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Choose a test function $\chi$ that equals one on the unit ball and vanishes outside the ball of radius two. Sinceintegration by parts givesThe first term on the right is supported where $1\le |x|\le 2$, where $u^2\le |x{|}^{\alpha }{u}^2$. The Cauchy-Schwarz inequality and Young inequality bound the second term byAfter absorbing the first integral,

This estimate also proves completeness. Indeed, a Cauchy sequence in $\Sigma$ is Cauchy in $H^1$ locally, while its gradients and the functions $|x{|}^{\alpha /2}{u}_n$ converge in $L^2$. The limits agree locally with a function $u$, so $u\in \Sigma$ and $u_n\to u$ in the energy norm. Thus $\Sigma$ is a Hilbert space; it is the confining-potential energy space for $V(x)=|x{|}^{\alpha }$.

WriteDifferentiating $\log J(u+th)$ at $t=0$ for a real test function $h$ givesAfter integration by parts,Rescaling the dependent and independent variables reduces this Euler-Lagrange equation to the ground-state equation for $Q$. The uniqueness of its positive radial solution and the equality cases in rearrangement show that all minimizers areThis is the classification of Weinstein-functional minimizers.

For real $h\in \mathcal{D}({\mathbb{R}}^d)$, differentiation under the integral givesThe ground-state equation $\Delta Q-Q+Q^{1+4/d}=0$ and integration by parts reduce this to

In particular, along the amplitude direction $h=Q$ the derivative is $-\Vert Q{\Vert }_2^2<0$. Hence $u_0=(1+\delta )Q$ has negative energy for every sufficiently small $\delta >0$, whilewhen $\delta$ is chosen small enough. The ground state has finite variance, so negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation shows that the corresponding solution blows up in finite time.

Write $e^x=1+xg(x)$, where $g=(e^x-1)/x$ is smooth at zero. Thenand a uniformly integrable expansion givesBy integration by parts,Therefore

Choose the Sobolev space $V$ encoding the homogeneous essential boundary conditions, setafter the appropriate integration by parts, and define the energy functionalIts first variation is $J^{\prime }(u)v=a(u,v)-\ell (v)$, so its stationary points are exactly the solutions of the weak formulationIf $a$ is bounded, symmetric, and coercive and $\ell \in V^{\prime }$, the Lax-Milgram theorem supplies a unique weak solution $u$. Moreover, for every $w\in V$,unless $w=0$. Thus $J$ is strictly convex, and $u$ is its unique global minimizer. This proves existence and uniqueness of the minimizer and of the weak solution simultaneously.

Let $\Omega =(-1,1{)}^2$ and use the clamped energy space $V=H_0^2(\Omega )$, whose traces satisfy $u={\partial }_{n}u=0$ on $\partial \Omega$. Two applications of integration by parts giveIf equality holds, then $\Delta u=0$. The maximum principle for harmonic functions and the zero Dirichlet boundary condition imply $u=0$. The biharmonic operator is therefore positive definite. The same identity and conclusion hold for the simply supported conditions $u=\Delta u=0$; either standard interpretation of the paper's phrase “zero boundary conditions” gives the result.

For an ultrarelativistic particle, $\epsilon =pc$. Including the two spin states of an electron, counting wavevectors in a sphere gives the density of statesThe grand canonical partition function of the ideal Fermi gas is thereforeSince the grand potential satisfies $pV=k_{B}T\log \mathcal{Z}$, integration by parts givesThusThis is the equation of state of an ideal ultrarelativistic gas.

Take a variation $y+\epsilon \eta$, where the differentiable function $\eta$ obeysbecause both $y$ and its first derivative have fixed endpoint values. The first variation of the functional isApplying integration by parts once to the second term and twice to the third givesThe endpoint conditions on $\eta$ and ${\eta }^{\prime }$ make every boundary term zero. Since the remaining integral vanishes for every admissible variation, the fundamental lemma of the calculus of variations yields the higher-order Euler-Lagrange equation

The magnetic field is $B=\nabla \times A$. The magnetostatic Ampère-Maxwell equation is $\nabla \times B={\mu }_{0}J$, so the curl of the curl identity givesA gauge transformation $A\mapsto A+\nabla \chi$ does not change $B$. We may therefore choose the Coulomb gauge $\nabla \cdot A=0$ by solving the Poisson equation ${\nabla }^2\chi =-\nabla \cdot A$. It follows component by component in a Cartesian coordinate system thatThe free-space Green function of the Laplacian then givesFor a localized steady current density, conservation of electric charge gives ${\nabla }^{\prime }\cdot J=0$; integration by parts confirms directly that this integral has zero divergence and hence obeys the Coulomb gauge.

A linear operator $\alpha$ on a real inner-product space is self-adjoint whenfor all $v,w$.

To prove the finite-dimensional spectral theorem, first note that the continuous quadratic function $⟨\alpha v,v⟩$ has a maximum on the unit sphere. The Lagrange multiplier equation at a maximizing vector $v$ gives $\alpha v=\lambda v$, so $\alpha$ has a real unit eigenvector. Its orthogonal complement is invariant becauseInduction on the dimension supplies an orthonormal eigenbasis of that complement and hence of $V$.

For $f\ne 0$,while symmetry and bilinearity are immediate, so the displayed formula defines an inner product on $P_n$. Sinceintegration by parts givesThe boundary term vanishes for polynomials, proving that $\alpha$ is self-adjoint.

On the monomial $x^k$,Thus the matrix of $\alpha$ in the monomial basis is triangular with diagonalThese are therefore its eigenvalues. For $n=2$, corresponding eigenvectors arefor eigenvalues $0,-1,-2$, respectively. They are the first Laguerre polynomials up to normalization, as described by the Laguerre differential operator on polynomials.

Let $u\mapsto u+\epsilon \eta$ and $v\mapsto v+\epsilon \xi$, where the variations vanish on the boundary. The first variation isApplying integration by parts to the four derivative terms and discarding the boundary contributions givesThe variations $\eta$ and $\xi$ are independent and arbitrary in the interior. The fundamental lemma of the calculus of variations therefore gives the two Euler-Lagrange equations for two fields

The zero-boundary Sobolev space isOn it defineIf this quadratic form vanishes, then $\nabla u=0$. The Poincaré inequality givesso $u=0$. It is therefore an inner product, and its norm is equivalent to the usual $H^1$ norm.

A function $u\in H_0^1(\Omega )$ is a weak solution whenThis follows from integration by parts and incorporates the homogeneous Dirichlet boundary condition through membership in $H_0^1$.

For a compactly supported variation $v+t\varphi$, differentiation of the area functional at $t=0$ gives the first variationAfter integration by parts, this is the minimal surface equation for a graphFor $v_R(x)=R^{-1}v(Rx)$ one has $D{v}_R(x)=Dv(Rx)$. The equation is invariant under this scaling, so $v_R$ solves it on $B_1$ for every $R>0$.

For a smooth compactly supported $u$, expand the nonnegative squareSince $\nabla \cdot (x/|x{|}^2)=|x{|}^{-2}$ in three dimensions, integration by parts turns the cross term into $-\frac{1}{2}\int |u{|}^2/|x{|}^2$. HenceDensity extends this Hardy inequality in Euclidean space to $H^1({\mathbb{R}}^3)$.

Differentiate the energy and use the chain rule:After integration by parts, the middle term is $-\int (\Delta u)u_t$. Thereforeby the defocusing semilinear wave equation. Thus the total energy is conserved.

At late time $f(k,{\tau }_0)\to H/\sqrt{2k^3}$ and $a=-1/(H\tau )$. Define the elementary symmetric polynomialsThen part ii reduces towhere a prime removes the momentum-conserving Dirac delta distribution andExpanding the product givesRepeated integration by parts reduces every negative power to the supplied logarithmic integral. The power divergences are real and disappear when the imaginary part is taken. Writingone obtainsConsequently the late-time parity-odd primordial trispectrum isIts factor of $i$ is required by reality of a momentum-space scalar correlator: reversing all momenta complex-conjugates the correlator, while the scalar triple product changes sign.

The background spatial metric is $g_{ij}=a^2{\delta }_{ij}$. For the large spatial diffeomorphismhomogeneity and isotropy imply that the purely spatial background Christoffel symbols vanish. HenceThe shift is constant, transverse and traceless, and is therefore the zero-momentum adiabatic tensor mode.

A scalar transforms by its Lie derivative:Fourier transformation and integration by parts in momentum space giveThe term proportional to ${\omega }_i{}^i$ vanishes because the deformation is traceless.

Vertical hydrostatic equilibrium givesDefine vertically integrated pressure and the density-weighted disk scale height byMultiply the hydrostatic equation by $z$ and integrate. Since $zp\to 0$ at both boundaries, integration by parts givesTherefore

Write the Rayleigh equation for inviscid shear flow asMultiply by ${\varphi }^{\ast }$, integrate, and use integration by parts:Its imaginary part isFor an unstable mode $c_i>0$, the integral can vanish only if $U^{\prime \prime }$ changes sign somewhere in the flow. Thus the velocity profile must have an inflection point. This is Rayleigh's inflection-point theorem; it is necessary, not sufficient, for inviscid instability.

The chain rule andgiveApplying the same rule to $L_1$ and $L_2$ in the Euler-Lagrange field equation yields the exact modulated equation

Multiply this equation by ${\Phi }_{\theta }$. Since $k$ and $\omega$ are independent of the fast phase,The product rule then rearranges the field equation into the exact modulated-wave first integralAverage this identity over one $2\pi$ period in $\theta$. Periodicity kills the first term, while differentiation of the averaged Lagrangian givesConsequently

The same two equations follow directly from the modulated variational principle. Varying $\Phi$ giveswhich is the exact field equation above. For a variation of $\Theta$, use $\delta k={\partial }_X\delta \Theta$ and $\delta \omega =-{\partial }_T\delta \Theta$. Integration by parts in $X$ and $T$ gives the averaged equation. Thus variation with respect to the periodic profile reproduces the local wave equation, whereas variation with respect to its slow phase gives the Whitham modulation equation for wave action.