PDE: Elliptic Theory¶

Job: develop the frontier of elliptic regularity — Schauder estimates (Gilbarg–Trudinger) for $C^{k,\alpha}$ coefficients, De Giorgi–Nash–Moser Hölder regularity for measurable coefficients, Calderón–Zygmund $L^p$ estimates, and the Krylov–Safonov / Evans–Krylov theory for fully nonlinear equations — with sharp hypotheses on coefficients and the regularity each hypothesis yields.

Core¶

Schauder theory¶

Let $L=a^{ij}(x)\partial_{ij}+b^i(x)\partial_i+c(x)$ be uniformly elliptic on $\Omega\subset\mathbb{R}^n$, with coefficients $a^{ij}\in C^{k,\alpha}(\overline\Omega)$ (Hölder continuous of exponent $\alpha\in(0,1)$) and lower-order coefficients in $C^{k,\alpha}$.

Theorem (Schauder interior estimates, Gilbarg–Trudinger). If $u\in C^2(\Omega)$ solves $Lu=f$ with $f\in C^{k,\alpha}(\Omega)$ and the coefficients are in $C^{k,\alpha}$, then $u\in C^{k+2,\alpha}_{\mathrm{loc}}(\Omega)$ and for every compact $K\subset\Omega$, $$ |u|{C^{k+2,\alpha}(K)}\le C\bigl(|f|+|u|}(\Omega){C^0(\Omega)}\bigr), $$ with $C$ depending on $(\lambda,\Lambda,\Omega,K,k,\alpha)$ and the $C^{k,\alpha}$ norms of the coefficients. The global / boundary Schauder estimate holds when $\partial\Omega\in C^{k+2,\alpha}$ and the boundary data is $C^{k+2,\alpha}$: $$ |u|\le C\bigl(|f|}(\overline\Omega){C^{k,\alpha}(\overline\Omega)}+|u|\bigr). $$}(\partial\Omega)}+|u|_{C^0

The Schauder theory is the smooth-coefficient regularity: it bootstraps from $f\in C^{k,\alpha}$ to $u\in C^{k+2,\alpha}$, gaining two derivatives in Hölder scale. The proof reduces to the model Laplacian on a ball via freezing coefficients, then uses the Poisson representation / Newtonian potential; the key input is the $C^{2,\alpha}$ estimate for the model problem, which comes from the singular-integral structure of the second derivatives of the Newtonian potential (see See it).

De Giorgi–Nash–Moser¶

For divergence-form equations $L=-\mathrm{div}(A(x)\nabla)$ with $A$ uniformly elliptic and bounded measurable (no continuity), Schauder does not apply. The De Giorgi–Nash–Moser theorem gives the substitute:

Theorem (De Giorgi 1957, Nash 1958; Moser 1960–61). Let $L=-\mathrm{div}(A(x)\nabla)$ be uniformly elliptic with $A\in L^\infty$ ($\lambda|\xi|^2\le A\xi\cdot\xi\le\Lambda|\xi|^2$ a.e.) and $u\in H^1$ solve $Lu=f$ with $f\in L^p$, $p>n/2$. Then $u\in C^\alpha_{\mathrm{loc}}(\Omega)$ for some $\alpha=\alpha(n,\lambda,\Lambda,p)>0$, with the estimate $$ [u]{C^\alpha(B\le C\bigl(|u|}){L^2(B_1)}+|f|\bigr), $$ and $\alpha$ depends only on $(n,\lambda,\Lambda,p)$, not on the smoothness of $A$. The result holds for both divergence form (De Giorgi) and non-divergence form (Nash; Krylov–Safonov for non-divergence). The Harnack inequality: for $u\ge 0$ an $L$-harmonic in $B_1$, $\sup_{B_{1/2}}u\le C\inf_{B_{1/2}}u$ with $C=C(n,\lambda,\Lambda)$.

The proof (Moser iteration): a Caccioppoli inequality $\int\eta^2|\nabla u|^2\le C\int|\nabla\eta|^2 u^2$ gives an $L^\infty$ bound via the John–Nirenberg inequality / BMO; the iteration boosts $L^p$ to $L^\infty$ at a rate determined by the ellipticity, and the Harnack inequality follows by applying the bound to $u$ and $1/u$. The result is sharp for measurable coefficients: there are counterexamples (Piccinini–Miranda) showing no better than $C^\alpha$ holds in general.

Calderón–Zygmund $L^p$ theory¶

Theorem (Calderón–Zygmund). For $-\Delta u=f$ on $\mathbb{R}^n$ (or $\Omega\subset\mathbb{R}^n$ with $C^{1,1}$ boundary and Dirichlet data), the second-derivative estimate is $$ |D^2 u|{L^p}\le C_p|f|,\qquad 1<p<\infty, $$ with $C_p\sim p$ as $p\to\infty$ and $C_p\sim 1/(p-1)$ as $p\to 1$. The estimate fails at $p=1$ and $p=\infty$: at $p=1$ the bound is replaced by $\|D^2u\|_{L\log L}$ (Zygmund); at $p=\infty$ the bound is replaced by BMO (the second derivatives of $u$ are in BMO when $f\in L^\infty$ — the Schauder-via-BMO of the model problem). The constant $C_p$ is the norm of the Calderón–Zygmund singular integral (the Hessian Riesz transform $\partial_{ij}\Delta^{-1}$), which is unbounded on $L^1$ and $L^\infty$.

Coefficient hypothesis	Regularity obtained	Reference
Smooth ($C^\infty$)	$C^\infty$ (Weyl's lemma); analytic if $f$ analytic	classical
$C^{k,\alpha}$	$C^{k+2,\alpha}$ (Schauder)	Gilbarg–Trudinger
Lipschitz ($C^{0,1}$)	$C^{1,\alpha}$ any $\alpha<1$ (Schauder / bootstrap)	GT
Bounded measurable, divergence form	$C^\alpha$ (De Giorgi–Nash–Moser)	De Giorgi, Moser
Bounded measurable, non-divergence	$C^\alpha$ (Krylov–Safonov)	Krylov–Safonov
$L^p$ RHS, $C^{1,1}$ domain	$W^{2,p}$ (Calderón–Zygmund), $1<p<\infty$	CZ
$L^\infty$ RHS	BMO for $D^2 u$; $C^{1,\alpha}$ via Campanato	CZ / Campanato

Fully nonlinear elliptic: Krylov–Safonov and Evans–Krylov¶

Theorem (Krylov–Safonov 1979). For the non-divergence-form equation $a^{ij}(x)u_{ij}=f$ with $a^{ij}$ bounded measurable uniformly elliptic, solutions are $C^\alpha$ — the non-divergence-form Harnack inequality $\sup_{B_{1/2}}u\le C\inf_{B_{1/2}}u$ holds (for $u\ge 0$ $L$-harmonic), with $\alpha$ depending only on $(n,\lambda,\Lambda)$.

The fully nonlinear case: $F(D^2u,x)=0$ with $F$ uniformly elliptic ($F$ concave or convex in $D^2u$ for the regularity theory).

Theorem (Evans 1978, Krylov 1982). If $F$ is uniformly elliptic and concave (or convex) in $D^2u$, and $u$ is a viscosity solution of $F(D^2u)=0$, then $u\in C^{2,\alpha}_{\mathrm{loc}}$ for some $\alpha>0$. The proof uses the Krylov–Safonov Harnack for the linearized equation (the difference quotient $w=u(\cdot+h)-u$ satisfies a uniformly elliptic linear equation with measurable coefficients, to which Krylov–Safonov applies) plus a Jensen-convexity argument exploiting the concavity of $F$. This is the frontier regularity for fully nonlinear elliptic — the Monge–Ampère equation $\det D^2u=f$ for $f>0$ (convex $u$) is the key application (Caffarelli–Gutiérrez, Caffarelli–Cabrè), giving the Calabi–Yau / affine geometry regularity (see complex-geometry/Kähler).

The open frontier: for non-concave $F$, the Evans–Krylov $C^{2,\alpha}$ fails in general — Nadirashvili–Vlăduţ constructed counterexamples (singular solutions of uniformly elliptic fully nonlinear equations) showing $C^{2,\alpha}$ fails without the concavity hypothesis. The optimal regularity for general $F$ is a current research problem.

See it¶

The $W^{2,p}$ Poisson estimate via the Newtonian potential and the Calderón–Zygmund singular integral; the $p=1,\infty$ failure. For $-\Delta u=f$ on $\mathbb{R}^n$ with $f\in L^p$ compactly supported, the solution is the Newtonian potential $u=\Phi\ast f$ with $\Phi(x)=c_n|x|^{2-n}$ ($n\ge 3$); differentiating twice,

\[ \partial_{ij}u(x)=\mathrm{p.v.}\int K_{ij}(x-y)f(y)\,dy\,+\,\frac{1}{n}\delta_{ij}f(x),\qquad K_{ij}(z)=c'_n\frac{z_i z_j}{|z|^n}. \]

The kernel $K_{ij}(z)$ is homogeneous of degree $-n$ (a Calderón–Zygmund singular kernel), and the principal-value operator $T_{ij}:f\mapsto\mathrm{p.v.}K_{ij}\ast f$ is a Calderón–Zygmund singular integral. The $L^p$ bound $\|D^2u\|_{L^p}\le C_p\|f\|_{L^p}$ for $1<p<\infty$ is the Calderón–Zygmund theorem: $T_{ij}$ is bounded on $L^p$ for $1<p<\infty$. The failure at $p=1,\infty$: $T_{ij}$ is unbounded on $L^1$ and $L^\infty$ — concretely, for the 1D analog (the Hilbert transform $Hf=\mathrm{p.v.}\frac{1}{\pi}\int\frac{f(y)}{x-y}dy$), $H\chi_{[0,1]}=\frac{1}{\pi}\log|\frac{x}{x-1}|$ is unbounded at $0,1$, so $\|H\chi_{[0,1]}\|_\infty=\infty$ while $\|\chi_{[0,1]}\|_1=1$. The same phenomenon occurs in higher dimensions: the Hessian Riesz transform amplifies $L^\infty$ functions to unbounded ones, so the $L^\infty$ bound fails, and is replaced by the BMO estimate ($D^2u\in\mathrm{BMO}$ when $f\in L^\infty$) via the John–Nirenberg inequality. The $p=1$ failure is replaced by $\|D^2u\|_{L\log L}\le C\|f\|_{L^1}$ (Zygmund class). This single computation — the Hessian Riesz transform is a Calderón–Zygmund singular integral, unbounded at $L^1,L^\infty$ but bounded on $L^p$ for $1<p<\infty$ — is the operational content of the Calderón–Zygmund $L^p$ theory and the reason for the $p\ne 1,\infty$ restriction.

Check¶

Why does the Schauder estimate require $C^{k,\alpha}$ coefficients rather than merely bounded measurable, and identify the regularity gain lost when passing to measurable coefficients.

The Schauder estimate $\|u\|_{C^{k+2,\alpha}}\le C(\|f\|_{C^{k,\alpha}}+\|u\|_{C^0})$ gains two Hölder derivatives from the coefficients: $f\in C^{k,\alpha}\Rightarrow u\in C^{k+2,\alpha}$. The proof freezes the coefficients at a point $x_0$, reduces to the constant-coefficient Laplacian $\Delta u_{x_0}=f-a^{ij}(x_0)\partial_{ij}u$ on a small ball, and uses the Newtonian-potential / Poisson representation — the error is controlled by the $C^{k,\alpha}$ modulus of the coefficients, which requires the $C^\alpha$ modulus. With merely bounded measurable coefficients, the error cannot be controlled by a modulus, the Newtonian representation breaks, and the gain is only $C^\alpha$ (De Giorgi–Nash–Moser): two derivatives in Hölder scale are lost (one gets $C^\alpha$ instead of $C^{2,\alpha}$). The lost regularity is the price of the measurable-coefficient hypothesis; it is intrinsic, not technical (counterexamples show no better than $C^\alpha$ holds in general for measurable coefficients).

Sketch why the Evans–Krylov $C^{2,\alpha}$ estimate for fully nonlinear concave $F$ reduces to the Krylov–Safonov Harnack for a linear equation, and identify the role of concavity.

The strategy: take two solutions $u$ and $v$ of $F(D^2u)=0$; the difference $w=u-v$ satisfies a linear equation $a^{ij}(x)w_{ij}=0$ where $a^{ij}(x)=\int_0^1\partial_{ij}F(tD^2u+(1-t)D^2v)\,dt$ (by the mean-value theorem, using the concavity / convexity of $F$ to ensure the $a^{ij}$ are uniformly elliptic — the ellipticity of the interpolated Hessian is controlled by the ellipticity of $F$ and the concavity ensures the integral is a uniformly elliptic matrix). The linear equation for $w$ has measurable coefficients (depending on $u,v$), so Krylov–Safonov gives $w\in C^\alpha$. To upgrade to $u\in C^{2,\alpha}$, one applies this to second difference quotients $w_h=u(\cdot+h)+u(\cdot-h)-2u(\cdot)$ (which satisfy a related linear equation) and uses a Jensen-convexity argument exploiting the concavity of $F$ to obtain a uniform $C^{2,\alpha}$ bound on $u$. The concavity is essential: it guarantees the interpolated Hessian matrix $a^{ij}$ is uniformly elliptic (for non-concave $F$ the interpolation can degenerate, and the argument fails — the Nadirashvili–Vlăduţ counterexamples exploit this).

Pitfalls¶

Schauder needs $C^{k,\alpha}$ coefficients: bounded measurable is insufficient for Schauder; the substitute is De Giorgi–Nash–Moser, giving only $C^\alpha$ (a loss of $2$ derivatives in Hölder scale). Conflating the two regimes is the standard error.
Calderón–Zygmund fails at $p=1,\infty$: the Hessian Riesz transform is a CZ singular integral unbounded on $L^1,L^\infty$; the substitutes are $L\log L$ (for $p=1$) and BMO (for $p=\infty$). Citing $\|D^2u\|_{L^\infty}\le C\|f\|_{L^\infty}$ is wrong.
De Giorgi–Nash–Moser gives only $C^\alpha$ for measurable coefficients: the exponent $\alpha$ depends on $(n,\lambda,\Lambda)$ and is typically small; do not expect $C^1$ or higher without further coefficient regularity.
Fully nonlinear regularity needs concavity: the Evans–Krylov $C^{2,\alpha}$ requires $F$ concave or convex; non-concave $F$ has counterexamples (Nadirashvili–Vlăduţ) to $C^{2,\alpha}$. The Monge–Ampère $\det D^2u=f$ regularity uses the concavity of $\det^{1/n}$ on positive matrices.
Krylov–Safonov is non-divergence form: it is the non-divergence analog of De Giorgi–Nash–Moser; the divergence-form Harnack (Moser) and non-divergence Harnack (Krylov–Safonov) are different theorems with different proofs.

Refs¶

Gilbarg–Trudinger, Elliptic PDE of Second Order (Schauder, De Giorgi–Nash–Moser, Krylov–Safonov)
Stein, Singular Integrals; Grafakos, Classical / Modern Fourier Analysis (Calderón–Zygmund)
Caffarelli–Cabrè, Fully Nonlinear Elliptic Equations; Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order
Han–Lin, Elliptic PDE (modern textbook treatment)

Next¶

Dispersive & Wave · Navier–Stokes · Calculus of Variations · PDE index · Foundations · Functional analysis · Microlocal analysis · Complex geometry / Kähler · Function spaces — Sobolev/Besov/TL spaces and interpolation for PDE