Uniformly elliptic operator

A second-order operator with coefficient matrix $A(x)=(a^{ij}(x))$ is uniformly elliptic when some $\lambda >0$ satisfies $a^{ij}(x){\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ everywhere.

The quadratic form of a matrix only sees its symmetric part. Thus $\lambda |\xi {|}^2\le {\xi }^{T}A\xi \le \Lambda |\xi {|}^2$ need not bound the skew part of a measurable coefficient matrix. For $A=\left(\begin{array}{cc}1& k\\ -k& 1\end{array}\right)$ the quadratic form is exactly $|\xi {|}^2$, regardless of $k$. On a square, take $k(x)=1/|x_1|$ off the line $x_1=0$, with a finite arbitrary value on that line. Two compactly supported smooth functions that equal $x_2$ and $x_1$ near the origin have $(A\nabla u)\cdot \nabla v=k$ there; the purported energy pairing is not integrable. A Lax-Milgram theorem proof therefore requires bounded full coefficients or another hypothesis that makes the bilinear form bounded. Symmetry supplies the missing matrix bound from the quadratic upper bound.

A nondivergence-form elliptic operator applies its principal coefficients directly to the Hessian matrix: $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$. Only the symmetric part of $a$ contributes to the second-derivative term. Uniform ellipticity bounds its quadratic form below by $\lambda |\xi {|}^2$ with a constant $\lambda >0$ independent of position.

A divergence-form elliptic operator is $Lu=D_i(a^{ij}{D}_{j}u)$, with bounded measurable coefficients whose symmetric part has a uniform positive lower bound. Its weak formulation involves only first weak derivatives, so differentiability of the coefficients is unnecessary.

On an interval, the weak equation $(a{u}^{\prime }{)}^{\prime }=0$ makes $a{u}^{\prime }$ constant almost everywhere. If $\lambda \le a\le \Lambda$, the derivative energy on concentric subintervals satisfies $E(r)\le (\Lambda /\lambda {)}^2(r/R)E(R)$. This supplies dyadic energy decay even though a one-dimensional annulus is disconnected.

If $Lu=\mathrm{div}F+g$ and $L$ contains $\mathrm{div}(bu)+du$, then $L(u-k)=\mathrm{div}(F-kb)+(g-kd)$. Thus shifting a weak solution to make it nonnegative changes the forcing unless the operator annihilates constants. This bookkeeping is essential in applying the Weak Harnack inequality to upper and lower oscillations.

For a bounded measurable coefficient matrix $A$, an $H^1$ weak solution of $\mathrm{div}(A\nabla u)=0$ satisfies $\int A\nabla u\cdot \nabla \phi =0$ for every compactly supported smooth test function. By density this identity extends to compactly supported $H^1$ tests. If $A$ is symmetric and $\lambda I\le A\le \Lambda I$, Caccioppoli inequalities follow without differentiating $A$.

For $Lu=D_i(a^{ij}{D}_{j}u)$, a weak supersolution satisfies $Lu\le 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \ge 0$ for every nonnegative compactly supported test function $\varphi$. The Weak Harnack inequality applies to nonnegative weak supersolutions with this sign convention.

For the sign convention $Lu=D_i(a^{ij}{D}_{j}u)$, a weak subsolution satisfies $Lu\ge 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \le 0$ for every nonnegative compactly supported test function $\varphi$. A weak supersolution has the reversed inequality.

For a uniformly elliptic divergence-form elliptic operator with lower-order coefficients in the supercritical Lebesgue spaces, truncation tests and Moser iteration bound the positive part of a weak subsolution on an interior ball by its Lp norm on a larger ball and the forcing norms. In particular, every weak subsolution of the homogeneous equation is locally bounded above. This is the preliminary estimate needed when a strong maximum principle is phrased using essential suprema.

Let $B[u,v]={\int }_U{a}^{ij}{\partial }_{i}u{\partial }_{j}v+cuv$ on $H_0^1(U)$ for a bounded smooth domain, smooth symmetric uniformly positive coefficients, and bounded real $c$. If $B[u,u]>0$ for every nonzero $u$, then $B[u,u]\ge \alpha \Vert u{\Vert }_{H^1}^2$ for some $\alpha >0$. If this failed, choose $\Vert u_j{\Vert }_{H^1}=1$ with $B[u_j,u_j]\to 0$. The Rellich-Kondrachov compactness theorem gives a subsequence weakly converging in $H_0^1$ and strongly in $L^2$. The principal energy has weak lower semicontinuity, and the potential term converges, so strict positivity forces the limit to be zero. Uniform ellipticity then gives $\Vert D{u}_j{\Vert }_2\to 0$, contradicting normalization. Merely being a positive semidefinite operator is insufficient: $-\Delta -{\lambda }_1$ has a nonzero Dirichlet eigenfunction of zero energy.

If the constant symmetric matrix $A$ satisfies $A\xi \cdot \xi \ge \theta |\xi {|}^2$, then every smooth function $u$ with compact support satisfiesIndeed, the Fourier transform of a derivative and the Plancherel theorem turn the squared norms into integrals with Fourier multipliers $|\xi {|}^4$ and $(A\xi \cdot \xi {)}^2$.

If $\Vert a^{ij}-A^{ij}{\Vert }_{\infty }<\epsilon$ for every $i,j$, then the Cauchy-Schwarz inequality givesConsequently $\epsilon \le \theta /(2n)$ preserves half of the estimate's lower bound.

For a uniformly elliptic operator $Lu=a^{ij}(x)D_{ij}u$ with continuous coefficients and $W⋐U$,Uniform continuity lets one freeze $a^{ij}$ on sufficiently small open balls and use the perturbation of a constant-coefficient elliptic second-derivative estimate. A partition of unity and cutoff functions reduce the global interior estimate to finitely many such balls; derivatives of the cutoffs produce only the displayed lower-order norm.

Let $L=a^{ij}(x)D_{ij}$ be uniformly elliptic with continuous coefficients. The coefficient-freezing interior second-derivative estimate extends from smooth functions to the graph norm closure of $L$ on $H^1(U)$: if $u_k\to u$ in $H^1(U)$ and $L{u}_k\to f$ in $L^2(U)$, then $(u_k)$ is Cauchy in $H^2(W)$ for every $W⋐U$. Hence $u\in H_{\mathrm{loc}}^2(U)$ and $Lu=f$. Equivalently, the standard local regularization argument gives the same conclusion for weak solutions.

A second-order operator is strictly elliptic at a point when its symmetric principal coefficient matrix is positive definite there. Uniform ellipticity strengthens this pointwise condition by requiring one positive lower bound throughout the domain.

For a second-order elliptic operator $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$ with $c\le 0$, the inequality $Lu\ge 0$ on a bounded domain impliesAn exponential barrier reduces the non-strict inequality to a contradiction at a positive interior maximum.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, odd, strictly increasing and onto. Suppose $u\in C^2({\mathbb{R}}^d)$ tends to zero at infinity and satisfies $-\Delta u+h(u)=f$, with bounded $f$. At a positive maximum $m$ of $u$, $-\Delta u\ge 0$, so $h(m)\le f\le \Vert f{\Vert }_{\infty }$. Apply the same argument to $-u$ to obtain the displayed bound, using the maximum-point argument of the weak maximum principle for elliptic operators. For $h(s)=s+\sin s$, strict monotonicity follows by integrating $h^{\prime }=1+\cos s$, whose zeros are isolated. Thus $\Vert u{\Vert }_{\infty }\le h^{-1}(\Vert f{\Vert }_{\infty })\le \Vert f{\Vert }_{\infty }+1$. The stronger bound by $\Vert f{\Vert }_{\infty }$ holds when that norm is at most $\pi$, but need not hold for larger sources.

If $c\le 0$, $\Delta u+cu=f$, and $u=\psi$ on the boundary of a bounded domain inside $\{|x_1|<d\}$, let $M={\sup }_{\partial \Omega }|\psi |$ and $F={\sup }_{\Omega }|f|$. The nonnegative barrier $v=M+(e^{2d}-e^{x_1+d})F$ satisfies $(\Delta +c)v\le -F$. Apply the weak maximum principle for elliptic operators to $u-v$ and $-u-v$ to get $\Vert u{\Vert }_{\infty }\le M+e^{2d}F$.

For a locally uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with locally bounded coefficients and $c\le 0$, a $C^2$ function satisfying $Lu\ge 0$ on a connected domain cannot attain a nonnegative global maximum inside unless it is constant. If $c=0$, the maximum can have either sign. If the maximum set were proper, choose an interior ball in its complement tangent to that set. The Hopf boundary point lemma gives a nonzero normal derivative at the contact point, contradicting the vanishing gradient at an interior maximum.

The strong minimum principle for elliptic operators follows by applying this to $-u$. For nonnegative $u$ with $Lu\le 0$, replacing $c$ by $\min (c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.

For $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u$ with $a^{ij}{\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ and $|b^1|<\lambda \ell$, the function $q(x)=e^{-\ell x_1}$ satisfiesIf $Lu\ge 0$ had a strict interior maximum above the boundary maximum, then $u+\epsilon q$ would retain an interior maximum for small $\epsilon >0$, where ellipticity gives $L(u+\epsilon q)\le 0$. This contradicts its strictly positive image under $L$.

A positive zeroth-order coefficient can destroy the weak maximum principle. On $(0,1)$, the function $u(x)=\sin (\pi x)$ has zero boundary values and a positive interior maximum, but

The Alexandrov–Bakelman–Pucci estimate bounds the positive maximum of a function by its boundary maximum and an $L^n$ norm of the negative part of a uniformly elliptic operator applied to it. It yields a maximum principle and a sup-norm estimate for elliptic equations with measurable principal coefficients.

Let $L$ be uniformly elliptic with bounded coefficients and nonpositive zeroth-order coefficient. If $u\ge 0$, $Lu\le 0$, and $u$ vanishes at an interior point of a connected domain, then $u$ vanishes identically. This is the minimum form of the strong maximum principle.

A Schauder estimate bounds a solution in a Hölder norm two derivatives stronger than the forcing term. For a uniformly elliptic Dirichlet problem with $C^{\alpha }$ coefficients,

For $u\in C^{l,\alpha }(B_R)$, $0\le j<l$, and every $\epsilon >0$,with analogous estimates for lower Hölder seminorms. Scaling reduces the result to the unit ball.

For ${\Omega }^{\prime }⋐\Omega$ and a uniformly elliptic operator with $C^{0,\alpha }$ coefficients,The constant deteriorates as ${\Omega }^{\prime }$ approaches the boundary.

The bounded-forcing norm alone cannot control the Hölder seminorm of the Hessian matrix. On a unit Euclidean ball take $u_m=m^{-2}\sin (m{x}_1)$ and $f_m=\Delta u_m=-\sin (m{x}_1)$. The derivative supremum norm $|u_m{|}_2$ and $\Vert f_m{\Vert }_{\infty }$ stay bounded, but the Hessian matrix difference between zero and $\pi /(2m)e_1$ gives $[D^2{u}_m{]}_{\alpha }\ge (2m/\pi {)}^{\alpha }$ on $B_{1/2}$.

A compactness proof of the interior Schauder estimate can magnify a pair of points where the Hölder seminorm of the Hessian matrix is nearly attained. If the proposed estimate fails, normalize that seminorm to one while lower norms and forcing tend to zero. The separation $r_j$ of the chosen pair tends to zero. Subtract the quadratic Taylor polynomial and divide by $r_j^{2+\alpha }$ after rescaling space by $r_j$.

The resulting functions have uniformly bounded Hessian Hölder seminorms on expanding domains and vanishing Taylor data at zero. Compact embedding of Hölder spaces yields an entire harmonic function whose second derivatives have finite global $\alpha$-Hölder seminorms, with $0<\alpha <1$. The polynomial-growth Liouville theorem for harmonic functions makes those derivatives constant, contradicting the normalized nonzero oscillation. The Simon absorption lemma removes the residual small multiple of the larger-ball seminorm.

Subtracting the quadratic Taylor polynomial at the rescaling centre makes the function, gradient and Hessian matrix vanish there. Dividing by $A{r}^{2+\mu }$ preserves the normalized Hölder oscillation of the Hessian matrix while turning a small forcing-to-oscillation ratio into a harmonic limiting equation. The three vanished Taylor coefficients provide local bounds needed by the Arzelà-Ascoli theorem.

For a uniformly elliptic equation on a half-ball with $C^{0,\alpha }$ coefficients and $C^{2,\alpha }$ Dirichlet data $\phi$ on its flat face,

On a smooth bounded domain, interior and boundary Schauder estimates combine into a $C^{2,\alpha }$ estimate up to the full boundary. The constant depends on the domain regularity, coefficient norms, and ellipticity constants.

On a fixed smooth bounded domain, zero-boundary solutions with forcing in a bounded subset of $C^{0,\mu }$ are uniformly bounded in $C^{2,\mu }$ by a maximum-principle barrier and the global Schauder estimate. The Arzelà-Ascoli theorem gives convergence in $C^2$. A closed Hölder-norm constraint passes to the limit by lower semicontinuity of a Hölder seminorm, so the constrained solution set is compact rather than merely relatively compact.

For a fixed uniformly elliptic operator with $C^{0,\mu }$ coefficients on a bounded smooth connected domain, nonnegative solutions with zero Dirichlet boundary condition satisfy the displayed estimate when ${\Omega }^{\prime }$ is nonempty and compactly contained in $\Omega$. If it failed, normalize the global supremum to one. The forcing and the interior infimum then tend to zero. The global Schauder estimate and compact embedding of Hölder spaces produce a nonzero homogeneous limit with an interior zero, contradicting the strong minimum principle for elliptic operators. An arbitrary bounded zeroth-order coefficient is allowed: replace $c$ by $\min (c,0)$ for the positivity argument. Keeping global boundary control prevents the normalized maximum from disappearing in the limit.

The Simon absorption lemma turns a scale-local estimate containing a sufficiently small multiple of the same quantity on a larger ball into a uniform estimate. Its boundary form uses balls intersected with a half-space and is the covering step that absorbs local boundary Hölder seminorms.

If $u\in W^{1,2}(B_1)$ weakly solves $D_i(a^{ij}{D}_{j}u)=0$ with bounded measurable uniformly elliptic coefficients, then $u$ is locally Hölder continuous. For $0<\theta <1$, some $\alpha \in (0,1)$ and $C$ depending only on the dimension, ellipticity bounds, and $\theta$ satisfy

For a fixed bounded measurable uniformly elliptic operator in divergence form, solutions bounded by one have uniform local Caccioppoli inequalities and Hölder continuity estimates. A diagonal subsequence argument yields a local weak $W^{1,2}$ limit, while the Arzelà-Ascoli theorem gives uniform convergence on a fixed smaller closed Euclidean ball. The weak equation passes to the limit because the coefficients are fixed and bounded. A global supremum normalization does not force the interior limit to be nonzero.

Moser iteration combines a Caccioppoli inequality with the Sobolev embedding theorem to raise an integrability exponent geometrically. On nested balls it turns an $L^p$ bound for a nonnegative subsolution of a uniformly elliptic divergence-form equation into a local supremum bound.

For $q_i=2{\alpha }^i$, $r_i=1/2+2^{-i-1}$, a uniform power gain for elliptic solutions bounds the successive norms by factors $[DR{2}^{i+2}{]}^{{\alpha }^{-i}}$. Their infinite product isIt is finite because both $\sum {\alpha }^{-i}$ and $\sum i{\alpha }^{-i}$ converge. Uniform control of the $L^{q_i}$ norms on $B_{1/2}$ then controls the essential supremum, by comparing with the measure of a set where the function exceeds the proposed bound. A constant independent of $q$ left outside the $2/q$ power at every step would instead give a divergent product.

For $n\ge 3$, a nonnegative entire weak subsolution of a bounded measurable uniformly elliptic divergence-form equation that belongs to $L^p({\mathbb{R}}^n)$ for some $p>1$ is zero almost everywhere. Rescaling the local Moser iteration estimate gives $\Vert u{\Vert }_{L^{\infty }(B_{R/2})}\le C{R}^{-n/p}\Vert u{\Vert }_{L^p(B_R)}$. Let $R\to \infty$.

The weak estimate for supersolutions is distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions.

For a nonnegative weak supersolution of a uniformly elliptic divergence-form equation, a positive $L^q$ mean on a ball is bounded by a constant times the infimum on a smaller concentric ball. The admissible exponent and constant depend only on dimension and the ellipticity bounds.

For a nonnegative weak supersolution with zero forcing, the Weak Harnack inequality makes zero essential infimum on an interior ball imply vanishing almost everywhere on that ball. An overlapping ball then has zero essential infimum too. Chains of overlapping interior balls propagate the vanishing through a connected domain.

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the Weak Harnack inequality implyfor a universal $0<\theta <1$. Iteration gives local Hölder continuity.

If the essential oscillation on concentric balls obeys $\omega (R/2)\le \theta \omega (R)+A{R}^{\alpha }$ with $0<\theta <1$ and $\alpha >0$, dyadic iteration gives $\omega (r)\le C{r}^{\mu }$ for every $0<\mu <\min \{\alpha ,-{\log }_2\theta \}$. The resulting decay gives a Hölder continuous function. Taking the exponent strictly below both decay rates avoids a logarithmic factor when the rates coincide.

A nonnegative weak solution of a uniformly elliptic divergence-form equation satisfieswhenever a fixed larger concentric ball lies in the domain. Combine the subsolution bound from Moser iteration with the Weak Harnack inequality for supersolutions.