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Here Morrey's inequality has Hölder exponent $1-3/6=1/2$. Every $u\in W^{1,6}({\mathbb{R}}^3)$ has a unique continuous representative $u^{\ast }$ such thatWe use the Hölder norm $\Vert f{\Vert }_{C^{0,1/2}}=\Vert f{\Vert }_{\infty }+[f{]}_{C^{0,1/2}}$, where the Hölder seminorm is ${\sup }_{x\ne y}|f(x)-f(y)|/|x-y{|}^{1/2}$. The second bound is the homogeneous form of Morrey's inequality; the first includes control of the values themselves.

Let $u_j={\rho }_{1/j}\ast u$ be the mollifications from (i). They converge to $u$ in the Sobolev norm. Apply the assumed smooth version of Morrey's inequality to the differences $u_j-u_{\ell }$:Thus $(u_j)$ is a Cauchy sequence in the complete Hölder space $C^{0,1/2}({\mathbb{R}}^3)$ and converges there to a function $w$. In particular the convergence is uniform. On every bounded set $K$ of finite volume,Exhausting ${\mathbb{R}}^3$ by balls proves $w=u$ almost everywhere. Set $u^{\ast }=w$. Taking limits in the smooth Hölder norm bound gives the first boxed estimate. For any fixed $x\ne y$, taking limits ingives the second boxed estimate after taking the supremum. Finally, two continuous representatives equal almost everywhere must coincide everywhere, since a nonzero continuous difference would remain nonzero on an open ball of positive volume.

For $\psi \in C_c^{\infty }(B_1)$, the functions $\psi$ and ${\partial }_i\psi$ are bounded. Hence $\psi u$ and $\psi {\partial }_{i}u+u{\partial }_i\psi$ belong to $L^p(B_1)$. For a test function $\varphi$, the classical product rule and the weak derivative identity for $u$, tested with $\psi \varphi$, giveThereforeIn fact $\psi u$ has compact support in $B_1$ and belongs to the corresponding zero-boundary Sobolev space, by mollification within that support.

For arbitrary $\psi \in C^{\infty }(B_1)$, the same identity still proves the local product rule for weak derivatives, but membership in the global Sobolev space can fail. To give a counterexample for every dimension and every $p$ in the question, takeHere $u\in W^{1,p}(B_1)$ and $\psi$ is a smooth function on the open ball, while polar coordinates showThus $\psi u\notin L^p(B_1)$, and the global assertion is false. Boundedness of $\psi$ and its first derivatives would suffice without compact support.

There is a small incompatibility in the printed cutoff function assumptions. No $\phi \in C_c^{\infty }(B_1)$ can equal one on $B_{1/2}$ and satisfy $|\nabla \phi |\le 2$: for a unit vector $e$, choose $t<1$ beyond its support. Continuity gives $\phi (e/2)=1$, while the fundamental theorem of calculus would give $1\le 2(t-1/2)<1$. Replace the number $2$ by a fixed sufficiently large absolute constant $C_0$; such a cutoff function exists, and the requested Caccioppoli inequality is unchanged apart from its constant. We keep $M=\Vert \nabla \phi {\Vert }_{\infty }$ in the calculation, which also yields the formal bound under the printed assumptions.

The weak formulation is ${\int }_{B_1}A\nabla u\cdot \nabla \zeta =0$ for test functions $\zeta$. Since $A$ is bounded, this extends by density of smooth functions in a Sobolev space to every $\zeta \in H_0^1(B_1)$. Part (a)(ii), followed by mollification inside the compact support, makes ${\phi }^2(u-k)$ an admissible choice. The product rule givesPut $X=\int {\phi }^2|\nabla u{|}^2$ and $Y=\int (u-k{)}^2|\nabla \phi {|}^2$. The lower and upper uniform ellipticity bounds, together with Cauchy-Schwarz inequality, implyIf $X=0$ the conclusion is immediate. Otherwise divide by $X^{1/2}$ and square to obtain $X\le 4{\Lambda }^2{\lambda }^{-2}Y$. Since $\phi =1$ on $B_{1/2}$,For a fixed valid cutoff function, $M\le C_0$, this has the required constant independent of $k$. Formally inserting the printed value $M=2$ gives $16{\Lambda }^2/{\lambda }^2$. No derivative of $A$ was taken, so differentiability of the coefficients is unnecessary.

For $u\in C_c^{\infty }({\mathbb{R}}^d)$, the fundamental theorem of calculus along the $j$th coordinate line givesTake $f=|u|$ and $s_j=|{\partial }_{j}u|$ in the introductory result. It follows thatThe last step is the arithmetic-geometric mean inequality. This bounds the target norm by the first-order Sobolev space norm. Approximate any $u\in W^{1,1}({\mathbb{R}}^d)=L_1^1({\mathbb{R}}^d)$ by compactly supported smooth functions, using cutoffs and mollification as in Question 4. The estimate applied to differences gives convergence in $L^{d/(d-1)}$, and subsequences of the simultaneous $L^1$ and target-norm convergence identify the two limits almost everywhere. Thus $W^{1,1}$ embeds continuously into $L^{d/(d-1)}$. The coordinate-derivative product bound also passes to the limit. For $d=1$, the line argument instead gives $\Vert u{\Vert }_{\infty }\le \Vert u^{\prime }{\Vert }_1$ and the embedding into $L^{\infty }$.

For smooth $u$, the fundamental theorem along the segment in direction $e_k$ gives, for either sign of $h$,The entire segment lies in $\Omega$ under the distance hypothesis. The Cauchy-Schwarz inequality on $[0,1]$, followed by integration and translation, yieldsFor general $u\in W^{1,2}(\Omega )$, extend $u$ and its weak gradient separately by zero as $L^2$ functions on ${\mathbb{R}}^n$, and mollify $u$ with radius $\epsilon$. At points farther than $\epsilon$ from the boundary the mollified gradient equals the convolution of the zero-extended gradient, because the convolution kernel is supported wholly inside $\Omega$. Choose $\epsilon <\mathrm{dist}({\Omega }^{\prime },\partial \Omega )-|h|$. All the segments then lie in this region, and the preceding calculation, followed by the $L^2$ contraction bound for convolution, gives the same right side $\Vert D_{k}u{\Vert }_{L^2(\Omega )}$. The mollifications converge locally in $L^2$, as do their fixed-$h$ translated differences, so passage to the limit preserves the estimate. HenceThe argument proves the sharper estimate with $D_{k}u$ on the right. It is the basic Sobolev difference quotient bound.

No additional hypothesis on $F$ is needed here for scalar interior $C^{1,\beta }$ regularity. In the elliptic sense of uniform convexity, the assumptions already give constants $0<\lambda \le M<\infty$ with $\lambda I\le D^{2}F(p)\le MI$ for all $p$. The following argument gives interior gradient regularity for uniformly convex autonomous energies.

The first variation at the minimizer exists because $|DF(p)|\le C(1+|p|)$, and its weak Euler-Lagrange equation isFor a coordinate difference quotient ${\delta }_{h}u$, subtract the translated equation and the original equation. The averaged linearization of a nonlinear divergence-form equation givesThese measurable symmetric matrices satisfy $\lambda I\le A_h\le MI$ independently of $h$. The Caccioppoli inequality and the standard bound $\Vert {\delta }_{h}u{\Vert }_2\le \Vert D_{k}u{\Vert }_2$ bound $D{\delta }_{h}u$ on every smaller interior set. Quoting the Sobolev characterization by bounded difference quotients yields $u\in H_{\mathrm{loc}}^2$.

Now $DF$ is globally Lipschitz continuous, so the Sobolev chain rule and commutation of distributional derivatives allow differentiation of the weak equation. Each $v_k=D_{k}u\in H_{\mathrm{loc}}^1$ satisfiesThe De Giorgi-Nash-Moser theorem states that weak solutions of this scalar divergence-form equation with bounded measurable uniformly elliptic coefficients are locally Hölder continuous. It gives a common $\beta \in (0,1)$ depending only on $n$ and the ellipticity ratio $M/\lambda$, with $D_{k}u\in C_{\mathrm{loc}}^{0,\beta }$ for every $k$. A Sobolev function whose weak gradient is continuous has a $C^1$ representative, as follows by local mollification and integration of its gradient. HenceThe higher-regularity conclusion comes from differentiating an autonomous scalar equation; it does not rely on boundary regularity or a higher derivative such as $D^{3}F$.

For the Hurwitz proof of the planar isoperimetric inequality, take a positively oriented regular simple closed curve of length $L$ and enclosed area $A$. Write its complex position as $z(t)=x(t)+iy(t)$, with $0\le t\le 2\pi$ proportional to arc length. Then $|z^{\prime }(t)|=L/(2\pi )$. Translate the curve to make its mean position zero, and write its Fourier coefficients as $c_n$, with $c_0=0$.

Green's theorem gives the signed area, and Parseval's identity computes it:Periodic integration by parts and Parseval's identity applied to $z^{\prime }$ giveSince $n\le n^2$ for every integer $n$,The sums converge absolutely: $\sum |n||c_n{|}^2\le (\sum |c_n{|}^2{)}^{1/2}(\sum n^2|c_n{|}^2{)}^{1/2}$. Equality forces $c_n=0$ unless $n=0$ or $1$; after the mean translation, $z(t)=c_1{e}^{it}$ is a circle. Conversely a circle attains equality. Thus circles uniquely attain equality, up to translation and orientation.

The same proof applies to a rectifiable simple closed curve using its Lipschitz arc length parametrization. Its derivative exists almost everywhere and belongs to $L^2$; periodic mollification converges to the curve in the function and derivative $L^2$ norms. This justifies the derivative coefficient identity, Parseval's identity and area integral by approximation. Reversing orientation, if necessary, makes the enclosed area positive. The printed name “Hurewitz” is read as Hurwitz.

Use a bounded BV extension operator for the bounded Lipschitz domain. Extend $u_k$ to $w_k\in BV({\mathbb{R}}^n)$, supported in one fixed compact set, with $\Vert w_k{\Vert }_{BV({\mathbb{R}}^n)}\le C_{\Omega }\Vert u_k{\Vert }_{BV(\Omega )}$. Such an extension is obtained by reflection in Lipschitz boundary charts, a partition of unity and a fixed cutoff function. A naive zero extension must also charge its boundary-trace jump; it cannot discard that contribution.

Let $M$ bound these norms and let $w_{k,ϵ}=w_k\ast {\rho }_{ϵ}$ be their mollifications. The BV mollification error estimate givesThe variation measure here is on ${\mathbb{R}}^n$. The printed lemma's $|Dw|(\Omega )$ needs this correction unless it also assumes all the derivative mass lies in $\Omega$: a nonconstant bump supported outside $\overline{\Omega }$ disproves its literal wording. The correct estimate follows by averaging the BV translation estimate $\Vert w(\cdot -h)-w{\Vert }_1\le |h||Dw|({\mathbb{R}}^n)$ over a mollifier supported in $|h|\le ϵ$.

For each fixed $ϵ>0$, convolution gives uniform boundsThe mollifications have common compact support and are uniformly equicontinuous. The Arzelà-Ascoli theorem supplies a uniformly convergent subsequence at each scale $ϵ=1/m$. A diagonal subsequence converges at every one of those scales. For two late members of that subsequence,First choose large $m$, then late $k,\ell$; the sequence is a Cauchy sequence in $L^1$. Its limit $w$ belongs to the BV space because the total variation seminorm on a domain is sequentially lower semicontinuous. Uniform derivative bounds and integration by parts give $D{w}_k\stackrel{\ast }{\rightharpoonup }Dw$. Restriction to $\Omega$ proves the required weak-star convergence in BV.

For $1\le p<\infty$, an integer $k\ge 0$, and a normalized nonnegative mollifier $\rho \in C_c^{\infty }({\mathbb{R}}^n)$, put ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$. If $u\in W^{k,p}({\mathbb{R}}^n)$, its mollification is smooth and converges to $u$ in the Sobolev norm. The definition of a weak derivative and Fubini's theorem give ${\partial }^{\alpha }({\rho }_{\epsilon }\ast u)={\rho }_{\epsilon }\ast {\partial }^{\alpha }u$ for every multi-index with $|\alpha |\le k$. The approximate identity theorem in the Lp space $L^p$ applied to these finitely many weak derivatives proves the claim. On a domain with boundary this direct convolution argument applies only away from the boundary unless an extension is first supplied.

On a connected open set, a Sobolev space function with zero weak gradient is constant almost everywhere. Local mollification gives constant smooth representatives on interior balls; overlaps and connectedness identify their constants. Without connectedness, the constant may differ between components. This explains the constant ambiguity in a Neumann Poisson problem.

Strict convergence combines strong convergence in $L^1$ with convergence of the total variation seminorm. It implies weak-star convergence in BV but does not make jump sets stable: mollifications of an interior step can converge strictly to the step while every approximant has an empty jump set of a bounded-variation function.