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The backward heat equation $u_t=-D{u}_{xx}$, $D>0$, amplifies a spatial Fourier mode by $e^{D{k}^{2}t}$. On a periodic domain, fix $T>0$ and take initial data $f_n=e^{-D{n}^{2}T/2}\sin (nx)$. Their L2 norms tend to zero, while the corresponding solutions at $T$ have L2 norms growing without bound. Thus the solution map is not continuous in the ordinary L2 norm; this is an ill-posed problem.^[ref]

Real analyticity alone does not resolve the issue. The periodic analytic function $f(x)={\sum }_{n\ge 1}{e}^{-n}\cos (nx)$ has exponentially decaying coefficients, but after any positive time the proposed coefficients $e^{D{n}^{2}t-n}$ are not square summable. The required data restriction must control growth on the Gaussian frequency scale. For example, a fixed-horizon normgives $\Vert u(T){\Vert }_2=\Vert f{\Vert }_{X_T}$ when the Fourier basis is normalized. This very strong data norm changes the stability question. In inverse heat reconstruction, an explicit regularization or frequency cutoff is needed when measured data carry ordinary small-scale noise.

Let $B$ be a Banach space and let $f$ be a strongly measurable function on a measure space. It is Bochner integrable exactly whenStrong measurability means almost-everywhere norm approximation by simple vector-valued functions. The Bochner integral is the norm limit of their integrals when the approximation also tends to zero in the integral norm; the limit is independent of the approximating sequence.^[ref]

For $f(t)=(t,t^2)$ on $[0,1]$ with values in ${\mathbb{R}}^2$, the integral is $(1/2,1/3)$. In an infinite-dimensional Banach space, norm convergence is part of the construction and cannot be replaced by coordinatewise convergence alone. The boundfollows from the triangle inequality for simple functions and passage to the norm limit. A bounded linear operator commutes with the integral by the same argument. Strong measurability alone does not imply integrability: the scalar example $f(t)=1/t$ on $(0,1)$ fails the norm-integral condition.

The direct method in the calculus of variations proves existence by passing to a limit of a minimizing sequence. A useful abstract version takes a nonempty weakly sequentially closed admissible subset $\mathcal{A}$ of a reflexive Banach space $X$ and a functional $E:X\to \mathbb{R}\cup \{+\infty \}$. Assume $E$ is finite at some admissible point, bounded below, coercive on $\mathcal{A}$, and has weak lower semicontinuity.

Choose $u_n\in \mathcal{A}$ with $E(u_n)\to m={\inf }_{\mathcal{A}}E$. Coercivity bounds $\Vert u_n{\Vert }_X$. Reflexivity supplies a weakly convergent subsequence, and closure puts its limit $u$ in $\mathcal{A}$. Finally,so $u$ attains the infimum. Each hypothesis has a distinct role: compactness must preserve the constraints, and lower semicontinuity must hold for the convergence actually obtained.

For the Dirichlet energy with linear forcing on $H_0^1(\Omega )$ over a bounded domain, the Poincaré inequality controls the forcing and gives coercivity, while the squared gradient norm is weakly lower semicontinuous. The resulting minimizer solves the weak Poisson equation by its first variation.^[ref] Strict convexity gives uniqueness when the admissible class is convex. The oscillating minimizing sequence with no minimizer shows what can fail without suitable lower semicontinuity.

Distribution theory extends ordinary functions by letting them act on smooth test functions. A locally integrable function $f$ acts by $⟨f,\varphi ⟩=\int f\varphi$, while the Dirac delta distribution acts by $⟨{\delta }_a,\varphi ⟩=\varphi (a)$. The latter is a point source, rather than an ordinary function with a very large value. The space of test functions and its topology make distributions continuous linear functionals.^[ref]

Integration by parts supplies a distributional derivative even when ordinary differentiability fails:For the Heaviside function $H$, $⟨H^{\prime },\varphi ⟩=-{\int }_0^{\infty }{\varphi }^{\prime }(x)dx=\varphi (0)$, so $H^{\prime }={\delta }_0$. A jump has become a point source under differentiation. This is useful for impulses in differential equations, concentrated loads and interface jump conditions.

The extension has limits: multiplying a distribution by a smooth function is always defined, but arbitrary products of distributions are not. The theory controls which operations are meaningful through their action on test functions, rather than by manipulating symbolic point values.

An elliptic boundary value problem couples an elliptic differential equation in a domain to specified boundary conditions. An important example is the Poisson equationHere the Dirichlet boundary data fix the boundary value. Prescribing outward flux instead gives a Neumann boundary-value problem, which has different compatibility and normalization requirements. Thus the differential expression alone does not specify the problem.

For zero Dirichlet boundary conditions, multiply the Poisson equation by a smooth compactly supported test function and apply integration by parts:This weak formulation requires only first weak derivatives, so it makes sense for $u\in H_0^1(\Omega )$ and $f\in L^2(\Omega )$. On a bounded domain, the Poincaré inequality makes its bilinear form coercive, and the Lax-Milgram theorem gives a unique weak solution. The right side also extends to suitable forcing in the continuous dual space of $H_0^1(\Omega )$.^[ref]

The energy method explains the value of this formulation: subtracting two solutions and testing with their difference proves uniqueness. Elliptic regularity asks whether additional smoothness of coefficients, forcing and boundary upgrades the weak solution to a classical one. A Schauder estimate quantifies this gain in Hölder spaces; rough coefficients instead lead to results such as the De Giorgi-Nash-Moser theorem.

Suppose $u\in L_{\mathrm{loc}}^2$ solvesas a distribution, with $b,c$ and $f$ smooth on the open domain. The divergence-forcing interior H1 estimate first gives $u\in H_{\mathrm{loc}}^1$. If $u\in H_{\mathrm{loc}}^m$, smooth multiplication puts $\mathrm{div}(bu)+cu$ in $H_{\mathrm{loc}}^{m-1}$, and the smooth forcing belongs there too. Interior Poisson regularity then yields $u\in H_{\mathrm{loc}}^{m+1}$. Iteration and the Sobolev embedding theorem give a smooth representative.^[ref]

The regularity of $f$ is part of this argument. With only $f\in H_{\mathrm{loc}}^r$ for a nonnegative integer $r$, the same process gives $u\in H_{\mathrm{loc}}^{r+2}$, rather than unlimited smoothness. Smooth coefficients cannot remove a rough forcing term from the right-hand side.

For $u,v\in C^2(\overline{\Omega })$ on a bounded domain with a smooth boundary and outward unit normal $n$, Green's second identity statesThe product rule gives $\mathrm{div}(u\nabla v-v\nabla u)=u\Delta v-v\Delta u$; the divergence theorem then proves the identity. This is a bilinear identity with no complex conjugation. To compare complex inner products, substitute $\overline{u}$ for $u$.^[ref]

When both functions are harmonic functions, the boundary integral is zero. With homogeneous matching Dirichlet boundary conditions or Neumann boundary conditions, it also supplies the symmetry of the Laplace operator. Taking one function to be a singular fundamental solution, and removing a small ball before taking a limit, yields Green's third identity.

Every $C^2$ harmonic function on an open set in ${\mathbb{R}}^n$ is infinitely differentiable, in every dimension. Let ${\rho }_{\epsilon }$ be a smooth radial mollifier supported in a radius-$\epsilon$ ball, with integral1. Radially integrating the mean value property for harmonic functions giveswherever that ball stays inside the domain. The left side is smooth, so $u$ is smooth locally.^[ref]

In two dimensions there is also a complex analysis proof: locally $u$ is the real part of a holomorphic function, whose power series can be differentiated to every order. That proof is planar; it is not a justification for arbitrary $n$. More generally, the Weyl lemma supplies a smooth representative even when the initial equation $\Delta u=0$ holds only in distributions.

For $D>0$ and $t>0$, the Euclidean heat kernel for $u_t=D\Delta u$ on ${\mathbb{R}}^d$ isIt is the fundamental solution from a unit Dirac delta distribution. Its integral is one, its coordinate variances are $2Dt$, and $K_t\ast K_s=K_{t+s}$. For suitable initial data, $u(\cdot ,t)=K_t\ast u_0$ solves the heat equation. The approximate identity property recovers the initial data in Lp norm for $1\le p<\infty$, and locally uniformly for bounded continuous data.^[ref]

The formula quantifies spreading: the width grows like $\sqrt{Dt}$ while the peak falls like $t^{-d/2}$. Its strict positivity yields infinite propagation speed for nonzero nonnegative initial data. On an interval, boundary conditions require a different kernel, such as the Dirichlet heat kernel on an interval or Neumann heat kernel on an interval; their mass behavior records whether the boundary absorbs or reflects. The Gaussian heat kernel for standard Brownian motion uses $D=1/2$, which explains its different factor of two.

A Sobolev space measures a function together with specified weak derivatives in Lp spaces. For an open set $U\subseteq {\mathbb{R}}^n$, an integer $k\ge 0$ and $1\le p\le \infty$, defineThe derivatives are understood through integration by parts against test functions, rather than by requiring classical differentiability at every point. The Sobolev norm makes this a Banach space; for $p=2$, write $H^k(U)=W^{k,2}(U)$, a Hilbert space.^[ref]

For example, $u(x)=|x|$ on $(-1,1)$ belongs to $H^1$ even though it is not differentiable at zero: its weak derivative is $\mathrm{sgn}(x)$ and both functions are square-integrable. A step with a jump at zero does not belong to $H^1(-1,1)$, because its distributional derivative is a Dirac delta, which is not an L2 function. This distinction admits corners while excluding jumps at first-order square-integrable regularity.

The Fourier transform extends the square-integrable scale to every real $s$. With a unitary Fourier convention, $H^s({\mathbb{R}}^n)$ consists of tempered distributions satisfyingFor integer $s\ge 0$, this norm is equivalent to the weak-derivative Sobolev norm. Such spaces are natural for weak formulations of partial differential equations: a finite gradient energy can exist even when the prospective solution lacks classical second derivatives.

For $u\in C^2(\Omega )$, subharmonic means $\Delta u\ge 0$, using the Laplacian $\Delta ={\sum }_j{\partial }_{x_j}^2$. The strict inequality $\Delta u>0$ defines strict subharmonicity and is stronger. Every harmonic function is therefore subharmonic, even though its Laplacian vanishes.^[ref]

The general definition allows an upper-semicontinuous function $u:\Omega \to [-\infty ,\infty )$, not identically $-\infty$ on any connected component, satisfying the ball mean inequalityFor locally integrable functions, distributional nonnegativity of $\Delta u$ produces this canonical representative. An arbitrary change at one point need not preserve the mean inequality, which is why the representative matters.

The example $u(x)=|x{|}^2$ has $\Delta u=2n>0$, while $|z{|}^p$ is subharmonic for every $p>0$. More generally, $|f(z){|}^p$ is subharmonic for a holomorphic function $f$ and $p>0$. At a strict interior maximum of a twice differentiable function, the Hessian matrix is negative semidefinite; the maximum principle for subharmonic functions extends this elementary obstruction to the non-strict and nonsmooth setting.

For a real second-order principal coefficient matrix $A(x)=(a^{ij}(x))$, uniform ellipticity means that one constant $\lambda >0$, independent of $x$, satisfiesFor measurable coefficients the condition is imposed almost everywhere; for a classical pointwise operator it holds at every relevant point. The quadratic form depends only on the symmetric part of $A$. Many estimates also assume a finite uniform upper bound or bounded full coefficients; the lower bound alone does not supply them.

For example, $A=\mathrm{diag}(1,4)$ has lower bound one. On $0<x_1<1$, the matrix $A(x)=\mathrm{diag}(x_1^2,1)$ is positive definite at each point but has no positive uniform lower bound: test $\xi =(1,0)$ as $x_1\downarrow 0$. On the restricted strip $x_1>\delta >0$, a bound $\min ({\delta }^2,1)$ works. This distinguishes a strictly elliptic operator from a uniformly elliptic operator on the whole domain.

In a divergence-form elliptic operator, the same inequality controls the principal gradient energy. Together with bounded coefficients and the Poincaré inequality, it underlies coercivity and weak existence. A nonsymmetric coefficient matrix needs separate boundedness control, as shown by quadratic ellipticity does not bound a nonsymmetric coefficient matrix.^[ref]

For a real semilinear wave equation $u_{tt}-\Delta u+V^{\prime }(u)=0$, its energy density and energy flux areMultiply the equation by $u_t$ and use the product rule to obtain the local balance ${\partial }_{t}e+\mathrm{div}j=0$. On a fixed regular domain $\Omega$ this givesThus total wave energy is conserved when the boundary term vanishes: for example homogeneous Dirichlet boundary conditions, homogeneous Neumann boundary conditions, paired periodic boundary conditions, or suitable decay on all of space. A driven boundary can instead inject energy.^[ref]

For the linear wave equation, take $V=0$ and obtain $E_{\mathrm{lin}}=\frac{1}{2}\int (u_t^2+|\nabla u{|}^2)dx$. A defocusing potential such as $V(u)=|u{|}^{p+1}/(p+1)$ adds a nonnegative term. A focusing negative potential still obeys the balance law but need not control positive norms; conservation and positivity are separate properties.