Harnack inequality

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The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case, for nonlocal equations one needs to assume that the function is nonnegative in the full space.

The Harnack inequality is tightly related to Holder estimates. Whenever one holds, the other is expected to hold as well, either as a direct consequence or by applying the same proof method.

The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not valid, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.

Elliptic case

In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then \[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + ||f|| \right). \]

The norm $||f||$ may depends on the type of equation.

Parabolic case

In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies an elliptic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then \[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + ||f|| \right). \]

The norm $||f||$ may depend on the type of equation.

Concrete examples

The Harnack inequality as above is known to hold in the following situations.

• Generalizad elliptic Krylov-Safonov. If $L_x u(x)$ is a symmetric integro-differential operator of the form

\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \] with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.

In this case the elliptic Harnack inequality is known to hold with a constant $C$ which does not blow up as $s\to 2$, and $||f||$ refers to $||f||_{L^\infty(B_1)}$ . It is a generalization of Krylov-Safonov theorem. The corresponding parabolic Harnack inequality with a uniform constant $C$ is not known.

• Elliptic equations with variable order (but strictly less than 2). If $L_x u(x)$ is an integro-differential operator of the form

\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \] with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then the elliptic Harnack inequality holds if $f \equiv 0$. The constants in this result blow up as $s_2 \to 2$, so it does not generalize Krylov-Safonov theorem. The proof uses probability and was based on a previous result with fixed order .

It is conceivable that a purely analytic proof could be done using the method of the corresponding Holder estimate , but such proof has never been done.

• Gradient flows of symmetric Dirichlet forms with variable order. If $u_t - L_x u(x)=0$ is the gradient flow of a Dirichlet form:

\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 k(x,y)\,dy. \] for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori blows up as $s_2 \to 2$ .

It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates.