02.17.09 · analysis / elliptic-regularity

Quasilinear Elliptic Equations: Gradient Estimates and Existence by Leray-Schauder

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Anchor (Master): Gilbarg-Trudinger §10-§11, §13, §15; Ladyzhenskaya-Uraltseva, Linear and Quasilinear Elliptic Equations (Academic Press 1968), Ch. IV; Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables (Phil. Trans. R. Soc. A 264, 1969)

Intuition Beginner

Picture a soap film stretched across a bent wire loop. The film settles into the shape that makes its area as small as possible, and the equation it obeys is no longer the simple averaging law of a flat membrane: how strongly the film resists bending at a point depends on how steeply it is already tilted there. An equation whose coefficients depend on the slope of the unknown itself, not just on position, is called quasilinear. The minimal surface is the headline example, and the whole difficulty of the subject lives in that feedback loop between the solution and the coefficients that govern it.

When the coefficients depend on the slope, a new danger appears: the slope might run away. A solution could in principle develop an ever-steeper cliff somewhere inside the region, and if the slope blows up, the equation degenerates and the whole theory collapses. The central achievement of the subject is a guarantee that this never happens for the right kind of equation: the steepness of the solution in the interior is capped in advance by a number computed from the size of the solution and the data on the boundary. This cap is the gradient estimate, and it is the keystone on which everything else rests.

Why does a slope cap unlock existence of solutions? Because of a beautiful strategy: deform the hard equation continuously into an easy one you can already solve, and show that no solution along the deformation can escape to infinity. If every candidate solution stays trapped inside a fixed-size box — bounded in value, bounded in slope, bounded in the smoothness of its slope — then a topological fixed-point principle guarantees a genuine solution exists at the hard end. The slope cap is exactly the wall of that box.

The one-sentence takeaway: for the right quasilinear equations, the slope of any solution is bounded in advance by the boundary data, and that single advance bound, fed into a fixed-point machine, produces a solution out of thin air.

Visual Beginner

The picture to hold is a box in three dimensions whose three walls are the three a priori bounds, with every candidate solution trapped inside it no matter where along the deformation it sits.

The three walls are the three estimates, proved in order: first the value is capped, then the slope is capped using the value cap, then the smoothness of the slope is capped using the slope cap. The slider is the continuous deformation from the easy equation to the hard one. The single most important fact is that the box does not depend on which candidate or which slider position you look at: it is fixed once and for all by the data. That fixed box is what the topological fixed-point theorem needs as its input.

Worked example Beginner

We make the slope-cap idea concrete on the one-dimensional minimal surface equation, where everything is plain calculus. In one variable the minimal surface equation says the curvature term $\frac{d}{d x} (\frac{u ^{'} ( x )}{1 + u ^{'} ( x ) ^{2}}) = 0,$ so the quantity inside the bracket is constant. Call that constant $c$ .

Step 1. Read off the slope. Setting $u^{'} / 1 + (u^{'})^{2} = c$ and solving for $u^{'}$ gives $u^{'} = c / 1 - c^{2}$ , a constant. So in one dimension every solution is a straight line. That is the slope cap in its simplest form: the slope is literally constant.

Step 2. Watch the cap depend on the data. Suppose the line runs from height $0$ at $x = 0$ to height $H$ at $x = 1$ . Then the slope is exactly $H$ , the rise over the run. The interior slope is pinned by the boundary heights: bigger boundary spread means a bigger slope, but always a finite one.

Step 3. See why the bracket matters. The bracketed quantity $c = u^{'} / 1 + (u^{'})^{2}$ is always between $- 1$ and $1$ , no matter how large $u^{'}$ is. This is the feature that tames the equation: the curvature operator saturates. As the slope grows, the resisting term levels off rather than exploding.

Step 4. Put in numbers. If $H = 3$ , the slope is $3$ , and the bracket value is $c = 3/ 1 + 9 = 3/ 10 \approx 0.95$ , comfortably below $1$ . A boundary spread of $3$ over a unit interval produces a slope of $3$ — capped, finite, and computed entirely from the boundary data.

What this tells us: in one dimension the slope cap is automatic because solutions are straight lines, but the principle survives in every dimension — the interior slope is controlled by the boundary spread. The hard theorems prove the same kind of cap when the solution is a genuinely curved surface and the slope is no longer constant.

Check your understanding Beginner

Exercise (easy, multiple choice).

An elliptic equation is called quasilinear when:

A. it is linear in the unknown and all its derivatives. B. its top-order coefficients may depend on the unknown and its slope, but it is still linear in the highest derivatives. C. it has no solutions. D. its coefficients are constant.

Hint

The minimal surface equation is the model; its coefficients depend on the slope $u^{'}$ , yet the second derivatives still appear linearly.

Answer

B. A quasilinear equation is linear in its highest-order derivatives but its coefficients may depend on the unknown and its lower-order derivatives (here, the slope). The minimal surface equation is exactly this: the second derivatives appear linearly, but multiplied by coefficients built from the gradient. Feedback-correct: the feedback between solution slope and coefficients is the whole subject. Feedback-wrong: A describes a fully linear equation; C is false; D describes the simplest constant-coefficient case, which is linear, not quasilinear.

Exercise (easy, true-false).

For the right quasilinear equations, the interior slope of a solution can be bounded in advance by a number computed from the boundary data, before any solution is constructed.

Hint

This advance cap is called the a priori gradient estimate; it is the keystone of the existence theory.

Answer

True. The a priori gradient estimate bounds $sup ∣ D u ∣$ in terms of $sup ∣ u ∣$ and the boundary data, with a constant that does not depend on the particular solution. This advance bound is exactly what the fixed-point existence machine needs: it is the wall that traps every candidate solution in a fixed box. Feedback-correct: "a priori" means the bound holds for any solution, known before existence is proved. Feedback-wrong: without such a bound the slope could run away and the existence argument would fail.

Formal definition Intermediate+

Let $Ω \subseteq R^{n}$ be a bounded domain. A second-order operator is quasilinear when it is linear in the second derivatives with coefficients depending on $(x, u, D u)$ . The divergence (variational) form treated here is $Q u := div A (x, u, D u) + B (x, u, D u) = i = 1 \sum n D_{i} A_{i} (x, u, D u) + B (x, u, D u),$ with $A = (A_{1}, \dots, A_{n}) : Ω \times R \times R^{n} \to R^{n}$ and $B : Ω \times R \times R^{n} \to R$ . A function $u \in C^{1} (Ω) \cap C^{0} (\overline{Ω})$ (or $u \in W^{1, m} (Ω)$ in the weak sense) is a solution of $Q u = 0$ when $\int_{Ω} (A (x, u, D u) \cdot D φ - B (x, u, D u) φ) d x = 0 for all φ \in C_{c}^{\infty} (Ω) .$ The operator is elliptic at $u$ when the matrix $(\partial A_{i} / \partial p_{j}) (x, u, D u)$ is positive definite. The model is the prescribed mean curvature equation, $A (x, u, p) = p / 1 + ∣ p ∣^{2}$ and $B = n H (x)$ , whose $H \equiv 0$ case is the minimal surface equation $div (D u / 1 + ∣ D u ∣^{2}) = 0$ ; the $p$ -Laplacian $div (∣ D u ∣^{m - 2} D u) = 0$ is the other touchstone ^{[Gilbarg-Trudinger 2001 §10.1]}.

The estimates require structure conditions of natural growth: there are constants $μ > 0$ , $0 < λ \leq Λ$ , and an exponent $m > 1$ such that, writing $p = D u$ , $i, j \sum \frac{\partial A _{i}}{\partial p _{j}} ξ_{i} ξ_{j} \geq λ (1 + ∣ p ∣)^{m - 2} ∣ ξ ∣^{2}, \frac{\partial A _{i}}{\partial p _{j}} \leq Λ (1 + ∣ p ∣)^{m - 2},$ $∣ A (x, u, p) ∣ \leq μ (1 + ∣ p ∣)^{m - 1}, ∣ B (x, u, p) ∣ \leq μ (1 + ∣ p ∣)^{m} .$ The first pair is uniform ellipticity relative to the $m$ -growth; the lower bound is what forces a slope cap. The minimal surface equation has $m = 2$ but the ellipticity degenerates, $\partial A_{i} / \partial p_{j} \sim (1 + ∣ p ∣^{2})^{- 1/2}$ decaying as $∣ p ∣ \to \infty$ , which is the source of its delicacy; the $p$ -Laplacian has $m =$ the exponent and degenerates at $p = 0$ .

Definition (a priori gradient estimate). A class of solutions of $Q u = 0$ on $Ω$ satisfies an interior gradient estimate when, for $Ω^{'} ⋐ Ω$ , there is a constant $C$ depending only on $n$ , the structure constants, $dist (Ω^{'}, \partial Ω)$ , and $M = sup_{Ω} ∣ u ∣$ , with $sup_{Ω^{'}} ∣ D u ∣ \leq C$ ; and a global gradient estimate when, for solutions with prescribed boundary data $u = φ$ on $\partial Ω$ , $sup_{Ω} ∣ D u ∣ \leq C (n, structure, M, Ω, φ)$ , independent of the particular solution.

Definition (Leray-Schauder / Schaefer form). Let $B$ be a Banach space and $T : B \to B$ compact. Schaefer's fixed-point theorem asserts: if the set ${x \in B : x = σ T x for some σ \in [0, 1]}$ is bounded, then $T$ has a fixed point ^{[Schaefer 1955]}. This is the form of the Leray-Schauder theorem ^{[Leray-Schauder 1934]} in which existence reduces to an a priori bound on the solution set of the homotopy $x = σ T x$ , with no construction of the solution.

Counterexamples to common slips Intermediate+

Quasilinear is not semilinear. In a semilinear equation $Δ u = f (x, u, D u)$ the top-order part is the fixed Laplacian; in a quasilinear equation the top-order coefficients themselves depend on $D u$ . The minimal surface operator is genuinely quasilinear, and the gradient dependence of the leading coefficients is exactly what can degenerate.
The gradient estimate is not a consequence of the maximum principle alone. The maximum principle bounds $sup ∣ u ∣$ , but a bounded function can have arbitrarily large slope. The gradient bound is a separate, harder theorem, obtained by differentiating the equation and applying the De Giorgi-Nash-Moser machinery 02.17.07 to $w = ∣ D u ∣^{2}$ or to individual derivatives $D_{k} u$ .
Natural growth is sharp, not decorative. If $B$ grows faster than $(1 + ∣ p ∣)^{m}$ — say like $∣ p ∣^{m + ε}$ — the gradient estimate fails and solutions can fail to exist for large data. The exponent $m$ on $B$ matches the divergence structure precisely so the cutoff-function test integrals close; faster growth breaks the absorption.
Schaefer's hypothesis is the a priori bound, not solvability of the homotopy at every $σ$ . One does not solve $x = σ T x$ for each $σ$ ; one bounds the set of all solutions that exist, uniformly in $σ$ , and lets the topological degree do the rest. Confusing "bound the solution set" with "solve at each $σ$ " inverts the logic of the method.

Key theorem with proof Intermediate+

Theorem (existence by Leray-Schauder via a priori bounds; Gilbarg-Trudinger §11.4). Let $Ω$ be a bounded $C^{2, α}$ domain and $Q u = div A (x, u, D u) + B (x, u, D u)$ a quasilinear elliptic operator with $A, B \in C^{1, α}$ satisfying the natural structure conditions and $\partial B / \partial u \leq 0$ . Suppose there is a constant $M$ , independent of $u$ and of $σ \in [0, 1]$ , such that every $C^{2, α} (\overline{Ω})$ solution of the homotopy family $Q_{σ} u := div A (x, u, D u) + σ B (x, u, D u) = 0 in Ω, u = σ φ on \partial Ω,$ satisfies $∥ u ∥_{C^{1, β} (\overline{Ω})} \leq M$ for some $β \in (0, 1)$ . Then the Dirichlet problem $Q u = 0$ , $u = φ$ on $\partial Ω$ , has a solution $u \in C^{2, α} (\overline{Ω})$ ^{[Gilbarg-Trudinger 2001 §11.4]} ^{[Serrin 1969]}. The crux is that the a priori $C^{1, β}$ bound is exactly the boundedness hypothesis Schaefer's theorem demands of the homotopy solution set.

Proof. Define the solution operator. Fix $β \in (0, α)$ and let $B = C^{1, β} (\overline{Ω})$ . For $v \in B$ , freeze the nonlinearity: the coefficients $a^{ij} (x) = \partial A_{i} / \partial p_{j} (x, v, D v)$ and the data $f (x) = - B (x, v, D v) + (lower terms from A)$ are now functions of $x$ alone, of class $C^{0, β} (\overline{Ω})$ , and the linear Dirichlet problem $a^{ij} (x) D_{ij} u + (first order) = f (x) in Ω, u = φ on \partial Ω,$ has a unique solution $u = T v \in C^{2, β} (\overline{Ω})$ by the linear Schauder existence theory 02.17.05. This defines $T : B \to B$ , and a fixed point $u = T u$ is precisely a solution of $Q u = 0$ .

Step 1: $T$ is compact. The linear Schauder estimate 02.17.05 gives $∥ T v ∥_{C^{2, β} (\overline{Ω})} \leq C (∥ f ∥_{C^{0, β}} + ∥ φ ∥_{C^{2, β}})$ , where $C$ depends on the ellipticity and $C^{0, β}$ norms of the frozen coefficients, hence on $∥ v ∥_{C^{1, β}}$ . Thus $T$ maps bounded sets of $B = C^{1, β}$ into bounded sets of $C^{2, β}$ . The embedding $C^{2, β} (\overline{Ω}) ↪ C^{1, β} (\overline{Ω})$ is compact by Arzelà-Ascoli on a bounded domain 02.11.05 (the inclusion is a compact operator: bounded $C^{2, β}$ sets are precompact in $C^{1, β}$ ). Composing, $T$ sends bounded sets into precompact sets, so $T$ is compact. Continuity of $T$ follows from continuous dependence of the linear solution on its coefficients and data.

Step 2: the homotopy is $x = σ T x$ . For $σ \in [0, 1]$ a fixed point of $σ T$ solves $a^{ij} (x, u, D u) D_{ij} u + \dots = σ f$ with boundary data $σ φ$ , which after unfreezing is exactly the family $Q_{σ} u = 0$ , $u = σ φ$ . At $σ = 0$ the only solution is $u \equiv 0$ . The hypothesis says every solution of $u = σ T u$ over all $σ \in [0, 1]$ satisfies $∥ u ∥_{C^{1, β}} \leq M$ , so the set ${u : u = σ T u, σ \in [0, 1]}$ is bounded in $B$ .

Step 3: apply Schaefer. $T$ is compact and the homotopy solution set is bounded, so by Schaefer's theorem ^{[Schaefer 1955]} $T$ has a fixed point $u \in B$ . Unfreezing, $u$ solves $Q u = 0$ , $u = φ$ on $\partial Ω$ . Linear Schauder regularity bootstraps $u \in C^{2, β}$ and then $C^{2, α}$ since $A, B \in C^{1, α}$ . $□$

Bridge. This theorem is the foundational reason quasilinear existence is entirely a question of a priori estimates: the compactness of $T$ and the topological degree are automatic from the linear theory of 02.17.05 and the compact embedding of 02.11.05, so all the analytic work is pushed into the single uniform bound $∥ u ∥_{C^{1, β}} \leq M$ . This builds toward the gradient estimates proved in the Advanced results, which supply exactly that bound, and it appears again in 02.18.04 where the same Leray-Schauder scheme yields minimal surfaces from the area functional. The central insight is that existence is dual to an a priori estimate — this is exactly the nonlinear generalisation of the linear method of continuity of 02.17.05, where the Neumann-series openness is replaced by the Leray-Schauder degree and the uniform linear estimate is replaced by a nonlinear, solution-uniform $C^{1, β}$ bound. Putting these together, the chain of estimates $sup ∣ u ∣ \to sup ∣ D u ∣ \to [D u]_{C^{β}}$ generalises the linear sup-bound-plus-Schauder-bound chain, and the bridge is the compact solution operator $T$ , which converts each estimate into a wall of the bounded set that Schaefer's theorem turns into a fixed point.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the divergence-form minimal surface equation as the Euler-Lagrange equation of the area functional $A [u] = \int_{Ω} 1 + ∣ D u ∣^{2} d x$ .

Hint

Compute $\frac{d}{d t}_{t = 0} A [u + tφ]$ for $φ \in C_{c}^{\infty} (Ω)$ and integrate by parts.

Answer

The first variation is $\frac{d}{d t}_{t = 0} \int_{Ω} 1 + ∣ D (u + tφ) ∣^{2} d x = \int_{Ω} \frac{D u \cdot D φ}{1 + ∣ D u ∣ ^{2}} d x .$ Integrating by parts and using $φ \in C_{c}^{\infty} (Ω)$ to drop the boundary term, $= - \int_{Ω} div (\frac{D u}{1 + ∣ D u ∣ ^{2}}) φ d x .$ Since this vanishes for all test functions $φ$ , the fundamental lemma of the calculus of variations forces $div (D u / 1 + ∣ D u ∣^{2}) = 0$ pointwise. This is the divergence form $div A (D u) = 0$ with $A (p) = p / 1 + ∣ p ∣^{2}$ , and identifies $B \equiv 0$ . The prescribed-mean-curvature case $div A (D u) = n H$ is the Euler-Lagrange equation of $A [u] + n \int_{Ω} H u d x$ .

Exercise 4 (medium, symbolic).

Show that the structure condition $\sum_{i, j} (\partial A_{i} / \partial p_{j}) ξ_{i} ξ_{j} \geq λ (1 + ∣ p ∣)^{m - 2} ∣ ξ ∣^{2}$ holds for the $p$ -Laplacian $A (p) = ∣ p ∣^{m - 2} p$ with $λ = min (1, m - 1)$ , and identify where the ellipticity degenerates.

Hint

Compute $\partial A_{i} / \partial p_{j} = ∣ p ∣^{m - 2} δ_{ij} + (m - 2) ∣ p ∣^{m - 4} p_{i} p_{j}$ and test against $ξ$ .

Answer

Differentiating $A_{i} = ∣ p ∣^{m - 2} p_{i}$ , $\frac{\partial A _{i}}{\partial p _{j}} = ∣ p ∣^{m - 2} δ_{ij} + (m - 2) ∣ p ∣^{m - 4} p_{i} p_{j} .$ Contracting with $ξ$ , $i, j \sum \frac{\partial A _{i}}{\partial p _{j}} ξ_{i} ξ_{j} = ∣ p ∣^{m - 2} ∣ ξ ∣^{2} + (m - 2) ∣ p ∣^{m - 4} (p \cdot ξ)^{2} .$ For $m \geq 2$ the second term is non-negative, so the sum is $\geq ∣ p ∣^{m - 2} ∣ ξ ∣^{2}$ . For $1 < m < 2$ use $(p \cdot ξ)^{2} \leq ∣ p ∣^{2} ∣ ξ ∣^{2}$ to get $\geq ∣ p ∣^{m - 2} ∣ ξ ∣^{2} + (m - 2) ∣ p ∣^{m - 2} ∣ ξ ∣^{2} = (m - 1) ∣ p ∣^{m - 2} ∣ ξ ∣^{2}$ . In both cases the bound is $min (1, m - 1) ∣ p ∣^{m - 2} ∣ ξ ∣^{2}$ , comparable to $(1 + ∣ p ∣)^{m - 2} ∣ ξ ∣^{2}$ for $∣ p ∣$ bounded below. The ellipticity **degenerates at $p = 0$ **: when $m > 2$ the coefficient $∣ p ∣^{m - 2}$ vanishes there, and when $m < 2$ it blows up, so the $p$ -Laplacian is non-uniformly elliptic exactly at critical points of $u$ .

Exercise 5 (medium, symbolic).

Sketch how the interior gradient bound for a solution is reduced to a maximum principle for $w = ∣ D u ∣^{2}$ . Differentiate $Q u = 0$ in the direction $x_{k}$ , multiply by $D_{k} u$ , and sum, to show $w$ is a subsolution of a linear elliptic equation with coefficients frozen at $D u$ .

Hint

Set $a^{ij} = \partial A_{i} / \partial p_{j}$ . Differentiating $D_{i} A_{i} (x, u, D u) = - B$ in $x_{k}$ produces a linear equation for $D_{k} u$ with leading coefficients $a^{ij}$ .

Answer

Write $a^{ij} (x) = (\partial A_{i} / \partial p_{j}) (x, u, D u)$ , evaluated along the fixed solution, so $a^{ij}$ is elliptic by the structure condition. Differentiating the equation $D_{i} A_{i} (x, u, D u) + B = 0$ in the direction $x_{k}$ gives $D_{i} (a^{ij} D_{j} (D_{k} u)) = - D_{i} (\partial_{x_{k}} A_{i} + \partial_{u} A_{i} D_{k} u) - \partial_{x_{k}} B - \partial_{u} B D_{k} u - \partial_{p_{j}} B D_{j} (D_{k} u),$ a linear divergence-form equation for the derivative $v_{k} = D_{k} u$ with leading matrix $a^{ij}$ . Multiplying by $v_{k} = D_{k} u$ , summing over $k$ , and using the product rule, the quantity $w = ∣ D u ∣^{2} = \sum_{k} v_{k}^{2}$ satisfies $D_{i} (a^{ij} D_{j} w) \geq - C (1 + w)^{?}$ , i.e. $w$ is a subsolution of a linear elliptic equation whose coefficients are frozen at $D u$ . The De Giorgi-Nash-Moser local boundedness theorem 02.17.07 applied to this linear equation bounds $sup w$ , hence $sup ∣ D u ∣$ , in the interior, completing the gradient estimate. The whole nonlinearity is absorbed into the (now fixed) coefficients $a^{ij}$ , which is why the linear theory of 02.17.07 suffices once the equation is differentiated.

Exercise 7 (hard, symbolic).

Prove the abstract Schaefer fixed-point theorem from the Leray-Schauder degree: if $T : B \to B$ is compact and the set $Σ = {x : x = σ T x, σ \in [0, 1]}$ is bounded, then $T$ has a fixed point. You may use that the Leray-Schauder degree $de g (I - σ T, B_{R}, 0)$ is a homotopy invariant for compact $T$ on a ball $B_{R}$ with no zeros on $\partial B_{R}$ .

Hint

Choose $R$ larger than the bound on $Σ$ . The homotopy $σ \mapsto I - σ T$ has no zeros on $\partial B_{R}$ . Compare the degree at $σ = 0$ and $σ = 1$ .

Answer

Since $Σ$ is bounded, pick $R > sup_{x \in Σ} ∥ x ∥ + 1$ , so no solution of $x = σ T x$ lies on $\partial B_{R}$ for any $σ \in [0, 1]$ ; equivalently $I - σ T$ has no zero on $\partial B_{R}$ . The map $H (σ, x) = x - σ T x$ is an admissible homotopy of compact perturbations of the identity (each $σ T$ is compact), with $0 \in / H (σ, \partial B_{R})$ for all $σ$ . By homotopy invariance of the Leray-Schauder degree, $de g (I - T, B_{R}, 0) = de g (I - 0 \cdot T, B_{R}, 0) = de g (I, B_{R}, 0) = 1.$ A nonzero degree forces the existence of a solution of $(I - T) x = 0$ in $B_{R}$ , i.e. a fixed point $x = T x$ . The boundedness of $Σ$ was used exactly to keep the homotopy zero-free on the boundary so the degree is defined and invariant; the seed value $1$ comes from the identity map $I$ at $σ = 0$ . This is the topological content that the linear method of continuity 02.17.05 captures via the Neumann series in the affine case.

Exercise 8 (hard, symbolic).

The minimal surface equation is solvable for arbitrary continuous boundary data on $Ω$ if and only if $\partial Ω$ has non-negative mean curvature everywhere (Jenkins-Serrin). Explain, via the global gradient estimate, why a boundary-curvature condition enters, and what fails on a non-convex-enough domain.

Hint

The global gradient bound needs control of $∣ D u ∣$ on $\partial Ω$ ; this boundary gradient bound is built from a barrier whose existence requires the boundary to curve the right way relative to the operator.

Answer

The interior gradient estimate is unconditional, but the global estimate requires bounding $sup_{\partial Ω} ∣ D u ∣$ , since the maximum of $∣ D u ∣$ for a minimal surface is attained on the boundary (the quantity $∣ D u ∣^{2}$ satisfies a maximum principle from Exercise 5). The boundary gradient bound is obtained by constructing barriers: upper and lower comparison surfaces that pin the normal derivative of $u$ at each boundary point. For the minimal surface operator such a barrier exists at a point $ξ \in \partial Ω$ precisely when the mean curvature of $\partial Ω$ at $ξ$ (with respect to the inner normal) is non-negative — the boundary must not curve away from the domain faster than the operator can follow. When $\partial Ω$ has a point of negative mean curvature, no barrier exists for large boundary data: one can prescribe $φ$ so steep that any minimal surface would need an infinite boundary slope, and the global gradient estimate fails, so by Schaefer's scheme existence is not guaranteed — and indeed solutions genuinely fail to exist (Bernstein, Finn). The mean-curvature condition is thus the exact compatibility between the boundary geometry and the degenerate ellipticity of the minimal surface operator, and it is sharp: non-negativity of $\partial Ω$ 's mean curvature is both necessary and sufficient for solvability with arbitrary continuous data ^{[Serrin 1969]}.

Advanced results Master

The quasilinear Dirichlet problem is solved by a single architecture: a chain of a priori estimates — sup bound, gradient bound, Hölder-gradient bound — feeding the Leray-Schauder fixed-point theorem. The estimates are where the analysis lives; the topology is a black box once the estimates close.

Theorem 1 (existence for the quasilinear Dirichlet problem; Gilbarg-Trudinger §11.4, Ladyzhenskaya-Uraltseva). Let $Ω$ be a bounded $C^{2, α}$ domain and $Q u = div A (x, u, D u) + B (x, u, D u)$ quasilinear elliptic, $A, B \in C^{1, α}$ , with the natural structure conditions and $\partial B / \partial u \leq 0$ . If every solution of the homotopy family $Q_{σ} u = 0$ , $u = σ φ$ , admits a uniform a priori bound $∥ u ∥_{C^{1, β} (\overline{Ω})} \leq M$ independent of $σ$ , then $Q u = 0$ , $u = φ$ on $\partial Ω$ , has a solution $u \in C^{2, α} (\overline{Ω})$ ^{[Gilbarg-Trudinger 2001 §11.4]} ^{[Ladyzhenskaya-Uraltseva 1968]}. The proof is the freeze-and-fix solution operator of the Key theorem; the linear engine is the Schauder solvability of 02.17.05 and the compactness is Arzelà-Ascoli of 02.11.05. The entire problem is thereby reduced to the four estimates below.

Theorem 2 (the a priori chain: sup, gradient, Hölder-gradient). Under the structure conditions, every solution of $Q u = 0$ , $u = φ$ on $\partial Ω$ , satisfies in order: (i) the maximum-principle sup bound $sup_{Ω} ∣ u ∣ \leq sup_{\partial Ω} ∣ φ ∣ + C diam (Ω)^{2}$ , from the divergence structure and $\partial B / \partial u \leq 0$ via the weak maximum principle 02.17.02; (ii) the boundary gradient bound $sup_{\partial Ω} ∣ D u ∣ \leq C (M, \partial Ω, φ)$ , from barriers adapted to the boundary curvature; (iii) the interior gradient bound, reducing $sup_{Ω^{'}} ∣ D u ∣$ to $sup w$ for $w = ∣ D u ∣^{2}$ , which satisfies a linear elliptic differential inequality whose local boundedness is the De Giorgi-Nash-Moser theorem 02.17.07; combining (ii) and (iii) gives the global gradient bound $sup_{Ω} ∣ D u ∣ \leq M_{1}$ ; (iv) the Hölder-gradient bound $[D u]_{C^{β} (\overline{Ω})} \leq M_{2}$ , applying the De Giorgi-Nash Hölder estimate 02.17.07 to the now uniformly-elliptic-with-bounded-coefficients differentiated equation ^{[Gilbarg-Trudinger 2001 §13.2]} ^{[Ladyzhenskaya-Uraltseva 1968]}. The chain is strictly ordered: each estimate consumes the previous one as a hypothesis.

Theorem 3 (interior gradient estimate for the minimal surface equation; Bombieri-De Giorgi-Miranda). For a solution $u$ of the minimal surface equation $div (D u / 1 + ∣ D u ∣^{2}) = 0$ on $B_{R} (x_{0})$ , $∣ D u (x_{0}) ∣ \leq C_{1} exp (C_{2} \frac{sup _{B_{R} (x_{0})} u - u ( x _{0} )}{R}),$ with $C_{1}, C_{2}$ depending only on $n$ ^{[Gilbarg-Trudinger 2001 §16.1]}. Strikingly, the interior slope is controlled by the oscillation of $u$ alone, with no boundary-gradient input: the minimal surface operator self-improves, a consequence of the equation governing the volume element of the graph and the resulting Sobolev inequality on the minimal hypersurface. This is the feature that distinguishes the model case from general quasilinear operators, for which the interior bound genuinely needs the global structure.

Theorem 4 (the Bernstein theorem and the role of dimension). An entire solution of the minimal surface equation on all of $R^{n}$ — a complete minimal graph — is an affine function (a hyperplane) for $n \leq 7$ , and this fails for $n \geq 8$ (Bombieri-De Giorgi-Giusti) ^{[Bernstein 1910]} ^{[Gilbarg-Trudinger 2001 §17]}. The proof for low dimensions runs the interior gradient estimate of Theorem 3 at every scale $R \to \infty$ : an entire solution with controlled oscillation has vanishing gradient oscillation, forcing affinity, exactly as the Liouville theorem follows from the Harnack inequality of 02.17.08. The dimensional ceiling $n = 7$ is the same threshold at which the Simons cone becomes a stable singular minimal hypersurface, tying scalar PDE rigidity to the geometry of minimal cones.

Theorem 5 (degenerate and singular cases: the $p$ -Laplacian; DiBenedetto, Uhlenbeck, Lewis). For the $p$ -Laplacian $div (∣ D u ∣^{p - 2} D u) = 0$ , $1 < p < \infty$ , solutions are of class $C_{loc}^{1, α}$ but generally **no better** than $C^{1, α}$ : second derivatives need not exist where $D u = 0$ , because ellipticity degenerates ( $p > 2$ ) or blows up ( $p < 2$ ) at critical points ^{[Gilbarg-Trudinger 2001 §13]}. The gradient estimate and the $C^{1, α}$ regularity survive the degeneration via a Caccioppoli estimate for $∣ D u ∣$ combined with a De Giorgi iteration on the level sets of $∣ D u ∣$ ; the optimal Hölder exponent $α = α (n, p)$ is not explicit. This is the canonical example where the natural-growth structure conditions with $m = p \neq = 2$ are not a technical convenience but the genuine generality the theory was built to handle.

Theorem 6 (uniqueness and comparison). If $A$ is monotone in $p$ ( $(A (x, u, p) - A (x, u, q)) \cdot (p - q) > 0$ for $p \neq = q$ ) and $B$ is non-increasing in $u$ , then solutions of the Dirichlet problem with the same data are unique, and the comparison principle holds: $u \leq v$ on $\partial Ω$ and $Q u \geq Q v$ in $Ω$ force $u \leq v$ in $Ω$ ^{[Gilbarg-Trudinger 2001 §10.1]} ^{[Serrin 1969]}. The proof tests the weak formulation of $Q u - Q v$ against $(u - v)^{+}$ and uses monotonicity to kill the leading term, the nonlinear analogue of the linear maximum principle of 02.17.02. Monotonicity of $A$ is the divergence-form expression of ellipticity integrated along the gradient, and it is what converts the local ellipticity into a global comparison.

Synthesis. The quasilinear theory is the foundational reason elliptic PDE has a unified existence architecture: every second-order elliptic Dirichlet problem, linear or nonlinear, is solved by pairing solvability of a model problem with an a priori estimate, and putting these together the Leray-Schauder scheme is exactly the nonlinear lift of the linear method of continuity of 02.17.05 — the Neumann-series openness becomes the topological degree, and the $t$ -uniform linear estimate becomes the solution-uniform $C^{1, β}$ chain. This is exactly the duality between estimates and existence read one level up, and the central insight is that the entire nonlinear difficulty is compressed into the single passage $sup ∣ u ∣ \to sup ∣ D u ∣$ , the gradient estimate, obtained by differentiating the equation and feeding the result to the De Giorgi-Nash-Moser machine of 02.17.07 and the Harnack-oscillation mechanism of 02.17.08. The bridge is the compact solution operator $T$ , whose derivative-gaining linear inverse and Arzelà-Ascoli compactness 02.11.05 turn each a priori bound into a wall of a bounded set, so the topology is inert once the estimates close and the subject is the art of the a priori bound. The minimal surface equation is the keystone, generalising the linear gradient theory because its interior estimate needs no boundary data — and it is dual to the geometry of minimal cones, where the dimension- $7$ Bernstein threshold of Theorem 4 is the same number governing the stability of the Simons cone, where scalar PDE rigidity and the singular structure of area-minimising hypersurfaces become one phenomenon.

Full proof set Master

Proposition 1 (Schaefer's fixed-point theorem). Let $B$ be a Banach space, $T : B \to B$ compact, and suppose $Σ = {x \in B : x = σ T x for some σ \in [0, 1]}$ is bounded. Then $T$ has a fixed point.

Proof. Choose $R > sup_{x \in Σ} ∥ x ∥$ , so that $x \neq = σ T x$ for all $∥ x ∥ = R$ and all $σ \in [0, 1]$ . The Leray-Schauder degree $de g (I - σ T, B_{R}, 0)$ is defined for each $σ$ since $I - σ T$ is a compact perturbation of the identity with no zero on $\partial B_{R}$ . The family $σ \mapsto I - σ T$ is a continuous homotopy of such maps (continuity in the operator norm, compactness of each $σ T$ ), zero-free on $\partial B_{R}$ throughout, so by homotopy invariance $de g (I - T, B_{R}, 0) = de g (I, B_{R}, 0) = 1 \neq = 0$ . The solution property of a nonzero degree yields $x \in B_{R}$ with $(I - T) x = 0$ , i.e. $x = T x$ . $□$

Proposition 2 (the freeze-and-fix operator is compact). Let $Ω$ be a bounded $C^{2, α}$ domain, $β \in (0, α)$ , and for $v \in C^{1, β} (\overline{Ω})$ let $T v \in C^{2, β} (\overline{Ω})$ be the unique solution of the linear Dirichlet problem with coefficients frozen at $(x, v, D v)$ and data $φ$ . Then $T : C^{1, β} (\overline{Ω}) \to C^{1, β} (\overline{Ω})$ is compact and continuous.

Proof. The frozen coefficients $a^{ij} (x) = (\partial A_{i} / \partial p_{j}) (x, v, D v)$ lie in $C^{0, β} (\overline{Ω})$ with norm controlled by $∥ v ∥_{C^{1, β}}$ (composition of the $C^{1, α}$ map $\partial A / \partial p$ with the $C^{0, β}$ argument $(x, v, D v)$ , using the Hölder chain rule), and they are uniformly elliptic by the structure conditions. The global linear Schauder estimate 02.17.05 then gives $∥ T v ∥_{C^{2, β} (\overline{Ω})} \leq C (∥ v ∥_{C^{1, β}}) (∥ f ∥_{C^{0, β}} + ∥ φ ∥_{C^{2, β}})$ . Hence $T$ maps each bounded set ${∥ v ∥_{C^{1, β}} \leq K}$ into a bounded subset of $C^{2, β} (\overline{Ω})$ . By Arzelà-Ascoli, bounded subsets of $C^{2, β} (\overline{Ω})$ are precompact in $C^{1, β} (\overline{Ω})$ on the bounded domain $Ω$ — the inclusion $C^{2, β} ↪ C^{1, β}$ is a compact operator 02.11.05. Thus $T$ sends bounded sets to precompact sets: $T$ is compact. For continuity, if $v_{k} \to v$ in $C^{1, β}$ the frozen coefficients converge in $C^{0, β^{'}}$ for $β^{'} < β$ , and continuous dependence of the linear Schauder solution on its coefficients gives $T v_{k} \to T v$ in $C^{1, β}$ . $□$

Proposition 3 (interior gradient estimate via the differentiated equation). Let $u \in C^{3} (Ω)$ solve $Q u = div A (x, u, D u) + B = 0$ with structure constants $λ, Λ, μ$ and $m = 2$ . Then for $Ω^{'} ⋐ Ω$ , $sup_{Ω^{'}} ∣ D u ∣ \leq C (n, λ, Λ, μ, dist (Ω^{'}, \partial Ω), sup_{Ω} ∣ u ∣)$ .

Proof. Write $a^{ij} = (\partial A_{i} / \partial p_{j}) (x, u, D u)$ , uniformly elliptic with constants $λ, Λ$ by the structure condition. Differentiating the equation in $x_{k}$ , the derivative $v_{k} = D_{k} u$ satisfies the linear divergence-form equation $D_{i} (a^{ij} D_{j} v_{k}) = - D_{i} g_{ik}, g_{ik} = \partial_{x_{k}} A_{i} + (\partial_{u} A_{i}) v_{k} - (terms from B),$ with $∣ g_{ik} ∣ \leq C (1 + ∣ D u ∣)$ by the growth bounds. Consider $w = ∣ D u ∣^{2} = \sum_{k} v_{k}^{2}$ . Multiplying the equation for $v_{k}$ by $v_{k}$ and summing, $D_{i} (a^{ij} D_{j} w) = 2 k \sum a^{ij} D_{j} v_{k} D_{i} v_{k} - 2 k \sum D_{i} g_{ik} v_{k} \geq 2 λ ∣ D^{2} u ∣^{2} - C (1 + ∣ D u ∣) ∣ D^{2} u ∣ - C ∣ D u ∣^{2} .$ By Young's inequality the cross term is absorbed: $D_{i} (a^{ij} D_{j} w) \geq λ ∣ D^{2} u ∣^{2} - C (1 + w)$ , so $w$ is a weak subsolution of a linear uniformly elliptic equation with right-hand side controlled by $1 + w$ . The local boundedness half of the De Giorgi-Nash-Moser theorem 02.17.07, applied on a ball $B_{2 r} (x_{0}) \subseteq Ω$ with $2 r = dist (Ω^{'}, \partial Ω)$ , gives $B_{r} sup w \leq C (\fint_{B_{2 r}} w d x + 1),$ and the energy (Caccioppoli) estimate for the equation $Q u = 0$ tested against $η^{2} u$ bounds $\int_{B_{2 r}} ∣ D u ∣^{2} \leq C (sup_{Ω} ∣ u ∣^{2} + 1) r^{n - 2}$ , controlling the average of $w$ by $sup_{Ω} ∣ u ∣$ and the geometry. Combining, $sup_{Ω^{'}} w \leq C (n, λ, Λ, μ, r, sup_{Ω} ∣ u ∣)$ , whence $sup_{Ω^{'}} ∣ D u ∣ = (sup_{Ω^{'}} w)^{1/2}$ is bounded as claimed. $□$

Proposition 4 (uniqueness by monotone testing). Let $A (x, u, p)$ be monotone in $p$ and $B (x, u, p)$ non-increasing in $u$ . If $u, v \in C^{1} (\overline{Ω})$ solve $Q u = Q v = 0$ in $Ω$ with $u = v$ on $\partial Ω$ , then $u \equiv v$ .

Proof. Subtract the weak formulations and test against $φ = (u - v)^{+} \in W_{0}^{1, 2} (Ω)$ (it vanishes on $\partial Ω$ since $u = v$ there): $\int_{Ω} (A (x, u, D u) - A (x, v, D v)) \cdot D (u - v)^{+} d x = \int_{Ω} (B (x, u, D u) - B (x, v, D v)) (u - v)^{+} d x .$ On the set ${u > v}$ where $(u - v)^{+} > 0$ , split the left integrand: $(A (x, u, D u) - A (x, u, D v)) \cdot D (u - v) \geq 0$ by monotonicity in $p$ (strictly positive where $D u \neq = D v$ ), while the remaining $u$ -dependence of $A$ and the $u$ -dependence of $B$ are controlled by the non-increase of $B$ and a Lipschitz bound, which after a standard absorption forces the left side $\geq 0$ and the right side $\leq 0$ . Hence both vanish, so $D (u - v) = 0$ on ${u > v}$ ; with $(u - v)^{+} = 0$ on $\partial Ω$ this gives $(u - v)^{+} \equiv 0$ , i.e. $u \leq v$ . By symmetry $v \leq u$ , so $u \equiv v$ . $□$

Proposition 5 (the homotopy is seeded at $σ = 0$ ). In the family $Q_{σ} u = div A (x, u, D u) + σ B = 0$ , $u = σ φ$ , the value $σ = 0$ has the unique solution $u \equiv 0$ , and the maximum-principle sup bound is uniform in $σ$ .

Proof. At $σ = 0$ the problem is $div A (x, u, D u) = 0$ , $u = 0$ on $\partial Ω$ . Testing against $u$ and using the lower structure bound $A (x, u, p) \cdot p \geq λ (1 + ∣ p ∣)^{m - 2} ∣ p ∣^{2} - C \geq \frac{λ}{2} ∣ p ∣^{2} - C$ gives $\int_{Ω} ∣ D u ∣^{2} \leq C \int_{Ω} 1$ on ${∣ D u ∣$ large $}$ , and with zero boundary data the Poincaré inequality forces $\int ∣ D u ∣^{2} = 0$ , so $u \equiv 0$ . For general $σ$ , the weak maximum principle 02.17.02 applied to the divergence structure with $\partial (σ B) / \partial u = σ \partial B / \partial u \leq 0$ yields $sup_{Ω} ∣ u ∣ \leq σ sup_{\partial Ω} ∣ φ ∣ + C diam (Ω)^{2} \leq sup_{\partial Ω} ∣ φ ∣ + C diam (Ω)^{2}$ , a bound independent of $σ \in [0, 1]$ . This $σ$ -uniform sup bound is the first link of the a priori chain and the seed of the Leray-Schauder homotopy. $□$

Connections Master

The De Giorgi-Nash-Moser theory of 02.17.07 is the analytic engine of the gradient estimate: differentiating the quasilinear equation produces a linear divergence-form equation for $w = ∣ D u ∣^{2}$ with bounded measurable coefficients, and the local boundedness theorem of that unit caps $sup ∣ D u ∣$ in the interior, while its Hölder estimate produces the $[D u]_{C^{β}}$ bound that closes the a priori chain — the entire nonlinearity is absorbed into frozen coefficients to which the linear measurable-coefficient theory applies verbatim.
The Harnack inequality and oscillation-decay mechanism of 02.17.08 are what convert the gradient bound into the Bernstein rigidity of Theorem 4: running the interior gradient estimate at every scale $R \to \infty$ is the exact analogue of reading the Harnack inequality at infinity to obtain the Liouville theorem, so the affineness of low-dimensional entire minimal graphs is the quasilinear shadow of the Liouville theorem proved there.
The linear method of continuity of 02.17.05 is the affine special case of the Leray-Schauder scheme used here: when the equation is linear the homotopy $L_{t} = (1 - t) Δ + t L$ replaces the nonlinear homotopy $Q_{σ}$ , the Neumann-series openness replaces the topological degree, and the $t$ -uniform Schauder estimate replaces the solution-uniform $C^{1, β}$ chain; the two units are the linear and nonlinear faces of the single principle that existence is dual to an a priori estimate, and the linear Schauder solvability proved there is the very engine that defines the compact operator $T$ here.
The compact-operator theory of 02.11.05 supplies the Arzelà-Ascoli compactness of the embedding $C^{2, β} ↪ C^{1, β}$ that makes the solution operator $T$ compact, without which the Leray-Schauder degree is undefined; the quasilinear existence theorem is the canonical PDE application that makes the abstract compactness of 02.11.05 indispensable.

Historical & philosophical context Master

The quasilinear theory grew from the calculus of variations: the minimal surface equation is the Euler-Lagrange equation of the area functional, and Sergei Bernstein's 1910 study of surfaces defined by their mean curvature ^{[Bernstein 1910]} introduced both the a priori estimate method and the rigidity theorem now bearing his name, proving that entire minimal graphs over $R^{2}$ are planes. Bernstein's insight — that bounds on a hypothetical solution, established before any solution is known to exist, are the route to existence — became the organising principle of the whole subject, and his method of differentiating the equation to estimate the gradient is the seed of every gradient estimate proved here.

The abstract existence machinery is due to Jean Leray and Juliusz Schauder, whose 1934 Annales de l'ENS paper ^{[Leray-Schauder 1934]} built the topological degree for compact perturbations of the identity in Banach spaces and reduced existence to an a priori bound; Helmut Schaefer's 1955 note ^{[Schaefer 1955]} isolated the clean fixed-point form used today. The analytic estimates that feed the scheme came in the 1950s and 1960s: Ennio De Giorgi's 1957 resolution of Hilbert's nineteenth problem ^{[De Giorgi 1957]} supplied the Hölder continuity of solutions of divergence-form equations with measurable coefficients, the precise tool needed for the differentiated quasilinear equation, and the comprehensive theory was assembled by Olga Ladyzhenskaya and Nina Uraltseva ^{[Ladyzhenskaya-Uraltseva 1968]} and, with sharp structure conditions and the definitive Dirichlet existence theorem, by James Serrin in his 1969 Philosophical Transactions memoir ^{[Serrin 1969]}. The Bernstein theorem's dimensional ceiling was settled by Bombieri, De Giorgi, and Giusti in 1969, who exhibited a non-affine entire minimal graph in dimension eight, linking the scalar PDE to the singular geometry of the Simons cone; Gilbarg and Trudinger's chapters 10-11, 13, and 15-17 ^{[Gilbarg-Trudinger 2001]} give the modern synthesis proved above.

Bibliography Master

@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}
}

@article{LeraySchauder1934,
  author  = {Leray, Jean and Schauder, Juliusz},
  title   = {Topologie et \'equations fonctionnelles},
  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {51},
  year    = {1934},
  pages   = {45--78}
}

@article{Schaefer1955,
  author  = {Schaefer, Helmut},
  title   = {\"Uber die Methode der a priori-Schranken},
  journal = {Mathematische Annalen},
  volume  = {129},
  year    = {1955},
  pages   = {415--416}
}

@book{LadyzhenskayaUraltseva1968,
  author    = {Ladyzhenskaya, Olga A. and Ural'tseva, Nina N.},
  title     = {Linear and Quasilinear Elliptic Equations},
  publisher = {Academic Press},
  year      = {1968}
}

@article{Serrin1969,
  author  = {Serrin, James},
  title   = {The problem of {D}irichlet for quasilinear elliptic differential equations with many independent variables},
  journal = {Philosophical Transactions of the Royal Society A},
  volume  = {264},
  year    = {1969},
  pages   = {413--496}
}

@article{DeGiorgi1957,
  author  = {De Giorgi, Ennio},
  title   = {Sulla differenziabilit\`a e l'analiticit\`a delle estremali degli integrali multipli regolari},
  journal = {Memorie dell'Accademia delle Scienze di Torino},
  volume  = {3},
  year    = {1957},
  pages   = {25--43}
}

@article{Bernstein1910,
  author  = {Bernstein, Serge},
  title   = {Sur les surfaces d\'efinies au moyen de leur courbure moyenne ou totale},
  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {27},
  year    = {1910},
  pages   = {233--256}
}

Prerequisites

02.17.07
02.17.08
02.17.05
02.11.05

Tier anchors

beginner: Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 2001), §10.1 and §11.3 (the soap-film / minimal-surface picture); physics-anchored as a stretched membrane whose slope can never run away in the interior
intermediate: Gilbarg-Trudinger §10.1-§10.5 (maximum and gradient bounds), §11.3 (Leray-Schauder existence), §13.1-§13.2 (interior gradient estimates); Han-Lin, Elliptic Partial Differential Equations, 2e (AMS/Courant 2011), §11
master: Gilbarg-Trudinger §10-§11, §13, §15; Ladyzhenskaya-Uraltseva, Linear and Quasilinear Elliptic Equations (Academic Press 1968), Ch. IV; Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables (Phil. Trans. R. Soc. A 264, 1969)

References

Gilbarg-Trudinger — Elliptic Partial Differential Equations of Second Order, 2e · Springer Grundlehren 224 (2001), §10-§11, §13, §15
Leray-Schauder — Topologie et équations fonctionnelles · Annales scientifiques de l'École Normale Supérieure 51 (1934), 45-78
Schaefer — Über die Methode der a priori-Schranken · Mathematische Annalen 129 (1955), 415-416
Ladyzhenskaya-Uraltseva — Linear and Quasilinear Elliptic Equations · Academic Press (1968), Ch. IV
Serrin — The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables · Philosophical Transactions of the Royal Society A 264 (1969), 413-496
De Giorgi — Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari · Memorie dell'Accademia delle Scienze di Torino 3 (1957), 25-43
Bernstein — Sur les surfaces définies au moyen de leur courbure moyenne ou totale · Annales scientifiques de l'École Normale Supérieure 27 (1910), 233-256

Estimated time

beginner: 25m
intermediate: 70m
master: 120m