $\Vert u{\Vert }_{L^2}$, which is the conserved mass.

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

A virial identity describes the time derivatives of a spatial moment of a solution to a dispersive equation. For the mass-critical nonlinear Schrödinger equation, the second derivative of the variance is a constant multiple of the conserved energy.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

For a finite-variance solution of the mass-critical focusing nonlinear Schrödinger equation, the virial identity givesIf $E(u_0)<0$, the right-hand side is a negative constant. The nonnegative variance would then become negative in finite time if the solution remained regular, so the solution blows up in finite time.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

For the mass-critical focusing nonlinear Schrödinger equation in dimension $d$, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

For $p=2+4/d$, the mass-critical Weinstein functional isIt is invariant under multiplication by a nonzero scalar, spatial dilation, translation, and constant phase. Its positive minimizers are rescalings and translations of the NLS ground state.

Normalize a minimizing sequence so that its $L^2$ norm and gradient norm are fixed. Symmetric decreasing rearrangement does not increase the gradient norm and preserves every $L^p$ norm, while radial compactness prevents translation loss. A weakly convergent subsequence therefore converges strongly in the nonlinear $L^{2+4/d}$ norm and yields a nonnegative minimizer.

Every minimizer of the Weinstein functional has the formwhere $Q$ is the positive radial NLS ground state. The Euler-Lagrange equation first reduces a minimizer to a rescaled ground-state equation; the equality cases in the diamagnetic and rearrangement inequalities give the constant phase, translation, and radial profile.

In dimension $d$, the mass-critical sharp inequality isand equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$. For $d=2$ this becomes $\Vert u{\Vert }_4^4\le 2\Vert Q{\Vert }_2^{-2}\Vert u{\Vert }_2^2\Vert \nabla u{\Vert }_2^2$.

The sharp inequality impliesThus energy controls the gradient whenever the mass is strictly below the mass of the NLS ground state.

Suppose $\Vert u_n{\Vert }_2=\Vert Q{\Vert }_2$, $\Vert \nabla u_n{\Vert }_2=\Vert \nabla Q{\Vert }_2$, and $E(u_n)\to 0$. The profile decomposition modulo translations and the sharp inequality force exactly one nonzero profile: splitting the mass would make the sharp inequality strict. Hence some translations $u_n(\cdot +x_n)$ converge strongly in $H^1$, and therefore in $L^{2+4/d}$, to an optimizer.

Every bounded sequence $(u_n)$ in $H^1({\mathbb{R}}^d)$ has, after passage to a subsequence, a decompositionwhere $|x_n^j-x_n^k|\to \infty$ for $j\ne k$, the $L^2$ and gradient norms decouple asymptotically, andfor every $2<p<2^{\ast }$.

Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

The trapped focusing cubic ground-state equation in two dimensions isA nonzero positive solution follows from minimizing the harmonic-oscillator quadratic form subject to a fixed $L^4$ norm and then rescaling by the positive Lagrange multiplier.

If $P_{\eta }$ solves the trapped focusing cubic ground-state equation, the ansatzsolves the two-dimensional mass-critical focusing nonlinear Schrödinger equation when

The initial conditions $\lambda (-1)=\sqrt{1+\eta }$ and $b(-1)=1$ selectThe scale grows linearly at infinity, yielding a finite critical spacetime norm for the corresponding lens solution.

Young's inequality for products states that for conjugate exponents $q,q^{\prime }>1$ and nonnegative $a,b$,Its weighted form absorbs a lower power into a coercive higher power at the cost of a constant.

For a nonnegative potential $V(x)\to \infty$ as $|x|\to \infty$, the confining-potential energy space isA local Poincare inequality controls the missing $L^2$ term, so this energy norm makes ${\Sigma }_V$ a Hilbert space.

The embedding ${\Sigma }_V\hookrightarrow L^2({\mathbb{R}}^d)$ is compact. The Rellich-Kondrachov compactness theorem gives compactness on each fixed ball, whilemakes the tails uniformly small.

In two dimensions, the quadratic functionalattains its minimum subject to $\Vert u{\Vert }_4^4=M>0$. A bounded minimizing sequence is weakly compact in the harmonic-oscillator energy space and strongly compact in $L^4$: local Rellich-Kondrachov compactness theorem and the weighted tail bound prevent escape to infinity.

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.

The two-dimensional defocusing cubic nonlinear Schrödinger equation is mass-critical: the scaling $u(t,x)\mapsto {\lambda }^{-1}u(t/{\lambda }^2,x/\lambda )$ preserves the $L^2$ norm. The defocusing sign makes its nonlinear energy positive.

The harmonic-oscillator energy space isThe spatial moment prevents mass from escaping to infinity.

The embedding $\Sigma \hookrightarrow L^2({\mathbb{R}}^d)$ is compact. Rellich compactness controls a fixed ball, whilecontrols the tail uniformly.

Critical points of the trapped defocusing cubic energysatisfy $-\Delta v+|x{|}^{2}v-\omega v+v^3=0$.