02.17.01 · analysis / elliptic-regularity

Interior and Boundary H^2 Regularity of Weak Elliptic Solutions

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Anchor (Master): Evans §6.3; Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 2001), §8.3-§8.4, §8.8-§8.13; Nirenberg, Remarks on strongly elliptic partial differential equations (Comm. Pure Appl. Math. 1955); Agmon-Douglis-Nirenberg I (Comm. Pure Appl. Math. 1959)

Intuition Beginner

A weak solution of an elliptic equation is found by a clever evasion. Instead of demanding that a function obey a differentiation rule at every point, the weak formulation only asks that it pass an averaged balance test against every probe function. This is much easier to satisfy, and the sibling unit shows it always has a unique answer. But it leaves a worry hanging: the function we found might be jagged, barely possessing one derivative, while the equation we wanted to solve speaks of second derivatives. Did we really solve the problem, or only a watered-down shadow of it?

The answer, for elliptic equations with smooth ingredients, is reassuring. The weak solution is secretly far smoother than the bare definition guarantees. If the data feeding the equation is square-summable, the solution turns out to have two square-summable derivatives. If the data is smoother still, the solution is smoother still, and the gain is always exactly two derivatives over the data. Push this all the way and a solution driven by infinitely smooth data is itself infinitely smooth. The watered-down weak solution was the real thing in disguise.

Why should averaging buy back smoothness? The secret is that elliptic equations control curvature in every direction at once; there is no favored thin direction along which roughness can hide. A non-elliptic equation, say one that diffuses only sideways, would let a solution stay rough in the unwatched direction. Ellipticity watches everything, so any roughness the solution might carry is immediately seen by the equation and forbidden by the data being smooth.

The tool that makes this precise is the divided difference: compare the solution's value at a point with its value a tiny step away, and divide by the step size. As the step shrinks this becomes a derivative. The trick is to show these divided differences stay bounded no matter how small the step, because a function whose divided differences stay bounded already has the next derivative. The whole proof is a careful accounting that the divided differences of the gradient cannot blow up, which hands us the second derivative for free.

Visual Beginner

The single picture to hold is a divided difference closing in on a derivative, paired with the idea that a uniform bound on those divided differences is exactly the next derivative waiting to be claimed.

Read the panels left to right. The left panel is the basic move: a divided difference is a secant slope, and shrinking the spacing tilts the secant toward the tangent. This is the everyday picture of a derivative as a limit of divided differences.

The middle panel is the engine of the whole subject. We do not take the limit first; we instead show that the secant slopes of the gradient stay trapped inside a fixed band no matter how small the spacing. A function whose divided differences never escape such a band is exactly a function that already owns the next derivative, and that derivative inherits the size of the band. Trapping the divided differences of the gradient therefore promotes the gradient to having its own derivatives, which are the second derivatives of the solution.

The right panel is the payoff. A weak solution starts life looking as though it might be rough, but feeding it through an elliptic equation with smooth data smooths it, gaining exactly two derivatives over whatever smoothness the data carries.

Worked example Beginner

We watch divided differences in action on a function we already understand, to see why a uniform bound on them is the same as owning the next derivative. Work on the line and take the smooth function $u (x) = x^{2}$ .

Step 1. Form the divided difference. Pick a small step $h$ and compute the divided difference of $u$ : take the value a step to the right, subtract the value here, and divide by the step. This is the quantity $(u (x + h) - u (x)) / h$ , which for $u (x) = x^{2}$ equals $((x + h)^{2} - x^{2}) / h$ .

Step 2. Expand and simplify. The top is $(x + h)^{2} - x^{2} = 2 x h + h^{2}$ . Dividing by $h$ leaves $2 x + h$ . So the divided difference of $x^{2}$ at spacing $h$ is exactly $2 x + h$ .

Step 3. Read the bound. On the interval from $0$ to $1$ , for any step $h$ between $0$ and $1$ , the value $2 x + h$ stays between $0$ and $3$ . The divided differences are bounded by $3$ , and this bound does not depend on how small $h$ is. That uniform bound is the whole point.

Step 4. Take the limit. As the step shrinks to zero, $2 x + h$ approaches $2 x$ , which is the genuine derivative of $x^{2}$ . The bounded divided differences converged to an honest derivative, and the size of that derivative is no bigger than the bound we found.

Step 5. Run it backward. The logic also works in reverse, and that is the form used in the proof. Suppose all we knew was that the divided differences of some function stayed below a fixed number for every spacing. That alone forces the function to have a square-summable derivative bounded by the same number. We never need the formula; the uniform bound by itself delivers the next derivative.

What this tells us: a divided difference is a stand-in for a derivative that we can manipulate before the derivative is known to exist, and a bound on the divided differences that ignores the step size is exactly a certificate that the next derivative exists and is no larger. The regularity proof manufactures such a certificate for the gradient of the solution, and that certificate is the second derivative.

Check your understanding Beginner

Exercise (easy, multiple choice).

What does the regularity theorem of this unit promise about a weak solution of an elliptic equation when the data is square-summable and the coefficients are smooth?

A. The solution is bounded but need not have any derivatives. B. The solution has two square-summable derivatives, even though the weak definition only guaranteed one. C. The solution is exactly as rough as the data. D. The solution exists only on the boundary of the region.

Hint

The headline is that the solution gains derivatives: the weak definition asks for one, the theorem delivers two.

Answer

B. The solution has two square-summable derivatives, even though the weak definition only guaranteed one. Elliptic regularity says the weak solution is smoother than its definition requires: with square-summable data and smooth coefficients it gains a full second derivative, landing in the space of functions with two square-summable derivatives. Feedback-correct: the gain is two derivatives over the data. Feedback-wrong: A and C deny the smoothing entirely, and D confuses a statement about smoothness with one about location.

Exercise (easy, true-false).

If the divided differences of a function stay below a fixed bound for every step size, no matter how small, then the function has a derivative no larger than that bound.

Hint

Recall the middle panel: a uniform bound on divided differences is a certificate for the next derivative.

Answer

True. A bound on the divided differences that does not depend on the step size is exactly the condition that the next derivative exists and is controlled by the same bound. This reverse reading is the heart of the difference-quotient method: bound the divided differences first, then collect the derivative they certify. Feedback-correct: uniform-in-step bounds yield the derivative. Feedback-wrong: a bound that grew as the step shrank would not certify a derivative, but a uniform one does.

Formal definition Intermediate+

Throughout, $Ω \subseteq R^{n}$ is open and bounded, and $L$ is the divergence-form operator of 02.16.04, $Lu = - i, j = 1 \sum n \partial_{i} (a^{ij} (x) \partial_{j} u) + i = 1 \sum n b^{i} (x) \partial_{i} u + c (x) u,$ uniformly elliptic with ellipticity constant $θ > 0$ , meaning $\sum_{i, j} a^{ij} (x) ξ_{i} ξ_{j} \geq θ ∣ ξ ∣^{2}$ for a.e. $x$ and all $ξ \in R^{n}$ . The Sobolev spaces $H^{k} (Ω) = W^{k, 2} (Ω)$ , the local spaces $H_{loc}^{k}$ , and the inequalities used below are those of 24.01.01 and 02.16.01. Given $f \in L^{2} (Ω)$ , a function $u \in H^{1} (Ω)$ is a weak solution of $Lu = f$ if $B [u, v] = \int_{Ω} (i, j \sum a^{ij} \partial_{j} u \partial_{i} v + i \sum b^{i} (\partial_{i} u) v + c u v) d x = \int_{Ω} f v d x for all v \in H_{0}^{1} (Ω),$ the bilinear form $B$ being that of 02.16.04; existence and uniqueness in $H_{0}^{1} (Ω)$ are supplied by the Lax-Milgram theory of that unit.

Definition (difference quotient). For a function $w$ on $Ω$ , a coordinate direction $e_{k}$ , and a step $h \neq = 0$ , the $k$ -th difference quotient of size $h$ is $D_{k}^{h} w (x) = \frac{w ( x + h e _{k} ) - w ( x )}{h},$ defined for $x$ in any subdomain $Ω^{'} ⋐ Ω$ once $∣ h ∣ < dist (Ω^{'}, \partial Ω)$ . Write $D^{h} w = (D_{1}^{h} w, \dots, D_{n}^{h} w)$ . The discrete product rule reads $D_{k}^{h} (φ ψ) (x) = φ (x + h e_{k}) D_{k}^{h} ψ (x) + (D_{k}^{h} φ (x)) ψ (x)$ , and the discrete integration-by-parts (summation by parts) identity reads $\int_{Ω} φ D_{k}^{h} ψ d x = - \int_{Ω} (D_{k}^{- h} φ) ψ d x$ whenever one factor is supported away from $\partial Ω$ at scale $∣ h ∣$ .

Definition (the $H^{2}$ estimate). The operator $L$ satisfies the interior $H^{2}$ estimate if for every $Ω^{'} ⋐ Ω^{''} ⋐ Ω$ there is $C = C (Ω^{'}, Ω^{''}, θ, coefficients)$ with $∥ u ∥_{H^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2} (Ω^{''})} + ∥ u ∥_{L^{2} (Ω^{''})})$ for every weak solution $u$ of $Lu = f$ . The same inequality with $Ω^{'}$ replaced by $Ω$ and $Ω^{''}$ by $Ω$ , valid when $\partial Ω$ is $C^{2}$ and $u \in H_{0}^{1} (Ω)$ , is the boundary (global) $H^{2}$ estimate.

Counterexamples to common slips Intermediate+

Regularity is not free of hypotheses on the coefficients. The leading coefficients $a^{ij}$ must be at least Lipschitz (here $C^{1}$ or better) for $H^{2}$ regularity; merely bounded measurable $a^{ij}$ give Hölder regularity through De Giorgi-Nash-Moser 02.17.07 but not $H^{2}$ . The difference-quotient argument differentiates the equation once, and that costs one derivative of the coefficients.
The estimate is a priori, not a stand-alone bound. The inequality $∥ u ∥_{H^{2}} \leq C (∥ f ∥_{L^{2}} + ∥ u ∥_{L^{2}})$ presumes $u$ is already a weak solution. It does not assert that any $H^{1}$ function with small $∥ f ∥, ∥ u ∥$ is in $H^{2}$ ; the $∥ u ∥_{L^{2}}$ term on the right is genuinely needed and cannot be dropped (constants solve $Lu = c u$ and require the lower-order term).
Interior regularity says nothing at the boundary by itself. A weak solution can lie in $H_{loc}^{2} (Ω)$ yet fail to be in $H^{2} (Ω)$ if $\partial Ω$ has a corner. The square domain's solution of the Dirichlet problem can have an unbounded second derivative at a reentrant corner; up-to-the-boundary $H^{2}$ requires $\partial Ω \in C^{2}$ (or convexity), and this is why the boundary argument is separated from the interior one.
The gain is exactly two derivatives, not more. With $f \in L^{2}$ one gets $u \in H^{2}$ , not $H^{3}$ . To reach $H^{3}$ one needs $f \in H^{1}$ and $a^{ij} \in C^{2}$ ; demanding $u \in H^{3}$ from $f \in L^{2}$ alone is false, as the Newtonian potential of an $L^{2}$ density shows.

Key theorem with proof Intermediate+

Theorem (interior $H^{2}$ regularity; Nirenberg). Let $a^{ij} \in C^{1} (Ω)$ with $a^{ij} = a^{j i}$ , let $b^{i}, c \in L^{\infty} (Ω)$ , let $L$ be uniformly elliptic with constant $θ$ , and let $f \in L^{2} (Ω)$ . If $u \in H^{1} (Ω)$ is a weak solution of $Lu = f$ , then $u \in H_{loc}^{2} (Ω)$ , the equation $Lu = f$ holds almost everywhere, and for every $Ω^{'} ⋐ Ω$ $∥ u ∥_{H^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{L^{2} (Ω)}),$ with $C$ depending only on $θ$ , the $C^{1}$ norms of the $a^{ij}$ , the $L^{\infty}$ norms of $b^{i}, c$ , and $Ω^{'}, Ω$ ^{[Nirenberg 1955]} ^{[Evans 2010 §6.3]}.

Proof. It suffices to bound the tangential and mixed second derivatives by difference quotients and then recover the remaining second derivative from the equation. Fix $Ω^{'} ⋐ Ω^{''} ⋐ Ω$ and a cutoff $ζ \in C_{c}^{\infty} (Ω^{''})$ with $0 \leq ζ \leq 1$ and $ζ \equiv 1$ on $Ω^{'}$ . Work first with the principal part; the lower-order terms are absorbed at the end. Write the weak form $\int_{Ω} \sum_{i, j} a^{ij} \partial_{j} u \partial_{i} v d x = \int_{Ω} \tilde{f} v d x$ , where $\tilde{f} = f - \sum_{i} b^{i} \partial_{i} u - c u \in L^{2} (Ω^{''})$ collects the source and the lower-order terms (these lie in $L^{2}$ because $u \in H^{1}$ ).

Fix a direction $e_{k}$ and a step $h$ with $∣ h ∣ < \frac{1}{2} dist (Ω^{''}, \partial Ω)$ , and insert the test function $v = - D_{k}^{- h} (ζ^{2} D_{k}^{h} u) \in H_{0}^{1} (Ω),$ legitimate because the difference quotients of an $H^{1}$ function are again $H^{1}$ and $ζ$ localises the support. Using summation by parts and the discrete product rule, the left side becomes $\int_{Ω} \sum_{i, j} D_{k}^{h} (a^{ij} \partial_{j} u) \partial_{i} (ζ^{2} D_{k}^{h} u) d x$ . Expanding $D_{k}^{h} (a^{ij} \partial_{j} u) (x) = a^{ij} (x + h e_{k}) D_{k}^{h} (\partial_{j} u) (x) + (D_{k}^{h} a^{ij} (x)) \partial_{j} u (x)$ and writing $w = D_{k}^{h} u$ , the left side splits as $I \int_{Ω} i, j \sum a^{ij} (\cdot + h e_{k}) \partial_{j} w \partial_{i} (ζ^{2} w) d x + I I \int_{Ω} i, j \sum (D_{k}^{h} a^{ij}) (\partial_{j} u) \partial_{i} (ζ^{2} w) d x .$ For $I$ , expand $\partial_{i} (ζ^{2} w) = ζ^{2} \partial_{i} w + 2 ζ (\partial_{i} ζ) w$ and use uniform ellipticity of the shifted matrix $a^{ij} (\cdot + h e_{k})$ on the leading piece: $\int_{Ω} ζ^{2} i, j \sum a^{ij} (\cdot + h e_{k}) \partial_{j} w \partial_{i} w d x \geq θ \int_{Ω} ζ^{2} ∣ D w ∣^{2} d x .$ The cross term $\int 2 ζ \sum a^{ij} (\partial_{j} w) (\partial_{i} ζ) w$ is bounded, by Cauchy-Schwarz and Young's inequality $∣ ab ∣ \leq \frac{θ}{4} a^{2} + \frac{1}{θ} b^{2}$ , by $\frac{θ}{4} \int ζ^{2} ∣ D w ∣^{2} + C \int_{Ω^{''}} ∣ w ∣^{2}$ , where $C$ depends on $θ$ , $∥ a^{ij} ∥_{L^{\infty}}$ , $∥ D ζ ∥_{L^{\infty}}$ . For $I I$ , the factor $D_{k}^{h} a^{ij}$ is bounded in $L^{\infty}$ by $∥ a^{ij} ∥_{C^{1}}$ (mean value theorem), and $\partial_{j} u \in L^{2} (Ω^{''})$ , so $∣ I I ∣ \leq \frac{θ}{4} \int ζ^{2} ∣ D w ∣^{2} + C \int_{Ω^{''}} (∣ D u ∣^{2} + ∣ w ∣^{2})$ after the same Young's inequality. The right side $\int_{Ω} \tilde{f} v d x$ is estimated by $∥ \tilde{f} ∥_{L^{2} (Ω^{''})} ∥ v ∥_{L^{2}}$ ; since $∥ v ∥_{L^{2}} \leq ∥ D_{k}^{- h} (ζ^{2} D_{k}^{h} u) ∥_{L^{2}} \leq C ∥ D (ζ^{2} D_{k}^{h} u) ∥_{L^{2}} \leq \frac{θ}{4} ∥ ζ D w ∥_{L^{2}}^{2} / x + C (\dots)$ , one more Young's inequality gives $\int \tilde{f} v \leq \frac{θ}{4} \int ζ^{2} ∣ D w ∣^{2} + C (∥ \tilde{f} ∥_{L^{2} (Ω^{''})}^{2} + \int_{Ω^{''}} ∣ D u ∣^{2})$ .

Collecting the four estimates and absorbing the three $\frac{θ}{4} \int ζ^{2} ∣ D w ∣^{2}$ terms into the left-hand $θ \int ζ^{2} ∣ D w ∣^{2}$ leaves $\frac{θ}{4} \int_{Ω} ζ^{2} ∣ D D_{k}^{h} u ∣^{2} d x \leq C (∥ f ∥_{L^{2} (Ω)}^{2} + ∥ u ∥_{H^{1} (Ω^{''})}^{2}),$ with $C$ independent of $h$ . The right side is finite and independent of $h$ , so the family ${ζ D D_{k}^{h} u}_{h}$ is bounded in $L^{2}$ uniformly in $h$ . By the Nirenberg difference-quotient lemma (Proposition 1 below), $\partial_{k} D u \in L^{2} (Ω^{'})$ for every $k$ , that is $u \in H^{2} (Ω^{'})$ , with $∥ D^{2} u ∥_{L^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{H^{1} (Ω^{''})}) .$ Finally the $H^{1}$ norm on the right is itself controlled by $∥ f ∥_{L^{2}} + ∥ u ∥_{L^{2}}$ through the Caccioppoli energy inequality (testing the weak form against $ζ^{2} u$ and using ellipticity), giving the stated estimate. Because $u \in H_{loc}^{2}$ , one may integrate the weak form by parts in reverse to find $\int_{Ω} (Lu - f) v = 0$ for all $v \in C_{c}^{\infty}$ , whence $Lu = f$ a.e. $□$

Bridge. This interior estimate is exactly the difference-quotient method of the sibling unit run in reverse: the translation estimate $∥ τ_{h} u - u ∥_{L^{2}} \leq ∣ h ∣ ∥ D u ∥_{L^{2}}$ of 02.16.01, read forwards, says an $H^{1}$ function has bounded difference quotients, and the foundational reason regularity works is that the converse also holds, so a uniform bound on the difference quotients of $D u$ promotes $D u$ to $H^{1}$ and $u$ to $H^{2}$ . The energy supplied by Lax-Milgram in 02.16.04 is the coercivity that, transplanted to the differentiated equation, produces the controlling bound $θ \int ζ^{2} ∣ D w ∣^{2}$ ; this is exactly the same ellipticity inequality powering existence, now powering smoothness. The estimate builds toward the bootstrap $f \in H^{k} \Rightarrow u \in H^{k + 2}$ and appears again in 24.01.01 as the regularity that makes finite-element error estimates of optimal order possible, and putting these together, the central insight is that ellipticity controls all second derivatives simultaneously, so existence and regularity are one coercivity estimate read at two different levels of differentiation.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Prove the difference-quotient translation estimate: if $w \in H^{1} (Ω)$ and $Ω^{'} ⋐ Ω$ , then for $∣ h ∣ < dist (Ω^{'}, \partial Ω)$ one has $∥ D_{k}^{h} w ∥_{L^{2} (Ω^{'})} \leq ∥ \partial_{k} w ∥_{L^{2} (Ω)}$ .

Hint

For smooth $w$ write $D_{k}^{h} w (x) = \frac{1}{h} \int_{0}^{h} \partial_{k} w (x + s e_{k}) d s$ , apply the Cauchy-Schwarz (Jensen) inequality to the average, integrate over $Ω^{'}$ , and use Fubini; extend to $H^{1}$ by density.

Answer

For $w \in C^{1}$ , the fundamental theorem of calculus gives $D_{k}^{h} w (x) = \frac{1}{h} (w (x + h e_{k}) - w (x)) = \frac{1}{h} \int_{0}^{h} \partial_{k} w (x + s e_{k}) d s$ . By the Cauchy-Schwarz inequality applied to the average over $[0, h]$ , $∣ D_{k}^{h} w (x) ∣^{2} \leq \frac{1}{h} \int_{0}^{h} ∣ \partial_{k} w (x + s e_{k}) ∣^{2} d s$ . Integrate over $x \in Ω^{'}$ and use Fubini, noting that for fixed $s \in [0, h]$ the map $x \mapsto x + s e_{k}$ sends $Ω^{'}$ into $Ω$ : $\int_{Ω^{'}} ∣ D_{k}^{h} w ∣^{2} d x \leq \frac{1}{h} \int_{0}^{h} \int_{Ω^{'}} ∣ \partial_{k} w (x + s e_{k}) ∣^{2} d x d s \leq \frac{1}{h} \int_{0}^{h} ∥ \partial_{k} w ∥_{L^{2} (Ω)}^{2} d s = ∥ \partial_{k} w ∥_{L^{2} (Ω)}^{2} .$ Density of $C^{1} \cap H^{1}$ in $H^{1}$ extends the inequality to all $w \in H^{1} (Ω)$ . This is the forward direction of the Nirenberg characterisation: $H^{1}$ membership forces bounded difference quotients.

Exercise 4 (medium, symbolic).

Prove the converse (Nirenberg lemma): if $w \in L^{2} (Ω)$ and there is $K$ with $∥ D_{k}^{h} w ∥_{L^{2} (Ω^{'})} \leq K$ for all $Ω^{'} ⋐ Ω$ and all small $∣ h ∣$ , then $\partial_{k} w \in L^{2} (Ω)$ exists (weakly) with $∥ \partial_{k} w ∥_{L^{2} (Ω)} \leq K$ .

Hint

Bounded sets in $L^{2}$ are weakly precompact: extract $h_{m} \to 0$ with $D_{k}^{h_{m}} w ⇀ v$ . Pass to the limit in $\int_{Ω} (D_{k}^{h} w) φ = - \int_{Ω} w D_{k}^{- h} φ$ for $φ \in C_{c}^{\infty} (Ω)$ to identify $v = \partial_{k} w$ .

Answer

The hypothesis bounds ${D_{k}^{h} w}$ in $L^{2} (Ω^{'})$ uniformly in $h$ . By weak compactness of bounded sets in the Hilbert space $L^{2}$ , there is a sequence $h_{m} \to 0$ and $v \in L^{2} (Ω)$ with $D_{k}^{h_{m}} w ⇀ v$ weakly, and $∥ v ∥_{L^{2}} \leq lim inf ∥ D_{k}^{h_{m}} w ∥_{L^{2}} \leq K$ by weak lower semicontinuity of the norm. For any $φ \in C_{c}^{\infty} (Ω)$ and $∣ h ∣$ small enough that $supp φ$ stays at distance $> ∣ h ∣$ from $\partial Ω$ , the summation-by-parts identity gives $\int_{Ω} (D_{k}^{h} w) φ d x = - \int_{Ω} w (D_{k}^{- h} φ) d x$ . As $h = h_{m} \to 0$ the left side tends to $\int_{Ω} v φ$ (weak convergence) and the right side tends to $- \int_{Ω} w \partial_{k} φ$ (since $D_{k}^{- h} φ \to \partial_{k} φ$ uniformly for smooth $φ$ ). Hence $\int_{Ω} v φ = - \int_{Ω} w \partial_{k} φ$ for all $φ$ , which is the definition of $\partial_{k} w = v$ in the weak sense, with $∥ \partial_{k} w ∥_{L^{2}} \leq K$ .

Exercise 5 (medium, symbolic).

For the Laplacian $L = - Δ$ (so $a^{ij} = δ^{ij}$ , $b = 0$ , $c = 0$ ) with $f \in L^{2}$ , recover the missing pure second derivative from the equation once the mixed and tangential ones are known to be in $L^{2}$ . Show that interior $H^{2}$ membership reduces to controlling $\partial_{11} u$ , and exhibit it from $- Δ u = f$ .

Hint

The difference-quotient argument directly controls $\partial_{k} (\partial_{j} u)$ for $k = 1, \dots, n$ in every direction, so all $\partial_{k j} u \in L^{2}$ . For a general operator one direction is recovered from the PDE; here all are already controlled, but write $\partial_{11} u = - f - \sum_{i \geq 2} \partial_{ii} u$ as the model of the boundary step.

Answer

The difference-quotient estimate of the Key theorem bounds $∥ ζ D D_{k}^{h} u ∥_{L^{2}}$ for every $k$ , so by the Nirenberg lemma $\partial_{k} \partial_{j} u \in L^{2} (Ω^{'})$ for all $k, j$ , giving $u \in H^{2} (Ω^{'})$ directly. To model the boundary step, note that almost everywhere $- Δ u = f$ reads $- \sum_{i = 1}^{n} \partial_{ii} u = f$ , so $\partial_{11} u = - f - i = 2 \sum n \partial_{ii} u .$ If the difference-quotient method had supplied only the derivatives $\partial_{ii} u$ for $i \geq 2$ (the tangential ones, as happens near a flat boundary where only tangential difference quotients are admissible), this identity recovers the remaining normal second derivative $\partial_{11} u$ in $L^{2}$ , because the right side is a sum of $L^{2}$ functions. The ellipticity $a^{11} \geq θ > 0$ is what makes the coefficient of $\partial_{11} u$ invertible in the general case; for the Laplacian it equals $1$ .

Exercise 7 (hard, symbolic).

Prove the bootstrap (higher interior regularity): if additionally $a^{ij} \in C^{k + 1} (Ω)$ , $b^{i}, c \in C^{k} (Ω)$ , and $f \in H^{k} (Ω)$ , then the weak solution $u \in H_{loc}^{k + 2} (Ω)$ , with the estimate $∥ u ∥_{H^{k + 2} (Ω^{'})} \leq C (∥ f ∥_{H^{k} (Ω)} + ∥ u ∥_{L^{2} (Ω)})$ .

Hint

Induct on $k$ . For the step, apply the base $H^{2}$ theorem to a difference quotient or weak derivative $\tilde{u} = \partial_{ℓ} u$ , which solves $L \tilde{u} = \tilde{f}$ with $\tilde{f} = \partial_{ℓ} f - (\partial_{ℓ} L) u \in H^{k - 1}$ , the corrector $(\partial_{ℓ} L) u$ being controlled by the inductive hypothesis.

Answer

Induct on $k$ ; the base case $k = 0$ is the Key theorem. Assume the statement for $k - 1$ . Differentiating the weak form $B [u, v] = (f, v)$ in a direction $e_{ℓ}$ (formally, test against $- \partial_{ℓ} v$ and integrate by parts; rigorously, use difference quotients and pass to the limit using the base estimate) shows that $\tilde{u} = \partial_{ℓ} u$ is a weak solution of $L \tilde{u} = \tilde{f}$ , where $\tilde{f} = \partial_{ℓ} f - i, j \sum (\partial_{ℓ} a^{ij}) \partial_{i} \partial_{j} u \dots - (\partial_{ℓ} b^{i}) \partial_{i} u - (\partial_{ℓ} c) u,$ the corrector terms being the $e_{ℓ}$ -derivatives of the coefficients applied to $u$ . By the inductive hypothesis $u \in H_{loc}^{k + 1}$ , so the corrector lies in $H_{loc}^{k - 1}$ , and $\partial_{ℓ} f \in H^{k - 1}$ ; hence $\tilde{f} \in H_{loc}^{k - 1}$ . Applying the inductive hypothesis at level $k - 1$ to $L \tilde{u} = \tilde{f}$ gives $\tilde{u} = \partial_{ℓ} u \in H_{loc}^{k + 1}$ , with the estimate $∥ \partial_{ℓ} u ∥_{H^{k + 1} (Ω^{'})} \leq C (∥ \tilde{f} ∥_{H^{k - 1}} + ∥ u ∥_{L^{2}})$ . Since $ℓ$ was arbitrary, every first derivative of $u$ lies in $H_{loc}^{k + 1}$ , so $u \in H_{loc}^{k + 2}$ . Chaining the estimates and using $∥ \tilde{f} ∥_{H^{k - 1}} \leq C (∥ f ∥_{H^{k}} + ∥ u ∥_{H^{k + 1}})$ together with the inductive bound on $∥ u ∥_{H^{k + 1}}$ closes the induction with the stated constant. $□$

Exercise 8 (hard, symbolic).

Establish the $C^{\infty}$ corollary: if $\partial Ω \in C^{\infty}$ , the coefficients $a^{ij}, b^{i}, c \in C^{\infty} (\overline{Ω})$ , and $f \in C^{\infty} (\overline{Ω})$ , then the weak solution $u \in H_{0}^{1} (Ω)$ is in $C^{\infty} (\overline{Ω})$ , hence a classical solution. Use the global bootstrap and Sobolev embedding.

Hint

The boundary version of the bootstrap gives $u \in H^{k + 2} (Ω)$ for every $k$ , since $f \in H^{k} (Ω)$ for all $k$ . Then apply the Sobolev embedding $H^{m} (Ω) ↪ C^{j} (\overline{Ω})$ for $m > j + n /2$ of 02.16.01.

Answer

Since $f \in C^{\infty} (\overline{Ω})$ and $Ω$ is bounded, $f \in H^{k} (Ω)$ for every $k \geq 0$ . The global (up-to-the-boundary) bootstrap, valid because $\partial Ω \in C^{\infty}$ and the coefficients are smooth, gives $u \in H^{k + 2} (Ω)$ for every $k$ , that is $u \in ⋂_{m \geq 0} H^{m} (Ω)$ . The Morrey-Sobolev embedding of 02.16.01 states that $H^{m} (Ω) ↪ C^{j} (\overline{Ω})$ whenever $m > j + n /2$ and $\partial Ω$ is Lipschitz. For each fixed $j$ , choosing $m > j + n /2$ shows $u \in C^{j} (\overline{Ω})$ ; as $j$ is arbitrary, $u \in ⋂_{j} C^{j} (\overline{Ω}) = C^{\infty} (\overline{Ω})$ . A function in $C^{2} (\overline{Ω})$ satisfying $Lu = f$ a.e. satisfies it pointwise by continuity, so $u$ is a classical solution with $u = 0$ on $\partial Ω$ . The weak solution produced by Lax-Milgram was the classical solution throughout.

Advanced results Master

The interior estimate sits at the base of a tower: the up-to-the-boundary estimate on smooth domains, the $L^{p}$ analogue of Calderón-Zygmund that replaces the Hilbert-space energy by singular-integral bounds, the Schauder $C^{2, α}$ companion on the Hölder scale, the elliptic-systems generalisation of Agmon-Douglis-Nirenberg, and the analytic-hypoellipticity refinement for analytic data. Each is the same principle — ellipticity converts smoothness of the data into smoothness of the solution with a fixed two-derivative gain — read on a different scale of function spaces.

Theorem 1 (global $H^{2}$ regularity up to the boundary). Let $\partial Ω \in C^{2}$ , $a^{ij} \in C^{1} (\overline{Ω})$ , $b^{i}, c \in L^{\infty}$ , $f \in L^{2} (Ω)$ , and let $u \in H_{0}^{1} (Ω)$ be the weak solution of $Lu = f$ with zero boundary data. Then $u \in H^{2} (Ω)$ and $∥ u ∥_{H^{2} (Ω)} \leq C (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{L^{2} (Ω)}),$ with $C = C (Ω, θ, coefficients)$ ^{[Evans 2010 §6.3]} ^{[Gilbarg-Trudinger 2001 §8.12]}. The proof flattens the boundary: near a boundary point a $C^{2}$ change of variables straightens $\partial Ω$ to a piece of the hyperplane ${x_{n} = 0}$ , the operator transforms to another uniformly elliptic operator with $C^{1}$ leading coefficients, and the difference quotients $D_{k}^{h} u$ are taken only in the $n - 1$ tangential directions $k = 1, \dots, n - 1$ — these preserve the homogeneous Dirichlet condition because translating tangentially keeps $u = 0$ on the flattened boundary. The interior argument then bounds all tangential and mixed second derivatives $\partial_{k} \partial_{j} u$ with $min (k, j) \leq n - 1$ in $L^{2}$ . The single remaining pure normal second derivative $\partial_{nn} u$ is recovered algebraically from the equation: $a^{nn} \partial_{nn} u = - f - \sum_{(i, j) \neq = (n, n)} a^{ij} \partial_{ij} u + (lower order)$ , and $a^{nn} \geq θ > 0$ makes the coefficient invertible, so $\partial_{nn} u \in L^{2}$ . A partition of unity glues the boundary charts to the interior estimate.

Theorem 2 ( $L^{p}$ (Calderón-Zygmund) regularity). Let $1 < p < \infty$ , $a^{ij} \in C (\overline{Ω})$ (or VMO), $f \in L^{p} (Ω)$ . Then the weak solution lies in $W_{loc}^{2, p} (Ω)$ with $∥ u ∥_{W^{2, p} (Ω^{'})} \leq C (∥ f ∥_{L^{p} (Ω)} + ∥ u ∥_{L^{p} (Ω)})$ . The energy method is unavailable for $p \neq = 2$ because $L^{p}$ is not a Hilbert space; the proof instead represents $D^{2} u$ through the second derivatives of the Newtonian potential, which is a Calderón-Zygmund singular integral, and invokes the $L^{p}$ boundedness of such operators. At $p = 2$ this reduces, by Plancherel, to the energy estimate of this unit; the difference-quotient method is the $p = 2$ shortcut that avoids the full singular-integral machinery. The borderline cases $p = 1$ and $p = \infty$ fail, which is why $L^{\infty}$ data does not give $C^{2}$ solutions and Schauder theory uses Hölder spaces instead.

Theorem 3 (Schauder $C^{2, α}$ companion). If instead $a^{ij}, b^{i}, c \in C^{α} (\overline{Ω})$ and $f \in C^{α} (Ω)$ for some $0 < α < 1$ , the weak solution is in $C_{loc}^{2, α} (Ω)$ with $∥ u ∥_{C^{2, α} (Ω^{'})} \leq C (∥ f ∥_{C^{α} (Ω)} + ∥ u ∥_{C^{0} (Ω)})$ . This is the Hölder-scale twin of the $L^{2}$ estimate developed in 02.17.04; the two scales are linked by the Sobolev embedding $W^{2, p} ↪ C^{1, α}$ for $p > n$ , so high-integrability $L^{p}$ regularity feeds Hölder regularity and conversely. The Schauder constant degenerates as $α \to 0$ or $α \to 1$ , mirroring the failure of $L^{p}$ theory at $p = \infty$ .

Theorem 4 (Agmon-Douglis-Nirenberg for elliptic systems). For a properly elliptic system $L u = f$ of order $2 m$ with smooth coefficients and complementing (Lopatinskii-Shapiro) boundary conditions, the a priori estimate $∥ u ∥_{H^{k + 2 m}} \leq C (∥ f ∥_{H^{k}} + ∥ u ∥_{L^{2}})$ holds, with the same fixed gain of $2 m$ derivatives over the data ^{[Agmon-Douglis-Nirenberg 1959]}. The scalar second-order case $m = 1$ of this unit is the prototype; the systems theory replaces uniform ellipticity by the invertibility of the principal symbol and replaces the single Dirichlet condition by the algebraic complementing condition that makes the boundary value problem Fredholm.

Theorem 5 (analytic and $C^{\infty}$ hypoellipticity). If the coefficients and $f$ are real-analytic on $Ω$ , the solution is real-analytic on $Ω$ (elliptic operators are analytic-hypoelliptic), strengthening the $C^{\infty}$ corollary; this is the elliptic case of the Petrowsky theorem. The $C^{\infty}$ version states that $L$ is hypoelliptic: $Lu \in C^{\infty}$ on an open set forces $u \in C^{\infty}$ there, regardless of how rough $u$ is globally. Ellipticity is what makes the operator invertible modulo smoothing on the symbol level, the pseudodifferential reading of regularity.

Synthesis. The difference-quotient estimate is the foundational reason the weak formulation loses nothing: existence in 02.16.04 and regularity here are the single coercivity inequality of ellipticity, read first at the level of the equation and then at the level of its derivative, and this is exactly why the rough averaged solution is the smooth classical solution in disguise. The translation estimate of 02.16.01 is dual to the Nirenberg lemma — one says $H^{1}$ implies bounded difference quotients, the other says bounded difference quotients imply $H^{1}$ — and putting these together the gain of exactly two derivatives is the central insight that organises every scale: the $L^{2}$ scale here, the $L^{p}$ Calderón-Zygmund scale of Theorem 2, and the Hölder $C^{2, α}$ scale of 02.17.04 are three readings of the same gain, generalising to the systems estimate of Agmon-Douglis-Nirenberg where the gain becomes the order $2 m$ of the operator. The boundary argument appears again wherever a finite-element or spectral method needs the solution to be smooth enough for its convergence rate, so the regularity proved here is dual to the numerical approximation theory of 24.01.01; the bridge is that ellipticity, expressed as invertibility of the principal symbol, is the structural fact that converts data regularity into solution regularity with a fixed gain on every function-space scale at once.

Full proof set Master

Proposition 1 (Nirenberg difference-quotient lemma). Let $w \in L^{2} (Ω)$ . Then $w \in H^{1} (Ω)$ with $∥ D w ∥_{L^{2} (Ω)} \leq K$ if and only if for every $Ω^{'} ⋐ Ω$ and every direction $k$ , $sup_{0 < ∣ h ∣ < dist (Ω^{'}, \partial Ω)} ∥ D_{k}^{h} w ∥_{L^{2} (Ω^{'})} \leq K$ .

Proof. ( $\Rightarrow$ ) is Exercise 3: for $w \in H^{1}$ , $∥ D_{k}^{h} w ∥_{L^{2} (Ω^{'})} \leq ∥ \partial_{k} w ∥_{L^{2} (Ω)} \leq K$ . ( $\Leftarrow$ ) is Exercise 4: the uniform bound makes ${D_{k}^{h} w}$ bounded in $L^{2}$ , so a subsequence $D_{k}^{h_{m}} w ⇀ v_{k}$ weakly with $∥ v_{k} ∥_{L^{2}} \leq K$ ; testing the summation-by-parts identity $\int (D_{k}^{h} w) φ = - \int w D_{k}^{- h} φ$ against $φ \in C_{c}^{\infty} (Ω)$ and letting $h_{m} \to 0$ identifies $v_{k}$ as the weak derivative $\partial_{k} w$ , so $w \in H^{1} (Ω)$ with $∥ \partial_{k} w ∥_{L^{2}} \leq K$ . $□$

Proposition 2 (Caccioppoli energy inequality). Let $u \in H^{1} (Ω)$ be a weak solution of $Lu = f$ with $f \in L^{2}$ , $L$ uniformly elliptic with bounded coefficients. For $Ω^{'} ⋐ Ω^{''} ⋐ Ω$ and a cutoff $ζ \in C_{c}^{\infty} (Ω^{''})$ , $ζ \equiv 1$ on $Ω^{'}$ , $∥ D u ∥_{L^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2} (Ω^{''})} + ∥ u ∥_{L^{2} (Ω^{''})}) .$

Proof. Test the weak form against $v = ζ^{2} u \in H_{0}^{1} (Ω)$ . The principal part gives $\int \sum a^{ij} \partial_{j} u \partial_{i} (ζ^{2} u) = \int ζ^{2} \sum a^{ij} \partial_{j} u \partial_{i} u + \int 2 ζ u \sum a^{ij} \partial_{j} u \partial_{i} ζ$ . Uniform ellipticity bounds the first term below by $θ \int ζ^{2} ∣ D u ∣^{2}$ . The second term, the drift, and the zeroth-order term are bounded above using Cauchy-Schwarz and Young's inequality with a small parameter $ϵ$ , absorbing $ϵ \int ζ^{2} ∣ D u ∣^{2}$ into the leading term; the source contributes $\int f ζ^{2} u \leq \frac{1}{2} ∥ f ∥_{L^{2} (Ω^{''})}^{2} + \frac{1}{2} ∥ ζ^{2} u ∥_{L^{2}}^{2}$ . Collecting and absorbing leaves $\frac{θ}{2} \int ζ^{2} ∣ D u ∣^{2} \leq C (∥ f ∥_{L^{2} (Ω^{''})}^{2} + ∥ u ∥_{L^{2} (Ω^{''})}^{2})$ ; restricting the left integral to $Ω^{'}$ where $ζ = 1$ gives the claim. $□$

Proposition 3 (interior $H^{2}$ estimate, assembled). Under the hypotheses of the Key theorem, $u \in H^{2} (Ω^{'})$ with $∥ u ∥_{H^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{L^{2} (Ω)})$ for every $Ω^{'} ⋐ Ω$ .

Proof. The difference-quotient estimate of the Key theorem yields $sup_{h} ∥ ζ D D_{k}^{h} u ∥_{L^{2}} \leq C (∥ f ∥_{L^{2}} + ∥ u ∥_{H^{1} (Ω^{''})})$ for each $k$ . Proposition 1 applied to $w = \partial_{j} u$ (whose difference quotients $D_{k}^{h} \partial_{j} u = \partial_{j} D_{k}^{h} u$ are controlled by the previous bound) gives $\partial_{k} \partial_{j} u \in L^{2} (Ω^{'})$ for all $k, j$ , hence $u \in H^{2} (Ω^{'})$ with $∥ D^{2} u ∥_{L^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2}} + ∥ u ∥_{H^{1} (Ω^{''})})$ . Proposition 2 replaces $∥ u ∥_{H^{1} (Ω^{''})}$ by $C (∥ f ∥_{L^{2}} + ∥ u ∥_{L^{2}})$ . Adding $∥ u ∥_{L^{2} (Ω^{'})}$ and $∥ D u ∥_{L^{2} (Ω^{'})}$ to the left gives the full $H^{2}$ norm with the stated bound. $□$

Proposition 4 (boundary $H^{2}$ via tangential difference quotients). Let $\partial Ω \in C^{2}$ and $u \in H_{0}^{1} (Ω)$ solve $Lu = f$ . Then $u \in H^{2} (Ω)$ with the global estimate of Theorem 1.

Proof. Cover $\partial Ω$ by finitely many charts; in each a $C^{2}$ diffeomorphism $Φ$ flattens $\partial Ω \cap U$ to ${y_{n} = 0}$ , carrying $L$ to a uniformly elliptic $\tilde{L}$ with $C^{1}$ leading coefficients and $u$ to $\tilde{u} \in H^{1}$ vanishing on ${y_{n} = 0}$ . For $k \leq n - 1$ the tangential difference quotient $D_{k}^{h} \tilde{u}$ still vanishes on ${y_{n} = 0}$ , so $v = - D_{k}^{- h} (ζ^{2} D_{k}^{h} \tilde{u})$ is an admissible test function in the half-ball, and the interior argument bounds $∥ ζ D D_{k}^{h} \tilde{u} ∥_{L^{2}}$ uniformly in $h$ . By Proposition 1 the second derivatives $\partial_{k} \partial_{j} \tilde{u}$ with $k \leq n - 1$ lie in $L^{2}$ . The pure normal derivative is recovered from the equation: $\tilde{a}^{nn} \partial_{nn} \tilde{u} = - f - \sum_{(i, j) \neq = (n, n)} \tilde{a}^{ij} \partial_{ij} \tilde{u} + (lower order)$ , and $\tilde{a}^{nn} \geq θ > 0$ gives $\partial_{nn} \tilde{u} \in L^{2}$ . Transforming back and summing over the charts with a partition of unity, together with the interior estimate of Proposition 3 on the inner region, gives $u \in H^{2} (Ω)$ and the global bound. $□$

Connections Master

The existence theory of 02.16.04 produces the weak solution whose smoothness this unit establishes: Lax-Milgram supplies $u \in H_{0}^{1}$ , and the difference-quotient method upgrades it to $H_{loc}^{2}$ , so the two units are the existence and regularity halves of a single elliptic boundary-value theory. The coercivity that powered existence is the same ellipticity inequality that, transplanted to the differentiated equation, powers the regularity estimate; this unit owns the regularity gain, 02.16.04 owns the solvability.
The Sobolev inequalities and embeddings of 02.16.01 provide both the translation estimate $∥ τ_{h} u - u ∥_{L^{2}} \leq ∣ h ∣∥ D u ∥_{L^{2}}$ that is the forward half of the Nirenberg characterisation and the embedding $H^{m} ↪ C^{j}$ that converts the $H^{k}$ bootstrap into the $C^{\infty}$ corollary. The sibling unit supplies the difference-quotient input and the smoothness-output dictionary that bracket the regularity argument at both ends.
The Sobolev-space framework $H^{k} (Ω)$ , $W^{k, p}$ , traces, and extension of 24.01.01 is the ambient setting in which the estimate $∥ u ∥_{H^{2} (Ω^{'})} \leq C (∥ f ∥_{L^{2}} + ∥ u ∥_{L^{2}})$ is even stated, and the up-to-the-boundary $H^{2}$ regularity proved here is exactly the hypothesis that makes finite-element error estimates of optimal order $O (h)$ in the energy norm and $O (h^{2})$ in $L^{2}$ rigorous, so this unit is the regularity backbone of the numerical theory of 24.01.01.
The maximum principle of 02.17.02 is the pointwise complement to this $L^{2}$ regularity theory: where the maximum principle controls the size and sign of $u$ from boundary data, the $H^{2}$ estimate controls the size of $D^{2} u$ from the source, and together they give both the qualitative and quantitative a priori control needed for the method of continuity in 02.17.05.
The Hölder-scale Schauder theory of 02.17.04 and the $L^{p}$ Calderón-Zygmund theory of 02.17.06 are the two siblings of this $L^{2}$ estimate on other function-space scales; all three deliver the same two-derivative gain, are linked by Sobolev embedding, and feed the existence-by-a-priori-estimates scheme, making this unit the energy-method entry to the three-scale regularity package of the chapter.

Historical & philosophical context Master

The question of whether a weak or generalised solution is automatically smooth — Hilbert's nineteenth problem in its analytic form — drove the development of elliptic regularity through the first half of the twentieth century. Hermann Weyl's 1940 lemma, proved by the method of orthogonal projection ^{[Weyl 1940]}, gave the first clean modern instance: a distribution that is weakly harmonic is smooth and classically harmonic, establishing that the averaged notion of a harmonic function loses nothing. Kurt Friedrichs extended the smoothness conclusion to general strongly elliptic operators in his 1953 paper on the differentiability of solutions ^{[Friedrichs 1953]}, introducing mollification and the systematic use of a priori estimates in Sobolev norms.

The difference-quotient method that makes the $L^{2}$ theory elementary is due to Louis Nirenberg, whose 1955 paper on strongly elliptic equations ^{[Nirenberg 1955]} isolated the characterisation of $H^{1}$ membership through uniformly bounded difference quotients and used it to prove interior regularity by differentiating the equation discretely rather than through the Fourier transform. The method's economy — it needs only the energy inequality and weak compactness, no singular integrals — is what made it the standard textbook route. Sergei Agmon, Avron Douglis, and Nirenberg then established the definitive $L^{p}$ and Schauder estimates up to the boundary for general elliptic systems with complementing boundary conditions in their 1959 paper ^{[Agmon-Douglis-Nirenberg 1959]}, fixing the now-standard form of the a priori estimate $∥ u ∥_{k + 2 m} \leq C (∥ f ∥_{k} + ∥ u ∥_{0})$ with its characteristic gain of the operator's order. The energy estimate of this unit is the Hilbert-space, second-order, scalar case of that general theory.

Bibliography Master

@article{Nirenberg1955,
  author  = {Nirenberg, Louis},
  title   = {Remarks on strongly elliptic partial differential equations},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {8},
  year    = {1955},
  pages   = {649--675}
}

@article{ADN1959,
  author  = {Agmon, Shmuel and Douglis, Avron and Nirenberg, Louis},
  title   = {Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {12},
  year    = {1959},
  pages   = {623--727}
}

@article{Friedrichs1953,
  author  = {Friedrichs, Kurt O.},
  title   = {On the differentiability of the solutions of linear elliptic differential equations},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {6},
  year    = {1953},
  pages   = {299--326}
}

@article{Weyl1940,
  author  = {Weyl, Hermann},
  title   = {The method of orthogonal projection in potential theory},
  journal = {Duke Mathematical Journal},
  volume  = {7},
  year    = {1940},
  pages   = {411--444}
}

@book{Evans2010,
  author    = {Evans, Lawrence C.},
  title     = {Partial Differential Equations},
  edition   = {2},
  series    = {Graduate Studies in Mathematics 19},
  publisher = {American Mathematical Society},
  year      = {2010}
}

@book{GilbargTrudinger2001,
  author    = {Gilbarg, David and Trudinger, Neil S.},
  title     = {Elliptic Partial Differential Equations of Second Order},
  edition   = {2},
  series    = {Grundlehren der mathematischen Wissenschaften 224},
  publisher = {Springer},
  year      = {2001}
}

Prerequisites

02.16.04
02.16.01
24.01.01

Tier anchors

beginner: Strogatz-style intuition that an averaged 'weak' solution of a smooth diffusion law is secretly as smooth as the law allows; 3Blue1Brown's picture of a finite difference (a divided difference of nearby values) approaching a derivative as the spacing shrinks
intermediate: Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §6.3 (interior and boundary regularity, the difference-quotient method); Brezis, Functional Analysis, Sobolev Spaces and PDEs (Springer 2011), §9.6
master: Evans §6.3; Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 2001), §8.3-§8.4, §8.8-§8.13; Nirenberg, Remarks on strongly elliptic partial differential equations (Comm. Pure Appl. Math. 1955); Agmon-Douglis-Nirenberg I (Comm. Pure Appl. Math. 1959)

References

Nirenberg — Remarks on strongly elliptic partial differential equations · Communications on Pure and Applied Mathematics 8 (1955), 649-675
Agmon, Douglis, Nirenberg — Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I · Communications on Pure and Applied Mathematics 12 (1959), 623-727
Evans — Partial Differential Equations, 2e · AMS Graduate Studies in Mathematics 19 (2010), §6.3
Gilbarg-Trudinger — Elliptic Partial Differential Equations of Second Order, 2e · Springer Grundlehren 224 (2001), §8.3-§8.4, §8.8-§8.13
Brezis — Functional Analysis, Sobolev Spaces and Partial Differential Equations · Springer Universitext (2011), §9.6
Friedrichs — On the differentiability of the solutions of linear elliptic differential equations · Communications on Pure and Applied Mathematics 6 (1953), 299-326
Weyl — The method of orthogonal projection in potential theory · Duke Mathematical Journal 7 (1940), 411-444

Estimated time

beginner: 25m
intermediate: 65m
master: 110m