Weak Elliptic Maximum Principles and Holder Regularity — Examples

Given: The Axiom of Choice and Countable Choice; the unit disc $\mathrm{\Omega }=B_1(0)\subset {\mathbb{R}}^2$; the coefficients $a^{ij}={\delta }^{ij}$, $b=c=0$; and the function $u(x)=\mathrm{\mid }x{\mathrm{\mid }}^2-1$.

Classical differentiation gives $D_{i}u=2x_i$ and $\mathrm{\Delta }u=4$ on $\mathrm{\Omega }$, so $\mathrm{\Delta }u\ge 0$: $u$ is subharmonic in the sense of Subharmonic and superharmonic functions in rn, and $u\in C^2(\bar{\mathrm{\Omega }})\cap H_0^1(\mathrm{\Omega })$ because $u\le 0$ on $\mathrm{\Omega }$ with equality exactly on $\mathrm{\partial }\mathrm{\Omega }$ and $u$ is a polynomial (Bounded C^k domains and boundary charts, The kernel of the trace is the closure of the test functions for the zero-trace identification).

Classical weak maximum principle for the Laplacian: for a bounded nonempty open $\mathrm{\Omega }$ and $u\in C^2(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$ with $\mathrm{\Delta }u\ge 0$, ${\max }_{\bar{\mathrm{\Omega }}}u={\max }_{\mathrm{\partial }\mathrm{\Omega }}u$ (Weak maximum principle for the laplacian).

Classical-to-weak consistency: if $u\in C^2(\bar{\mathrm{\Omega }})\cap H_0^1(\mathrm{\Omega })$ and $Lu=f$ with $f\in C(\bar{\mathrm{\Omega }})$, then $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}$ for every $v\in H_0^1(\mathrm{\Omega })$, for the sesquilinear form $a$ of Uniformly elliptic divergence-form operators and their sesquilinear forms (Classical solutions satisfy the weak formulation).

Alternative direct integration by parts: for $v\in H_0^1(\mathrm{\Omega })$ and $u\in C^2(\bar{\mathrm{\Omega }})$, the Sobolev Gauss-Green formula gives ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx=-{\int }_{\mathrm{\Omega }}v\mathrm{\Delta }udx$, the boundary term vanishing because $Tv=0$ (The Gauss-Green integration-by-parts formula with Sobolev traces, Weak subsolutions and supersolutions of a divergence-form equation).

Weak maximum principle for coercive divergence-form equations: under its hypotheses, a weak subsolution $u$ of $L_{0}u=0$ on a bounded $C^1$ domain satisfies ${\mathrm{ess\hspace{0.17em}sup}}_{\mathrm{\Omega }}u\le {\sup }_{\mathrm{\partial }\mathrm{\Omega }}{u}^{+}$ (Weak maximum principle for coercive divergence-form equations).

Given: The Axiom of Choice and Countable Choice; the annulus $\mathrm{\Omega }=\{x\in {\mathbb{R}}^2:\frac{1}{4}<\mathrm{\mid }x\mathrm{\mid }<1\}$; the radial coefficient $a$ equal to $1$ for $\mathrm{\mid }x\mathrm{\mid }<\frac{1}{2}$ and $4$ for $\mathrm{\mid }x\mathrm{\mid }>\frac{1}{2}$; and the radial function $u$ defined by the two displayed formulas.

Uniform ellipticity and boundedness: $a$ is measurable, $1\le a\le 4$ on $\mathrm{\Omega }$, and the matrix $A=a\mathrm{I}\mathrm{d}$ satisfies $\mathrm{\mid }\xi {\mathrm{\mid }}^2\le ⟨A(x)\xi ,\xi ⟩\le 16\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for all $\xi \in {\mathbb{R}}^2$, so the ellipticity constant is $\theta =1$ and the coefficient bound is $M_a=4$ in the convention of Uniformly elliptic divergence-form operators and their sesquilinear forms (The notation $H^k$ and the reserved zero-boundary symbol).

Regularity of the pieces: on each of the open annuli $U_1=\{\frac{1}{4}<\mathrm{\mid }x\mathrm{\mid }<\frac{1}{2}\}$ and $U_2=\{\frac{1}{2}<\mathrm{\mid }x\mathrm{\mid }<1\}$ the function $u$ is smooth and radial, with $\mathrm{\nabla }u=\frac{1}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }r}x$ and ${\mathrm{\partial }}_{r}u=1\mathrm{/}r$ on $U_1$, ${\mathrm{\partial }}_{r}u=1\mathrm{/}(4r)$ on $U_2$; the glued function lies in $H^1(\mathrm{\Omega })$ with these a.e. gradients. Indeed $u=G(\mathrm{\mid }x\mathrm{\mid })$, where $G(t)=-\log 4$ for $t\le 1\mathrm{/}4$, $G(t)=\log t$ for $1\mathrm{/}4<t\le 1\mathrm{/}2$, and $G(t)=\frac{1}{4}\log t-\frac{3}{4}\log 2$ for $t>1\mathrm{/}2$. This is a globally $4$-Lipschitz scalar function. Since $x\mapsto \mathrm{\mid }x\mathrm{\mid }$ is smooth on $\bar{\mathrm{\Omega }}$ with gradient $x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ and belongs to $H^1(\mathrm{\Omega })$, the Sobolev chain rule establishes the asserted membership and gradient (Chain rule for globally Lipschitz scalar maps of Sobolev functions) (Weak derivative of a locally integrable function, Holder's inequality for integrals, including the endpoint cases).

Continuity and differentiability across the interface: at $\mathrm{\mid }x\mathrm{\mid }=\frac{1}{2}$ the first formula gives $\log \frac{1}{2}=-\log 2$ and the second gives $\frac{1}{4}\log \frac{1}{2}-\frac{3}{4}\log 2=-\log 2$, so $u$ is continuous there; the radial derivative is $2$ from the inner side and $\frac{1}{2}$ from the outer side, so the derivative is discontinuous and $u\mathrm{\notin }{C}^1(\mathrm{\Omega })$, while $\mathrm{\Omega }$ is bounded away from the origin in polar coordinates, so $u$ is Lipschitz and hence Holder of every exponent $\alpha <1$ (Local Hölder and scaled C-two-alpha norms on balls).

The flux: $a\mathrm{\nabla }u=x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ a.e. on $\mathrm{\Omega }$, because $a(r){\mathrm{\partial }}_{r}u(r)=1\mathrm{/}r$ for both branches of $a$ and of $u$; the field $x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ is smooth on $\mathrm{\Omega }$, $\mathrm{div}(x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2)=0$ there, and $x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2=\mathrm{\nabla }\log \mathrm{\mid }x\mathrm{\mid }$ (Local weak solutions of a divergence-form operator).

Local weak solutions: $u\in H^1(\mathrm{\Omega })$ is a local weak solution of $-\mathrm{div}(a\mathrm{\nabla }u)=0$ when ${\int }_{\mathrm{\Omega }}a(x)\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx=0$ for every $v\in H_0^1(\mathrm{\Omega })$; the De Giorgi-Nash theorem gives, for such a solution with measurable uniformly elliptic coefficients, a Holder representative with exponent depending only on $n,\theta ,M_a$ (Local weak solutions of a divergence-form operator, De Giorgi-Nash interior Holder regularity for divergence-form equations).

Given: The Axiom of Choice and Countable Choice; the ball $\mathrm{\Omega }=B_{\pi }(0)\subset {\mathbb{R}}^3$; the coefficients $a^{ij}={\delta }^{ij}$, $b^i=0$, $c=-1$; and the function $u(x)=\sin \mathrm{\mid }x\mathrm{\mid }\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ for $x\mathrm{\ne }0$, $u(0)=1$.

The operator $L=-\mathrm{\Delta }-1$ is the divergence-form operator with $a^{ij}={\delta }^{ij}$, $b=0$, $c=-1$ in the convention of Uniformly elliptic divergence-form operators and their sesquilinear forms; the ball $B_{\pi }(0)$ is a bounded $C^{\mathrm{\infty }}$ domain with trace $T$ (Bounded C^k domains and boundary charts, Weak subsolutions and supersolutions of a divergence-form equation).

Assume the Axiom of Choice and Countable Choice. Classical-to-weak consistency: if $\mathrm{\Omega }$ is a bounded $C^1$ domain, $u\in C^2(\bar{\mathrm{\Omega }})\cap H_0^1(\mathrm{\Omega })$ and $Lu=f$ with $f\in C(\bar{\mathrm{\Omega }})$, then $u$ is a weak solution of $Lu=f$ in the sense of Weak Dirichlet solutions for a divergence-form operator, i.e. $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ for every $v\in H_0^1(\mathrm{\Omega })$ (Classical solutions satisfy the weak formulation).

Assume the Axiom of Choice. $\{w\in H^1(\mathrm{\Omega }):Tw=0\}=H_0^1(\mathrm{\Omega })$ (The kernel of the trace is the closure of the test functions); consequently a class in $C(\bar{\mathrm{\Omega }})\cap H^1(\mathrm{\Omega })$ vanishing on $\mathrm{\partial }\mathrm{\Omega }$ has zero trace and belongs to $H_0^1(\mathrm{\Omega })$.

The smooth radial function $r\mapsto \sin r\mathrm{/}r$ on $(0,\mathrm{\infty })$ has the convergent power series ${\sum }_{j\ge 0}(-1{)}^j{r}^{2j}\mathrm{/}(2j+1)!$, so $u$ extends to a $C^{\mathrm{\infty }}$ function on ${\mathbb{R}}^3$ with $u(0)=1$ (The notation $H^k$ and the reserved zero-boundary symbol for the Sobolev class notation).

Given: The Axiom of Choice and Countable Choice; the unit ball $B_1(0)\subset {\mathbb{R}}^2$; the linear function $u(x)=x_1$.

Harmonic linear functions are weak solutions: $\mathrm{\Delta }{x}_1=0$ classically, so ${\int }_{B_1}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx=0$ for every $v\in H_0^1(B_1)$ by the divergence theorem, and $u$ is a local weak solution of $-\mathrm{\Delta }u=0$ in the sense of Local weak solutions of a divergence-form operator with the coefficients of Uniformly elliptic divergence-form operators and their sesquilinear forms ($a^{ij}={\delta }^{ij}$, $b=c=0$).

On the open half-ball $B_{1\mathrm{/}2}(0)$, the values $u(x)=x_1$ approach $1\mathrm{/}2$ along $x_j=(1\mathrm{/}2-1\mathrm{/}j,0)$ and $-1\mathrm{/}2$ along $y_j=(-1\mathrm{/}2+1\mathrm{/}j,0)$ for $j>2$. Thus ${\sup }_{B_{1\mathrm{/}2}}u=1\mathrm{/}2$ and ${\inf }_{B_{1\mathrm{/}2}}u=-1\mathrm{/}2$, although neither boundary value is attained; by continuity these also equal the essential extrema (The average of a locally integrable function over a Euclidean ball, The essential supremum of a measurable function with respect to a measure).

The Harnack statement: for a nonnegative weak solution of $L_{0}u=-F$ one has ${\mathrm{ess\hspace{0.17em}sup}}_{B_{R\mathrm{/}2}}u\le C({\mathrm{ess\hspace{0.17em}inf}}_{B_{R\mathrm{/}2}}u+R^{2-n\mathrm{/}q}\mathrm{\parallel }F{\mathrm{\parallel }}_{L^q(B_{2R})})$; the sign hypothesis is used in the proof through the test functions with $u^{\beta }$ for negative exponents and through the weak Harnack inequality (Harnack inequality for nonnegative weak solutions).

Given: An open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, a function $u:\mathrm{\Omega }\to \mathbb{R}$ with finite oscillation on every compactly contained ball satisfying ${\mathrm{osc}}_{B_r(x)}u\le \theta {\mathrm{osc}}_{B_{2r}(x)}u$ whenever $B_{2r}(x)⋐\mathrm{\Omega }$, and the two values $\theta =\frac{1}{2}$ and $\theta =2^{-1\mathrm{/}3}$.

Oscillation-to-modulus conversion: under the stated finite-oscillation hypothesis, put ${\alpha }_0=\log (1\mathrm{/}\theta )\mathrm{/}\log 2$ and $\alpha =\min \{{\alpha }_0,1\mathrm{/}2\}$; then $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le 4^{\alpha }(\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}R{)}^{\alpha }{\mathrm{osc}}_{B_R(x_0)}u$ for all $x,y\in B_{R\mathrm{/}2}(x_0)$, with $4^{{\alpha }^{\mathrm{\prime }}}$ for every $0<{\alpha }^{\mathrm{\prime }}<\alpha$ (Geometric oscillation decay implies a Hölder modulus).

Real powers with positive base satisfy $2^{-a}=1\mathrm{/}{2}^a$, $\log (2^a)=a\log 2$ and $\log (1\mathrm{/}\theta )=-\log \theta$ for $\theta \in (0,1)$ (Real powers for positive bases, with the zero-base positive-exponent convention); the Hölder seminorm on balls is defined as in Local Hölder and scaled C-two-alpha norms on balls.

The De Giorgi oscillation reduction produces a ratio $\theta \in (0,1)$ on balls satisfying its doubled-ball condition. Reserving this interior margin permits dyadic iteration; the resulting power-law conversion is the one illustrated in [F1] (De Giorgi oscillation reduction: one half-level set is small).

Given: The Axiom of Choice and Countable Choice; the unit ball $B_1(0)\subset {\mathbb{R}}^3$; the coefficient matrix $A=\mathrm{diag}(1,0,0)$; and the function $u(x)=x_2^{+}$.

The coefficient matrix is bounded, measurable, symmetric and positive semidefinite, but not uniformly elliptic: ${\xi }^{T}A\xi ={\xi }_1^2\ge 0$ for all $\xi$, while for $\xi =e_2$ one has ${\xi }^{T}A\xi =0<\theta \mathrm{\mid }\xi {\mathrm{\mid }}^2$ for every $\theta >0$, so the ellipticity hypothesis of Uniformly elliptic divergence-form operators and their sesquilinear forms fails (The notation $H^k$ and the reserved zero-boundary symbol).

The function $u(x)=x_2^{+}$ is Lipschitz and nonnegative on $B_1(0)\subset {\mathbb{R}}^3$, nonconstant, and its zero set contains the lower half-ball $\{x\in B_1(0):x_2<0\}$, an open interior set of positive measure; moreover $u(x)>0$ on the upper half-ball (Weak derivative of a locally integrable function, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

The weak gradient of $u$ is $Du=(0,{\mathbf{1}}_{\{x_2>0\}},0)$ a.e. on $B_1(0)\subset {\mathbb{R}}^3$, so $u\in H^1(B_1(0))$. For $A=\mathrm{diag}(1,0,0)$ one has $ADu=0$ a.e., hence the degenerate form $a_0(u,v)={\int }_{B_1}ADu\cdot Dvdx$ is zero for every $v\in H_0^1(B_1)$ (Weak derivative of a locally integrable function, Local weak solutions of a divergence-form operator).

The conclusion that fails: for uniformly elliptic coefficients, a nonnegative weak solution of $L_{0}u=0$ on a connected open set is either zero a.e. or strictly positive a.e., and its continuous representative has no interior zero unless it vanishes identically (Strong maximum principle for weak elliptic solutions, Zero-set propagation for a nonnegative Holder weak solution).

Given: The Axiom of Choice and Countable Choice; an integer $n\ge 1$; $\mathrm{\Omega }=B_3(0)\subset {\mathbb{R}}^n$; the function $u(x)=\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(2n)$; and the constant source $f\equiv -1$.

Elementary differentiation: $D_{i}u=x_i\mathrm{/}n$ and $\mathrm{\Delta }u=2n\mathrm{/}(2n)=1$, so $-\mathrm{\Delta }u=-1=f$ classically on $B_3(0)$; the Laplacian is the operator $-\mathrm{\Delta }$ with $a^{ij}={\delta }^{ij}$, $b=c=0$ in the convention of Uniformly elliptic divergence-form operators and their sesquilinear forms (Local weak solutions of a divergence-form operator).

On the open half-ball $B_{1\mathrm{/}2}(0)$, $u$ has infimum $0$ attained at the origin, while its supremum $1\mathrm{/}(8n)$ is approached along $x_j=(1\mathrm{/}2-1\mathrm{/}j)e_0$ for $j>2$ and is not attained; continuity makes this equal to the essential supremum (The average of a locally integrable function over a Euclidean ball, The essential supremum of a measurable function with respect to a measure).

For $n\ge 2$, the displayed Harnack statements carry the forcing additively: ${\mathrm{ess\hspace{0.17em}sup}}_{B_{R\mathrm{/}2}}u\le C({\mathrm{ess\hspace{0.17em}inf}}_{B_{R\mathrm{/}2}}u+R^{2-n\mathrm{/}q}\mathrm{\parallel }F{\mathrm{\parallel }}_{L^q(B_{2R})})$ for weak solutions of $L_{0}u=-F$, and the analogous bound with the same additive structure holds for nonnegative supersolutions (Harnack inequality for nonnegative weak solutions, Weak Harnack inequality for nonnegative supersolutions, Weak subsolutions and supersolutions of a divergence-form equation). The polynomial counterexample itself is valid in every $n\ge 1$ by [F1]-[F2].

Given: The Axiom of Choice and Countable Choice; the unit disc $\mathrm{\Omega }=B_1(0)\subset {\mathbb{R}}^2$; the zero class $u\in H^1(\mathrm{\Omega })$ and the representative $\widehat{u}={\mathbf{1}}_{\{0\}}$.

The local weak formulation: $u$ is a local weak solution of $-\mathrm{\Delta }u=0$ on $\mathrm{\Omega }$ if ${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx=0$ for every $v\in H_0^1(\mathrm{\Omega })$; the zero class satisfies this identically (Local weak solutions of a divergence-form operator).

Weak derivatives depend only on the class: two $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ representatives of the same class have the same weak derivatives, and the set $\{0\}$ is Lebesgue-null, so $\widehat{u}$ and the zero function determine the same class (Weak differentiation ignores null-set changes, The space $L^p(\mu )$ as the quotient by null functions).

Essential versus pointwise suprema: the essential supremum of a class is the infimum of the essential bounds, hence ${\mathrm{ess\hspace{0.17em}sup}}_{\mathrm{\Omega }}u=0$ for the zero class, whereas the pointwise supremum of the particular function $\widehat{u}$ is ${\sup }_{\mathrm{\Omega }}\widehat{u}=1$ (The essential supremum of a measurable function with respect to a measure, Local Hölder and scaled C-two-alpha norms on balls).

The estimates of the page are stated for essential extrema of classes: the local boundedness theorem bounds ${\mathrm{ess\hspace{0.17em}sup}}_{B_{\rho R}}u$ by an $L^p$ mean of the class, and the Harnack inequality bounds ${\mathrm{ess\hspace{0.17em}sup}}_{B_{R\mathrm{/}2}}u$ by ${\mathrm{ess\hspace{0.17em}inf}}_{B_{R\mathrm{/}2}}u$ (De Giorgi local boundedness of homogeneous subsolutions, Harnack inequality for nonnegative weak solutions, De Giorgi-Nash interior Holder regularity for divergence-form equations).