02.17.04 · analysis / elliptic-regularity

Schauder Theory: Interior and Boundary C^{2,alpha} Estimates

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Anchor (Master): Gilbarg-Trudinger §4-6; Caffarelli-Cabré, Fully Nonlinear Elliptic Equations (AMS Colloquium 43, 1995), §4; Giaquinta-Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems (Edizioni della Normale 2012), §5 (Campanato)

Intuition Beginner

When you solve an equilibrium equation, the source you put in and the answer you get out are not equally smooth. The answer is always smoother. If you heat a metal plate with a source that is reasonably regular, the temperature that settles in is more regular still: it has two more degrees of smoothness than the source. Schauder theory is the precise bookkeeping of this gain.

The earlier Poisson story said that a continuous source gives a solution whose first derivatives behave nicely, but it stopped short of controlling the second derivatives. That gap matters, because the equation itself is a statement about second derivatives. The honest question is: if the source is smooth in a graded, fractional sense, how smooth are the second derivatives of the solution? Schauder's answer is the cleanest possible: the second derivatives are exactly as smooth as the source, measured on the same fractional scale.

The right scale is the one Hölder introduced. A function is Hölder smooth with exponent between zero and one if its values at nearby points differ by no more than a fixed multiple of the distance raised to that exponent. Exponent one would be an honest slope bound; smaller exponents allow gentler corners. This scale is the natural home for elliptic regularity, because the operators of equilibrium respect it perfectly, while they do not respect the cruder scale of plain continuity.

The headline estimate is a single inequality. It says the size of the solution together with all its derivatives up to second order, measured in the Hölder scale, is controlled by the size of the source in the same scale plus the size of the solution alone. In words: control the source and the raw solution, and you have automatically controlled all the derivatives the equation could ever ask about. Nothing wild can happen to the curvature of the solution that the source did not already announce.

The one-sentence takeaway: Schauder theory says an elliptic equilibrium gains exactly two fractional degrees of smoothness over its source, and it packages this gain as one master inequality bounding the solution's second derivatives by the source.

Visual Beginner

Picture two dials, each measuring smoothness on the same fractional scale from zero to one. The left dial reads the smoothness of the source you feed in; the right dial reads the smoothness of the second derivatives of the solution you get out. Schauder's theorem locks the two dials together: turn the source dial up, and the solution-derivative dial turns up by exactly the same amount. They never drift apart.

The left dial is the source measured in the fractional scale; the right dial is the second derivative of the solution measured in the same scale. The arrow marked "+2 orders" is the regularity gain: the solution carries two more whole derivatives than the source, and the leftover fractional smoothness passes through unchanged.

Worked example Beginner

We watch the two-orders gain happen in one variable, where everything is an honest calculation. Take the equation that asks for a function whose second derivative equals a given source on the interval from $0$ to $1$ , with the function pinned to zero at both ends. We feed in the source $f (x) = 12 x^{2}$ and read off how smooth the answer is compared to the source.

Step 1. Name the smoothness of the source. The function $f (x) = 12 x^{2}$ is a polynomial; it is as smooth as anything, so on the fractional scale it sits at the top with room to spare. We track derivatives by counting.

Step 2. Solve the equation $u^{''} (x) = 12 x^{2}$ by integrating twice. One antiderivative of $12 x^{2}$ is $4 x^{3}$ ; an antiderivative of that is $x^{4}$ . So $u (x) = x^{4} + A x + B$ for constants $A, B$ to be fixed by the endpoint conditions.

Step 3. Apply the endpoint conditions $u (0) = 0$ and $u (1) = 0$ . The first gives $B = 0$ . The second gives $1 + A = 0$ , so $A = - 1$ . The solution is $u (x) = x^{4} - x$ .

Step 4. Count the gain. The source $f$ is built from the power $x^{2}$ . The solution $u = x^{4} - x$ is built from the power $x^{4}$ , which is two powers higher. The second derivative of the solution is $u^{''} (x) = 12 x^{2}$ , which is exactly $f$ again. The solution carries two more whole derivatives than the source, and the second derivative landed back on the source with no loss of smoothness.

Step 5. Read the Schauder content. The master inequality would say: the size of $u$ and its first two derivatives, measured in the fractional scale, is bounded by the size of $f$ in that scale plus the size of $u$ . Here $u^{''} = f$ literally, so the second-derivative part of the bound is an equality, the sharpest possible instance.

What this tells us: solving the second-order equation integrates the source twice, which lifts it by two whole derivatives, and the smoothness of the source reappears intact in the second derivative of the solution. Schauder theory is the statement that this clean one-dimensional bookkeeping survives in every dimension, for variable-coefficient operators, once smoothness is measured on the fractional Hölder scale.

Check your understanding Beginner

Exercise (easy, multiple choice).

Schauder theory measures smoothness on the Hölder scale with an exponent $α$ . If an elliptic equation has a source of Hölder smoothness $α$ , how smooth are the second derivatives of the solution?

A. Smoothness $α$ (the same fractional smoothness as the source) B. Smoothness $α /2$ (half as smooth) C. Merely continuous, with no fractional smoothness D. Infinitely smooth regardless of the source

Hint

The headline of Schauder theory is that the second derivatives are exactly as smooth as the source, on the same scale.

Answer

A. Smoothness $α$ . The defining feature of Schauder theory is that the second derivatives of the solution inherit the same fractional Hölder smoothness as the source: the solution gains two whole derivatives, and the leftover fractional part passes through unchanged. Feedback-correct: this matched smoothness is exactly why the Hölder scale is the right home for elliptic regularity. Feedback-wrong: B and C understate the gain; D overstates it, since a merely-Hölder source cannot force infinite smoothness.

Formal definition Intermediate+

Fix an open set $Ω \subseteq R^{n}$ and an exponent $α \in (0, 1]$ . For a bounded function $u : Ω \to R$ the Hölder seminorm is $[u]_{α; Ω} = x, y \in Ω x \neq = y sup \frac{∣ u ( x ) - u ( y ) ∣}{∣ x - y ∣ ^{α}},$ and $u$ is Hölder continuous with exponent $α$ when $[u]_{α; Ω} < \infty$ . The space $C^{0, α} (\overline{Ω})$ consists of the bounded continuous $u$ with $[u]_{α; Ω} < \infty$ , normed by $∥ u ∥_{C^{0, α}} = ∥ u ∥_{C^{0}} + [u]_{α; Ω}$ , where $∥ u ∥_{C^{0}} = sup_{Ω} ∣ u ∣$ . For an integer $k \geq 0$ the Hölder space $C^{k, α} (\overline{Ω})$ is the set of $u \in C^{k} (\overline{Ω})$ all of whose derivatives up to order $k$ are bounded and whose top-order derivatives are Hölder continuous, with norm $∥ u ∥_{C^{k, α}; Ω} = ∣ β ∣ \leq k \sum ∥ D^{β} u ∥_{C^{0}; Ω} + ∣ β ∣ = k \sum [D^{β} u]_{α; Ω} .$ Each $C^{k, α} (\overline{Ω})$ is a Banach space ^{[Gilbarg-Trudinger 2001 §4]}. The completeness rests on the completeness of the underlying $L^{\infty}$ scale 02.07.06.

Let $L$ be a second-order operator in non-divergence form, $Lu = i, j = 1 \sum n a^{ij} (x) D_{ij} u + i = 1 \sum n b^{i} (x) D_{i} u + c (x) u,$ uniformly elliptic with constants $0 < λ \leq Λ$ , and with Hölder-continuous coefficients: $a^{ij}, b^{i}, c \in C^{0, α} (\overline{Ω})$ , with $∥ a^{ij} ∥_{C^{0, α}}, ∥ b^{i} ∥_{C^{0, α}}, ∥ c ∥_{C^{0, α}} \leq Λ$ . The structural data of a Schauder estimate are the dimension $n$ , the exponent $α$ , the ellipticity ratio $Λ/ λ$ , and the Hölder norms of the coefficients; the constant $C$ in every estimate below depends only on these (and, for global estimates, on the domain).

There is a second, equivalent route to the same Hölder scale, due to Campanato. For $u \in L_{loc}^{2} (Ω)$ write $\overset{u}{ˉ}_{x, r} = \fint_{B_{r} (x)} u$ for the average over the ball $B_{r} (x) \subseteq Ω$ . The Campanato seminorm of order $μ$ is $[u]_{L^{2, μ}; Ω} = (x \in Ω, 0 < r < r_{0} sup r^{- μ} \int_{B_{r} (x) \cap Ω} ∣ u - \overset{u}{ˉ}_{x, r} ∣^{2} d y)^{1/2} .$ For $μ = n + 2 α$ with $α \in (0, 1)$ , the Campanato space $L^{2, μ} (Ω)$ coincides with $C^{0, α} (\overline{Ω})$ , with equivalent norms, on domains satisfying a mild interior-cone condition ^{[Campanato 1963]}. Hölder continuity is thus an integral statement about the decay rate of mean oscillation, and this reformulation is what makes the regularity theory robust under the frozen-coefficient perturbation.

Counterexamples to common slips Intermediate+

The estimate fails on the continuous scale; the Hölder exponent is essential. There is a bounded continuous $f$ on $B_{1} \subseteq R^{2}$ whose Newtonian potential $u$ solves $- Δ u = f$ but has $D_{11} u - D_{22} u$ unbounded, so $u \in / C^{2}$ . Plain continuity of $f$ does not give bounded second derivatives; one needs $[f]_{α} < \infty$ for some $α > 0$ . This is precisely why the Poisson unit 02.13.02 stopped at $C^{1}$ for continuous data and forward-referenced the Hölder hypothesis here.
The endpoint $α = 1$ is not allowed. For $α = 1$ (Lipschitz data) the Schauder estimate breaks: bounded $D_{ij} u$ does not follow, and the correct endpoint statement involves the Zygmund space $C_{*}^{1, 1}$ , not $C^{2, 1}$ . The open interval $α \in (0, 1)$ is structural, not a technical convenience.
The lower-order term $∥ u ∥_{C^{0}}$ on the right cannot be dropped in general. For operators with $c \neq \leq 0$ the homogeneous problem $Lu = 0$ may have nonzero solutions (eigenfunctions), so $∥ u ∥_{C^{2, α}} \leq C ∥ Lu ∥_{C^{α}}$ alone is false; the $∥ u ∥_{C^{0}}$ term accounts for the kernel. When $c \leq 0$ and the maximum principle 02.17.02 supplies $∥ u ∥_{C^{0}} \leq C ∥ Lu ∥_{C^{0}}$ , the $∥ u ∥_{C^{0}}$ term may be absorbed.
Continuity of the coefficients alone is insufficient; they must be Hölder. If $a^{ij}$ is merely continuous, $C^{2, α}$ estimates fail and one falls back to $W^{2, p}$ estimates (Calderón-Zygmund) or to the De Giorgi-Nash-Moser $C^{0, β}$ theory. The matched Hölder regularity of source and coefficients is what produces matched Hölder regularity of the second derivatives.

Key theorem with proof Intermediate+

Theorem (interior Schauder estimate; Schauder 1934). Let $L$ be uniformly elliptic on $Ω \subseteq R^{n}$ with coefficients in $C^{0, α} (\overline{Ω})$ , ellipticity constants $λ, Λ$ , and coefficient Hölder norms bounded by $Λ$ . If $u \in C^{2, α} (Ω)$ satisfies $Lu = f$ with $f \in C^{0, α} (Ω)$ , then for every pair of concentric balls $B_{R} = B_{R} (x_{0}) ⋐ B_{2 R} (x_{0}) \subseteq Ω$ , $∥ u ∥_{C^{2, α} (B_{R})} \leq C (∥ u ∥_{C^{0} (B_{2 R})} + ∥ f ∥_{C^{0, α} (B_{2 R})}),$ where $C = C (n, α, λ, Λ, R)$ ^{[Gilbarg-Trudinger 2001 §6.1, Theorem 6.2]}.

Proof (frozen-coefficient / Korn perturbation). The argument reduces the variable-coefficient operator to the constant-coefficient model by freezing the coefficients at a point and treating the variation as a small perturbation. It runs in three movements: a model estimate for the Laplacian, a freezing step, and an absorption.

Model estimate. For the constant-coefficient operator $L_{0} = \sum a_{0}^{ij} D_{ij}$ with frozen coefficients $a_{0}^{ij} = a^{ij} (x_{0})$ , a linear change of variables $y = A x$ diagonalising $[a_{0}^{ij}]$ turns $L_{0}$ into a multiple of the Laplacian, and the ellipticity bounds control the distortion of the Hölder norms by a factor depending only on $Λ/ λ$ . For the Laplacian itself, the Newtonian-potential representation of 02.13.02 gives the model interior estimate: if $- Δ w = g$ on $B_{2 r}$ with $g \in C^{0, α}$ , then differentiating the convolution $w = Φ * (g χ) + h$ twice (with $χ$ a cutoff and $h$ harmonic) and estimating the singular kernel $D_{ij} Φ$ against the Hölder modulus of $g$ yields $[D^{2} w]_{α; B_{r}} \leq C (r^{- 2 - α} ∥ w ∥_{C^{0} (B_{2 r})} + [g]_{α; B_{2 r}}),$ the seminorm form of the model $C^{2, α}$ bound. This is the analytic heart, and it is exactly the second-derivative control of the Newtonian potential that 02.13.02 supplies and the Poisson unit deferred to Hölder data.

Freezing. Write $L = L_{0} + (L - L_{0})$ where $L_{0}$ freezes the second-order coefficients at $x_{0}$ and the first- and zeroth-order terms are moved to the right side. On a ball $B_{r} (x_{0})$ of radius $r$ , $L_{0} u = f - i, j \sum (a^{ij} (x) - a^{ij} (x_{0})) D_{ij} u - i \sum b^{i} D_{i} u - c u =: \tilde{f} .$ By Hölder continuity of the coefficients, $∣ a^{ij} (x) - a^{ij} (x_{0}) ∣ \leq [a^{ij}]_{α} r^{α}$ on $B_{r}$ , so the perturbation term obeys $[(a^{ij} (\cdot) - a^{ij} (x_{0})) D_{ij} u]_{α; B_{r}} \leq C r^{α} [D^{2} u]_{α; B_{r}} + C [a^{ij}]_{α} ∥ D^{2} u ∥_{C^{0} (B_{r})} .$ The factor $r^{α}$ is the gain: by choosing $r$ small the leading term carries a small coefficient.

Absorption. Apply the model estimate to $L_{0} u = \tilde{f}$ on $B_{r}$ and feed in the perturbation bound: $[D^{2} u]_{α; B_{r /2}} \leq C (r^{- 2 - α} ∥ u ∥_{C^{0} (B_{r})} + [f]_{α; B_{r}} + r^{α} [D^{2} u]_{α; B_{r}} + ∥ D^{2} u ∥_{C^{0} (B_{r})}) .$ Choose $r = r_{*}$ so small that $C r_{*}^{α} \leq \frac{1}{2}$ ; then the term $C r_{*}^{α} [D^{2} u]_{α}$ is absorbed into the left side at the cost of a factor of two, provided the seminorms on the two balls are reconciled by a standard interpolation inequality $∥ D^{2} u ∥_{C^{0}} \leq ε [D^{2} u]_{α} + C_{ε} ∥ u ∥_{C^{0}}$ , which trades the intermediate norm against a small multiple of the top seminorm and a large multiple of $∥ u ∥_{C^{0}}$ . The local estimate on $B_{r_{*}}$ is then covered by a finite chain of overlapping balls of radius $r_{*}$ filling $B_{R}$ , summing to the stated global-on- $B_{R}$ bound. $□$

Bridge. This interior estimate builds toward the boundary Schauder estimate and the existence theory of the Dirichlet problem in the Advanced results below, where the same frozen-coefficient perturbation is run against a flattened boundary chart, and it appears again in 02.17.02 through the maximum principle, which supplies the $∥ u ∥_{C^{0}}$ control that converts the a priori estimate into a closed regularity statement. The central insight is that a variable-coefficient elliptic operator is, at small scales, a constant-coefficient operator plus a perturbation that shrinks like $r^{α}$ , so the constant-coefficient Newtonian-potential bound of 02.13.02 is exactly the model whose error the Hölder modulus of the coefficients keeps under control. Putting these together, the Schauder estimate is dual to the Calderón-Zygmund $W^{2, p}$ estimate: one is the Hölder-scale and the other the Lebesgue-scale incarnation of the single fact that the second derivatives of an elliptic solution are no worse than its source, and this is exactly the foundational reason the method of continuity closes the existence theory.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Prove the product rule for Hölder seminorms: if $u, v \in C^{0, α} (\overline{Ω}) \cap C^{0} (\overline{Ω})$ are bounded, then $uv \in C^{0, α}$ and $[uv]_{α} \leq ∥ u ∥_{C^{0}} [v]_{α} + ∥ v ∥_{C^{0}} [u]_{α} .$

Hint

Write $u (x) v (x) - u (y) v (y) = u (x) (v (x) - v (y)) + v (y) (u (x) - u (y))$ and bound each piece.

Answer

For $x \neq = y$ , add and subtract $u (x) v (y)$ : $u (x) v (x) - u (y) v (y) = u (x) (v (x) - v (y)) + v (y) (u (x) - u (y)) .$ Taking absolute values and dividing by $∣ x - y ∣^{α}$ , $\frac{∣ u ( x ) v ( x ) - u ( y ) v ( y ) ∣}{∣ x - y ∣ ^{α}} \leq ∣ u (x) ∣ \frac{∣ v ( x ) - v ( y ) ∣}{∣ x - y ∣ ^{α}} + ∣ v (y) ∣ \frac{∣ u ( x ) - u ( y ) ∣}{∣ x - y ∣ ^{α}} \leq ∥ u ∥_{C^{0}} [v]_{α} + ∥ v ∥_{C^{0}} [u]_{α} .$ Taking the supremum over $x \neq = y$ gives the claimed bound; finiteness of the right side shows $uv \in C^{0, α}$ . This is the estimate that lets the frozen-coefficient proof move coefficient factors across the Hölder seminorm.

Exercise 4 (medium, symbolic).

Establish the interpolation inequality used in the absorption step: for $u \in C^{2, α} (\overline{B_{1}})$ and any $ε > 0$ , $∥ D^{2} u ∥_{C^{0} (B_{1})} \leq ε [D^{2} u]_{α; B_{1}} + C_{ε} ∥ u ∥_{C^{0} (B_{1})} .$ Sketch the scaling proof.

Hint

Argue by contradiction-and-compactness, or by a direct scaling: on a ball of radius $ρ$ , a Taylor expansion bounds $∥ D^{2} u ∥_{C^{0}}$ by $ρ^{α} [D^{2} u]_{α} + ρ^{- 2} ∥ u ∥_{C^{0}}$ , then optimise in $ρ$ .

Answer

Fix a point and a radius $ρ \leq 1$ . On $B_{ρ}$ , a second-order Taylor expansion of $u$ with integral remainder, combined with the Hölder modulus of $D^{2} u$ , gives at the centre $∣ D^{2} u ∣ \leq C (ρ^{- 2} ∥ u ∥_{C^{0} (B_{ρ})} + ρ^{- 1} ∥ D u ∥_{C^{0} (B_{ρ})} + ρ^{α} [D^{2} u]_{α; B_{ρ}}) .$ A first interpolation of the same type controls $∥ D u ∥_{C^{0}}$ by $ρ^{- 1} ∥ u ∥_{C^{0}} + ρ ∥ D^{2} u ∥_{C^{0}}$ , and absorbing the resulting $ρ ∥ D^{2} u ∥$ term yields $∥ D^{2} u ∥_{C^{0} (B_{1})} \leq C ρ^{α} [D^{2} u]_{α} + C ρ^{- 2} ∥ u ∥_{C^{0}} .$ Given $ε$ , choose $ρ$ with $C ρ^{α} = ε$ ; then $C ρ^{- 2} = C_{ε} = C (ε / C)^{- 2/ α}$ , giving the stated inequality. The mechanism is that a small ball makes the top seminorm cheap at the price of a large multiple of the bottom norm.

Exercise 6 (medium, symbolic).

Use the Campanato characterization to show that a function $u \in L^{2} (B_{1})$ with mean-oscillation decay $\int_{B_{r} (x)} ∣ u - \overset{u}{ˉ}_{x, r} ∣^{2} d y \leq M^{2} r^{n + 2 α} for all B_{r} (x) \subseteq B_{1}$ and some $α \in (0, 1)$ has a representative in $C^{0, α}$ . State which step uses the dimension-dependent exponent $n + 2 α$ .

Hint

Compare averages on dyadically shrinking concentric balls to show $\overset{u}{ˉ}_{x, r} \to u (x)$ at a Hölder rate, then compare averages on two nearby balls of comparable radius.

Answer

For concentric balls $B_{r} (x) \supseteq B_{r /2} (x)$ , the averages satisfy $∣ \overset{u}{ˉ}_{x, r /2} - \overset{u}{ˉ}_{x, r} ∣^{2} \leq \fint_{B_{r /2}} ∣ u - \overset{u}{ˉ}_{x, r} ∣^{2} \leq \frac{C}{r ^{n}} \int_{B_{r}} ∣ u - \overset{u}{ˉ}_{x, r} ∣^{2} \leq C M^{2} r^{2 α},$ using the hypothesis with exponent $n + 2 α$ to cancel the volume factor $r^{n}$ . Summing the geometric series over dyadic radii $r_{k} = 2^{- k} r$ shows $\overset{u}{ˉ}_{x, r_{k}}$ is Cauchy with limit $u (x)$ (a Lebesgue point) and $∣ u (x) - \overset{u}{ˉ}_{x, r} ∣ \leq C M r^{α}$ . For two points $x, y$ with $r = ∣ x - y ∣$ , comparing $\overset{u}{ˉ}_{x, 2 r}$ and $\overset{u}{ˉ}_{y, 2 r}$ on the overlapping ball $B_{2 r}$ by the same estimate gives $∣ u (x) - u (y) ∣ \leq C M ∣ x - y ∣^{α}$ . The exponent $n + 2 α$ is exactly the one for which the volume-normalised oscillation scales like $r^{2 α}$ , which is the engine of the whole argument.

Exercise 7 (hard, symbolic).

Carry out the frozen-coefficient reduction abstractly. Let $L = \sum a^{ij} (x) D_{ij}$ be uniformly elliptic with $a^{ij} \in C^{0, α}$ on $B_{1}$ , and suppose the model estimate $[D^{2} w]_{α; B_{r /2}} \leq C_{0} ([Δ w]_{α; B_{r}} + r^{- 2 - α} ∥ w ∥_{C^{0} (B_{r})})$ holds for the Laplacian. Show that there is $r_{*} > 0$ , depending only on $n, α, λ, Λ$ and the coefficient seminorms, such that the interior estimate holds on $B_{r_{*}}$ .

Hint

Freeze $a^{ij}$ at the centre, diagonalise to reduce $L_{0}$ to the Laplacian, move $(a^{ij} (x) - a^{ij} (0)) D_{ij} u$ to the right, bound it by $C r^{α} [D^{2} u]_{α} + C ∥ D^{2} u ∥_{C^{0}}$ , and absorb.

Answer

Freeze $a_{0}^{ij} = a^{ij} (0)$ and let $A$ be the linear map with $A^{⊤} A = [a_{0}^{ij}]^{- 1}$ , so that under $y = A x$ the operator $L_{0} = \sum a_{0}^{ij} D_{ij}$ becomes $Δ$ up to the bounded change-of-variable factors controlled by $Λ/ λ$ . Writing $L_{0} u = f - g$ with $g = \sum (a^{ij} (x) - a_{0}^{ij}) D_{ij} u$ , the model estimate gives $[D^{2} u]_{α; B_{r /2}} \leq C_{0} ([f]_{α; B_{r}} + [g]_{α; B_{r}} + r^{- 2 - α} ∥ u ∥_{C^{0} (B_{r})}) .$ By the product rule (Exercise 3) and $∣ a^{ij} (x) - a_{0}^{ij} ∣ \leq [a^{ij}]_{α} r^{α}$ on $B_{r}$ , $[g]_{α; B_{r}} \leq C r^{α} [D^{2} u]_{α; B_{r}} + C \sum [a^{ij}]_{α} ∥ D^{2} u ∥_{C^{0} (B_{r})} .$ Insert the interpolation $∥ D^{2} u ∥_{C^{0}} \leq δ [D^{2} u]_{α} + C_{δ} ∥ u ∥_{C^{0}}$ (Exercise 4). Choose $r_{*}$ with $C_{0} C r_{*}^{α} + C_{0} C δ \leq \frac{1}{2}$ (first fix $δ$ small, then $r_{*}$ ); the $[D^{2} u]_{α}$ terms on the right are absorbed into the left after reconciling $B_{r /2}$ and $B_{r}$ via a covering/iteration lemma, leaving $[D^{2} u]_{α; B_{r_{*} /2}} \leq C ([f]_{α; B_{r_{*}}} + ∥ u ∥_{C^{0} (B_{r_{*}})}) .$ Adding the lower-order norms by interpolation gives the full $C^{2, α}$ estimate on $B_{r_{*}}$ . The radius $r_{*}$ depends only on the listed structural data, as required.

Exercise 8 (hard, symbolic).

Deduce existence for the Dirichlet problem by the method of continuity. Let $L_{1} = L$ be the target operator (uniformly elliptic, $C^{α}$ coefficients, $c \leq 0$ ) and $L_{0} = Δ$ on a bounded $C^{2, α}$ domain $Ω$ . Set $L_{t} = (1 - t) L_{0} + t L_{1}$ . Assuming the global Schauder estimate $∥ u ∥_{C^{2, α}} \leq C ∥ L_{t} u ∥_{C^{0, α}}$ holds uniformly in $t$ , show $L_{1} : C_{0}^{2, α} (Ω) \to C^{0, α} (Ω)$ is surjective given that $L_{0}$ is.

Hint

The set of $t$ for which $L_{t}$ is invertible is open (perturbation) and closed (uniform estimate); use connectedness of $[0, 1]$ .

Answer

Let $X = C^{2, α} (Ω) \cap {u = 0 on \partial Ω}$ and $Y = C^{0, α} (Ω)$ , both Banach. Each $L_{t} : X \to Y$ is bounded, and the uniform a priori estimate $∥ u ∥_{X} \leq C ∥ L_{t} u ∥_{Y}$ (with $C$ independent of $t$ , available because $c \leq 0$ lets the maximum principle 02.17.02 absorb $∥ u ∥_{C^{0}}$ ) shows every $L_{t}$ is injective with closed range. Let $S = {t \in [0, 1] : L_{t} is onto}$ . If $t_{0} \in S$ then $L_{t_{0}}$ is a Banach-space isomorphism, and for $∣ t - t_{0} ∣$ small $L_{t} = L_{t_{0}} (I + L_{t_{0}}^{- 1} (L_{t} - L_{t_{0}}))$ is invertible by Neumann series since $∥ L_{t} - L_{t_{0}} ∥ = ∣ t - t_{0} ∣ ∥ L_{1} - L_{0} ∥$ is small and $∥ L_{t_{0}}^{- 1} ∥ \leq C$ ; so $S$ is open. If $t_{k} \to t$ with $t_{k} \in S$ , the uniform estimate makes $L_{t}^{- 1}$ bounded on the range and a limiting argument shows $L_{t}$ is onto; so $S$ is closed. Since $0 \in S$ ( $L_{0} = Δ$ solves the Dirichlet problem on a $C^{2, α}$ domain) and $[0, 1]$ is connected, $S = [0, 1]$ , so $1 \in S$ and $L_{1}$ is surjective: the Dirichlet problem $Lu = f$ , $u = 0$ on $\partial Ω$ is solvable for every $f \in C^{0, α}$ .

Advanced results Master

The Schauder apparatus organises around four pillars: the interior estimate and its frozen-coefficient proof, the Campanato integral characterization that makes the perturbation robust, the boundary estimate on $C^{2, α}$ domains, and the existence theory the estimates unlock through the method of continuity.

Theorem 1 (global interior estimate; Schauder 1934). Let $L$ be uniformly elliptic on a bounded $Ω$ with $a^{ij}, b^{i}, c \in C^{0, α} (\overline{Ω})$ and coefficient norms bounded by $Λ$ . If $u \in C^{2, α} (Ω) \cap C^{0} (\overline{Ω})$ solves $Lu = f$ with $f \in C^{0, α} (Ω)$ , then on any $Ω^{'} ⋐ Ω$ with $d = dist (Ω^{'}, \partial Ω)$ , $∥ u ∥_{C^{2, α} (Ω^{'})} \leq C (∥ u ∥_{C^{0} (Ω)} + ∥ f ∥_{C^{0, α} (Ω)}), C = C (n, α, λ, Λ, d)$ ^{[Gilbarg-Trudinger 2001 §6.1]}. The interior-distance weighting can be made explicit through the weighted Hölder norms $∥ u ∥_{*}^{(k)}$ of Gilbarg-Trudinger, which carry powers of $d_{x} = dist (x, \partial Ω)$ and render the estimate scale-covariant.

Theorem 2 (Campanato's characterization; Campanato 1963). For $α \in (0, 1)$ and $μ = n + 2 α$ , a function $u \in L^{2} (Ω)$ on a domain with the interior-cone property lies in $C^{0, α} (\overline{Ω})$ if and only if its Campanato seminorm $[u]_{L^{2, μ}}$ is finite, and the two norms are equivalent ^{[Campanato 1963]}. The proof is the dyadic mean-comparison of Exercise 6. The payoff is a coordinate-free, $L^{2}$ -based route to Schauder estimates (Campanato's method): one proves that a frozen-coefficient solution has Caccioppoli-type energy decay $\int_{B_{r}} ∣ D^{2} u - \overline{D^{2} u} ∣^{2} ≲ r^{n + 2 α}$ , and the characterization converts this decay directly into $D^{2} u \in C^{0, α}$ , bypassing the explicit singular-kernel estimates of the potential-theoretic proof.

Theorem 3 (boundary Schauder estimate; Agmon-Douglis-Nirenberg 1959). Let $Ω$ be a bounded domain with $C^{2, α}$ boundary, $L$ uniformly elliptic with $C^{0, α} (\overline{Ω})$ coefficients, and $u \in C^{2, α} (\overline{Ω})$ a solution of $Lu = f$ in $Ω$ with $u = φ$ on $\partial Ω$ for $φ \in C^{2, α} (\partial Ω)$ . Then $∥ u ∥_{C^{2, α} (\overline{Ω})} \leq C (∥ u ∥_{C^{0} (Ω)} + ∥ f ∥_{C^{0, α} (Ω)} + ∥ φ ∥_{C^{2, α} (\partial Ω)}),$ with $C = C (n, α, λ, Λ, Ω)$ ^{[Agmon-Douglis-Nirenberg 1959]}. The proof flattens the boundary by a $C^{2, α}$ diffeomorphism so that a neighbourhood of a boundary point becomes a half-ball $B_{R}^{+} = B_{R} \cap {x_{n} > 0}$ ; the operator transforms into another uniformly elliptic operator with $C^{0, α}$ coefficients (the chart's $C^{2, α}$ regularity is exactly what preserves the Hölder class of the coefficients), and the half-space model estimate for the Laplacian with the reflected Newtonian potential plays the role of the interior model. The boundary regularity of $Ω$ cannot be relaxed below $C^{2, α}$ without losing the conclusion at the same exponent.

Theorem 4 (existence by the method of continuity; Schauder 1934). Let $Ω$ be a bounded $C^{2, α}$ domain and $L$ uniformly elliptic with $C^{0, α} (\overline{Ω})$ coefficients and $c \leq 0$ . Then for every $f \in C^{0, α} (\overline{Ω})$ and $φ \in C^{2, α} (\partial Ω)$ the Dirichlet problem $Lu = f$ in $Ω$ , $u = φ$ on $\partial Ω$ has a unique solution $u \in C^{2, α} (\overline{Ω})$ ^{[Gilbarg-Trudinger 2001 §6.3]}. Uniqueness is the maximum principle 02.17.02; existence runs the continuity method of Exercise 8 from $Δ$ to $L$ , the global Schauder estimate of Theorem 3 supplying the uniform a priori bound that keeps the invertibility set both open and closed. This is the structural reason Schauder estimates are stated before existence is known: the estimate is an a priori inequality on hypothetical solutions, and existence is then bootstrapped from it.

Theorem 5 (higher regularity and bootstrap). If in addition $a^{ij}, b^{i}, c \in C^{k, α} (\overline{Ω})$ and $f \in C^{k, α}$ , then a solution $u$ of $Lu = f$ lies in $C_{loc}^{k + 2, α}$ , with the corresponding estimate. Differentiating the equation reduces each order to the base case: $D_{ℓ} u$ solves an elliptic equation with $C^{k - 1, α}$ data, so the interior estimate applies inductively. In particular $C^{\infty}$ coefficients and source force $u \in C^{\infty}$ , and real-analytic data force $u$ real-analytic (Morrey-Nirenberg), recovering the elliptic regularity that the De Giorgi-Nash-Moser theory 02.16.04 also reaches for divergence-form equations with measurable coefficients but only as far as the input regularity allows.

Theorem 6 (Schauder estimates for systems and the complementing condition). For elliptic systems $\sum_{β} L_{α β} u^{β} = f^{α}$ , the scalar Schauder estimate generalises provided the Legendre-Hadamard ellipticity condition and, near the boundary, the Agmon-Douglis-Nirenberg complementing (Lopatinski-Shapiro) condition on the boundary operators hold ^{[Agmon-Douglis-Nirenberg 1959]}. The complementing condition is the algebraic compatibility of the boundary symbol with the interior symbol that makes the half-space model problem uniquely solvable in Hölder spaces; without it the boundary estimate fails even for constant-coefficient systems.

Theorem 7 (scaling-invariant proof; Simon 1997). The interior estimate admits a proof with no explicit singular-integral kernel and no Campanato characterization, using only the scaling structure and a Liouville theorem: if the estimate failed, a blow-up sequence would converge to a global solution of a constant-coefficient equation with subquadratic growth, which a Liouville theorem forces to be a quadratic polynomial, contradicting the normalisation ^{[Simon 1997]}. This compactness-and-contradiction method is the most transferable, extending verbatim to fully nonlinear equations in the Caffarelli-Cabré theory and to minimal-surface and harmonic-map regularity.

Synthesis. Schauder theory is the foundational reason elliptic equations with Hölder data have Hölder second derivatives, and the entire structure rests on one mechanism: at small scales a variable-coefficient operator is a constant-coefficient operator plus an error that shrinks like the radius to the power $α$ , so the constant-coefficient Newtonian-potential bound of 02.13.02 propagates to the general operator with a controllable perturbation. This is exactly the statement that the second derivatives are no rougher than the source, and putting these together, the interior estimate, the Campanato integral reformulation, and the boundary estimate are three faces of the single regularity gain, each adapted to a different need: the potential-theoretic proof is explicit, the Campanato proof is robust under the perturbation, and the boundary proof flattens the geometry. The central insight is that the a priori estimate precedes existence: the method of continuity converts the uniform Schauder bound into surjectivity of the operator, so the estimate is dual to solvability, and this is exactly the foundational reason the Dirichlet problem for general second-order elliptic operators is well-posed in the Hölder scale.

The bridge is the matched pairing of source regularity and solution regularity on the fractional scale, which generalises in three directions. The Calderón-Zygmund theory of 02.13.02 is the Lebesgue-scale companion, trading $C^{0, α}$ for $L^{p}$ and $C^{2, α}$ for $W^{2, p}$ , the two estimates meeting through the embedding $W^{2, p} ↪ C^{0, α}$ for $p$ large. The De Giorgi-Nash-Moser theory 02.16.04 is the divergence-form sibling that survives merely measurable coefficients, where the explicit Schauder perturbation is unavailable and the Harnack inequality of 02.17.02 does the work instead. The Caffarelli-Cabré theory carries the frozen-coefficient and Campanato ideas to fully nonlinear elliptic equations, where the comparison principle replaces linearity and the same $C^{2, α}$ gain holds for viscosity solutions under a smallness condition on the oscillation of the operator.

Full proof set Master

Proposition 1 (Hölder spaces are Banach). For $k \geq 0$ and $α \in (0, 1]$ , $(C^{k, α} (\overline{Ω}), ∥ \cdot ∥_{C^{k, α}})$ is a Banach space.

Proof. Let $(u_{m})$ be Cauchy in $C^{k, α}$ . Each $∥ D^{β} u_{m} - D^{β} u_{m^{'}} ∥_{C^{0}} \to 0$ for $∣ β ∣ \leq k$ , so by completeness of $C^{0} (\overline{Ω})$ (the uniform norm on a compact set, an instance of $L^{\infty}$ completeness 02.07.06) there are continuous limits $u^{(β)} = lim_{m} D^{β} u_{m}$ , uniform in $x$ . Uniform convergence of functions together with uniform convergence of their derivatives gives $u^{(β)} = D^{β} u^{(0)}$ for $∣ β ∣ \leq k$ , so $u := u^{(0)} \in C^{k} (\overline{Ω})$ and $D^{β} u_{m} \to D^{β} u$ uniformly. It remains to control the top seminorms. For $∣ β ∣ = k$ and any $x \neq = y$ , $\frac{∣ D ^{β} u ( x ) - D ^{β} u ( y ) ∣}{∣ x - y ∣ ^{α}} = m lim \frac{∣ D ^{β} u _{m} ( x ) - D ^{β} u _{m} ( y ) ∣}{∣ x - y ∣ ^{α}} \leq m lim sup [D^{β} u_{m}]_{α} < \infty,$ since the Cauchy sequence has bounded seminorms; taking the supremum over $x \neq = y$ shows $u \in C^{k, α}$ . Finally, given $ε$ pick $N$ with $[D^{β} (u_{m} - u_{m^{'}})]_{α} < ε$ for $m, m^{'} \geq N$ ; letting $m^{'} \to \infty$ inside the difference quotient gives $[D^{β} (u_{m} - u)]_{α} \leq ε$ for $m \geq N$ , so $u_{m} \to u$ in $C^{k, α}$ . $□$

Proposition 2 (model $C^{2, α}$ estimate for the Laplacian). Let $w \in C^{2, α} (\overline{B_{2}})$ solve $- Δ w = g$ on $B_{2} \subseteq R^{n}$ with $g \in C^{0, α} (\overline{B_{2}})$ . Then $[D^{2} w]_{α; B_{1}} \leq C (n, α) (∥ w ∥_{C^{0} (B_{2})} + [g]_{α; B_{2}}) .$

Proof. Fix a cutoff $η \in C_{c}^{\infty} (B_{2})$ with $η \equiv 1$ on $B_{3/2}$ . The function $v = Φ * (η g)$ , with $Φ$ the Newtonian potential kernel of 02.13.02, satisfies $- Δ v = η g = g$ on $B_{3/2}$ , so $h := w - v$ is harmonic on $B_{3/2}$ . Interior derivative estimates for harmonic functions give $[D^{2} h]_{α; B_{1}} \leq C ∥ h ∥_{C^{0} (B_{3/2})} \leq C (∥ w ∥_{C^{0} (B_{2})} + ∥ v ∥_{C^{0} (B_{3/2})})$ , and $∥ v ∥_{C^{0}} \leq C ∥ g ∥_{C^{0}} \leq C (∥ g ∥_{C^{0}})$ . For the potential part, differentiate twice: $D_{ij} v (x) = \int_{B_{2}} D_{ij} Φ (x - y) η (y) g (y) d y$ , interpreted as a principal value with the correction term $- \frac{1}{n} δ_{ij} (η g) (x)$ . Writing the standard Hölder estimate for the second derivative of a Newtonian potential, one splits, for $∣ x - x^{'} ∣ = ρ$ , the integral over $B_{2 ρ} (x)$ and its complement: the near part is bounded by $C [η g]_{α} ρ^{α}$ using $\int_{B_{2 ρ}} ∣ D_{ij} Φ∣ d y \leq C ρ$ against the oscillation $g (y) - g (x)$ , and the far part by the mean-value bound $∣ D_{ij} Φ (x - y) - D_{ij} Φ (x^{'} - y) ∣ \leq C ρ ∣ x - y ∣^{- n - 1}$ integrated against $[η g]_{α} ∣ x - y ∣^{α}$ , giving $C [η g]_{α} ρ^{α}$ as well. Hence $[D^{2} v]_{α; B_{1}} \leq C [η g]_{α} \leq C (∥ g ∥_{C^{0}} + [g]_{α})$ by the product rule (Exercise 3). Combining the harmonic and potential parts yields the stated bound. $□$

Proposition 3 (interior estimate on a small ball). Let $L = \sum a^{ij} D_{ij} + \sum b^{i} D_{i} + c$ be uniformly elliptic on $B_{1} \subseteq R^{n}$ with coefficients in $C^{0, α} (\overline{B_{1}})$ of norm $\leq Λ$ . There is $r_{*} = r_{*} (n, α, λ, Λ) \in (0, 1]$ such that every $u \in C^{2, α} (\overline{B_{r_{*}}})$ with $Lu = f$ obeys $∥ u ∥_{C^{2, α} (B_{r_{*} /2})} \leq C (∥ u ∥_{C^{0} (B_{r_{*}})} + ∥ f ∥_{C^{0, α} (B_{r_{*}})}) .$

Proof. Fix the centre $0$ and freeze $a_{0}^{ij} = a^{ij} (0)$ . The constant-coefficient operator $L_{0} = \sum a_{0}^{ij} D_{ij}$ becomes $- Δ$ (up to a positive factor and a fixed linear change of variables bounded by $Λ/ λ$ ) under $y = A x$ with $A^{⊤} A = [a_{0}^{ij}]^{- 1}$ ; Proposition 2 transports to give, for $L_{0} u = F$ , $[D^{2} u]_{α; B_{r /2}} \leq C_{0} (r^{- 2 - α} ∥ u ∥_{C^{0} (B_{r})} + [F]_{α; B_{r}}),$ the $r$ -powers fixed by scaling $x \mapsto r x$ . Now $L_{0} u = f - \sum (a^{ij} (x) - a_{0}^{ij}) D_{ij} u - \sum b^{i} D_{i} u - c u =: F$ . By the product rule and $∣ a^{ij} (x) - a_{0}^{ij} ∣ \leq [a^{ij}]_{α} r^{α}$ on $B_{r}$ , $[F]_{α; B_{r}} \leq [f]_{α} + C r^{α} [D^{2} u]_{α; B_{r}} + C (∥ D^{2} u ∥_{C^{0} (B_{r})} + ∥ D u ∥_{C^{0, α} (B_{r})} + ∥ u ∥_{C^{0, α} (B_{r})}) .$ Insert the interpolation $∥ D^{2} u ∥_{C^{0}} + ∥ D u ∥_{C^{0, α}} + ∥ u ∥_{C^{0, α}} \leq δ [D^{2} u]_{α} + C_{δ} ∥ u ∥_{C^{0}}$ (Exercise 4, applied to each intermediate norm). Choosing first $δ$ with $C_{0} C δ \leq \frac{1}{4}$ and then $r_{*}$ with $C_{0} C r_{*}^{α} \leq \frac{1}{4}$ , the terms $C_{0} (C r_{*}^{α} + C δ) [D^{2} u]_{α} \leq \frac{1}{2} [D^{2} u]_{α}$ are absorbed after reconciling the radii $r_{*} /2$ and $r_{*}$ by the standard iteration/covering lemma. What remains is $[D^{2} u]_{α; B_{r_{*} /2}} \leq C ([f]_{α} + ∥ u ∥_{C^{0}})$ ; adding the lower-order norms by interpolation gives the full $C^{2, α}$ bound. $□$

Proposition 4 (the lower-order term cannot be dropped when $c \neq \leq 0$ ). There is a uniformly elliptic operator $L$ with smooth coefficients on a bounded domain $Ω$ for which no inequality $∥ u ∥_{C^{2, α} (Ω^{'})} \leq C ∥ Lu ∥_{C^{0, α} (Ω)}$ can hold with $C$ independent of $u$ .

Proof. Take $Ω = (0, π) \subseteq R$ and $Lu = u^{''} + u$ , with $c = 1 > 0$ , and $Ω^{'} = (π /4, 3 π /4) ⋐ Ω$ . The function $u (x) = sin x$ satisfies $Lu = - sin x + sin x = 0$ , so $∥ Lu ∥_{C^{0, α}} = 0$ , while $∥ u ∥_{C^{2, α} (Ω^{'})} \geq ∥ u ∥_{C^{0} (Ω^{'})} = sin (3 π /4) > 0$ . Any inequality $∥ u ∥_{C^{2, α}} \leq C ∥ Lu ∥_{C^{0, α}}$ would force $∥ u ∥_{C^{2, α}} \leq 0$ , a contradiction. The kernel of $L$ is nontrivial because $c = 1$ equals the principal Dirichlet eigenvalue of $- u^{''}$ on $(0, π)$ , the same threshold at which the maximum principle of 02.17.02 fails, and the $∥ u ∥_{C^{0}}$ term on the right of the Schauder estimate is exactly what accounts for this kernel. $□$

Connections Master

The Newtonian-potential second-derivative bound of 02.13.02 is the analytic engine of the entire theory: Proposition 2 is the Schauder estimate for the Laplacian, proved by differentiating the convolution $Φ * f$ twice and estimating the singular kernel $D_{ij} Φ$ against the Hölder modulus of the source. The Poisson unit explicitly deferred the $C^{α}$ -data case to Schauder theory; this unit closes that forward reference, supplying the $C^{2, α}$ regularity that continuous data alone could not provide.
The maximum principle for general elliptic operators 02.17.02 supplies the $∥ u ∥_{C^{0}}$ control that turns the a priori Schauder estimate into a closed regularity-and-existence statement: when $c \leq 0$ the maximum principle bounds $∥ u ∥_{C^{0}}$ by the data, and in the method of continuity it is the uniform a priori bound that keeps the invertibility set open and closed. The $sin x$ obstruction of Proposition 4 is the same principal-eigenvalue threshold at which both the maximum principle and the kernel-free Schauder estimate fail.
The $L^{p}$ -space theory of 02.07.06 underlies both the Banach completeness of the Hölder scale (Proposition 1, resting on $L^{\infty}$ completeness) and the Campanato characterization, whose seminorm is an $L^{2}$ mean-oscillation quantity; the Calderón-Zygmund $W^{2, p}$ estimates that run parallel to Schauder's live natively on the $L^{p}$ scale, and the Sobolev embedding $W^{2, p} ↪ C^{0, α}$ connects the two regularity scales quantitatively.
The De Giorgi-Nash-Moser regularity theory 02.16.04 is the divergence-form, measurable-coefficient sibling of Schauder theory: where Schauder needs Hölder coefficients and gains two full derivatives, De Giorgi-Nash-Moser tolerates merely bounded measurable coefficients and gains only $C^{0, β}$ for an uncontrolled small $β$ , using the Harnack inequality in place of the frozen-coefficient perturbation. Together the two theories cover the two normal forms of second-order elliptic operators.

Historical & philosophical context Master

The Hölder scale is named for Otto Hölder, whose 1882 Tübingen dissertation on potential theory ^{[Hölder 1882]} introduced the modulus-of-continuity condition $∣ u (x) - u (y) ∣ \leq M ∣ x - y ∣^{α}$ to study the second derivatives of the Newtonian potential, precisely the question Schauder theory later answered in full. Arthur Korn, in a sequence of papers around 1907-1909 ^{[Korn 1909]}, developed the freezing-of-coefficients technique for the second derivatives of potentials, establishing the model estimates that the perturbation argument rests on; the frozen-coefficient method is for this reason sometimes called the Korn trick.

Juliusz Schauder unified these strands in two 1934 papers, the Mathematische Zeitschrift article ^{[Schauder 1934]} establishing the a priori $C^{2, α}$ estimates and the Studia Mathematica note ^{[Schauder 1934b]} deriving existence for the Dirichlet problem by combining the estimates with the continuity method and his own fixed-point theory. Schauder's insight was that the a priori inequality, an estimate on hypothetical solutions, is logically prior to existence and converts into it through a topological argument; this inversion of the usual order remains the organising principle of linear elliptic theory. Schauder died in 1943, killed by the Gestapo in occupied Lwów.

The theory was recast twice. Sergio Campanato's 1963 paper in the Annali della Scuola Normale Superiore di Pisa ^{[Campanato 1963]} characterized Hölder continuity by the power-law decay of mean oscillation, replacing the singular-integral estimates with an $L^{2}$ energy-decay argument and making the method robust enough to extend to systems and to nonlinear problems. Agmon, Douglis, and Nirenberg in their 1959 Communications on Pure and Applied Mathematics paper ^{[Agmon-Douglis-Nirenberg 1959]} extended the boundary estimates to general elliptic systems under the complementing condition, fixing the algebraic compatibility between interior and boundary symbols that the scalar case had concealed. Leon Simon's 1997 scaling proof ^{[Simon 1997]} reduced the interior estimate to a Liouville theorem via blow-up compactness, the form in which the theory transferred to the Caffarelli-Cabré regularity theory for fully nonlinear equations.

Bibliography Master

@article{Schauder1934,
  author  = {Schauder, Juliusz},
  title   = {\"Uber lineare elliptische Differentialgleichungen zweiter Ordnung},
  journal = {Mathematische Zeitschrift},
  volume  = {38},
  year    = {1934},
  pages   = {257--282}
}

@article{Schauder1934b,
  author  = {Schauder, Juliusz},
  title   = {Numerische Absch\"atzungen in elliptischen linearen Differentialgleichungen},
  journal = {Studia Mathematica},
  volume  = {5},
  year    = {1934},
  pages   = {34--42}
}

@phdthesis{Hoelder1882,
  author  = {H\"older, Otto},
  title   = {Beitr\"age zur Potentialtheorie},
  school  = {Universit\"at T\"ubingen},
  year    = {1882}
}

@article{Korn1909,
  author  = {Korn, Arthur},
  title   = {\"Uber Minimalfl\"achen, deren Randkurven wenig von ebenen Kurven abweichen},
  journal = {Abhandlungen der K\"oniglich Preussischen Akademie der Wissenschaften},
  year    = {1909}
}

@article{Campanato1963,
  author  = {Campanato, Sergio},
  title   = {Propriet\`a di h\"olderianit\`a di alcune classi di funzioni},
  journal = {Annali della Scuola Normale Superiore di Pisa},
  volume  = {17},
  year    = {1963},
  pages   = {175--188}
}

@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{Simon1997,
  author  = {Simon, Leon},
  title   = {Schauder estimates by scaling},
  journal = {Calculus of Variations and Partial Differential Equations},
  volume  = {5},
  year    = {1997},
  pages   = {391--407}
}

@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}
}

@book{CaffarelliCabre1995,
  author    = {Caffarelli, Luis A. and Cabr\'e, Xavier},
  title     = {Fully Nonlinear Elliptic Equations},
  series    = {AMS Colloquium Publications 43},
  publisher = {American Mathematical Society},
  year      = {1995}
}

Prerequisites

02.13.02
02.17.02
02.07.06

Tier anchors

beginner: Han-Lin, Elliptic Partial Differential Equations, 2e (AMS/Courant Lecture Notes 1, 2011), §3; physics-anchored as the regularity gain from smooth-enough sources
intermediate: Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 2001), §4, §6.1; Han-Lin §3.1-3.4
master: Gilbarg-Trudinger §4-6; Caffarelli-Cabré, Fully Nonlinear Elliptic Equations (AMS Colloquium 43, 1995), §4; Giaquinta-Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems (Edizioni della Normale 2012), §5 (Campanato)

References

Schauder — Über lineare elliptische Differentialgleichungen zweiter Ordnung · Mathematische Zeitschrift 38 (1934), 257-282
Schauder — Numerische Abschätzungen in elliptischen linearen Differentialgleichungen · Studia Mathematica 5 (1934), 34-42
Hölder — Beiträge zur Potentialtheorie · Dissertation, Tübingen (1882)
Korn — Über Minimalflächen, deren Randkurven wenig von ebenen Kurven abweichen · Abhandlungen der Königlich Preussischen Akademie der Wissenschaften (1909)
Campanato — Proprietà di hölderianità di alcune classi di funzioni · Annali della Scuola Normale Superiore di Pisa 17 (1963), 175-188
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
Gilbarg-Trudinger — Elliptic Partial Differential Equations of Second Order, 2e · Springer Grundlehren 224 (2001), §4, §6
Caffarelli-Cabré — Fully Nonlinear Elliptic Equations · AMS Colloquium Publications 43 (1995), §4-5
Simon — Schauder estimates by scaling · Calculus of Variations and Partial Differential Equations 5 (1997), 391-407

Estimated time

beginner: 20m
intermediate: 60m
master: 95m