Harnack inequality
$$ \newcommand{\dd}{\mathrm{d}} \newcommand{\R}{\mathbb{R}} $$
The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either De Giorgi-Nash-Moser theorem or Krylov-Safonov theorem), 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 for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but Hölder estimates still hold true.
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 automatic, 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 depend 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 a parabolic 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 and parabolic 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 both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which does not blow up as $s\to 2$, and $\|f\|$ refers to $\|f\|_{L^\infty(B_1)}$ ^{[1]}^{[2]}. It is a generalization of Krylov-Safonov theorem.
- 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 $f \equiv 0$^{[3]}. 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 ^{[4]}.
It is conceivable that a purely analytic proof could be done using the method of the corresponding Holder estimate ^{[5]}, 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:
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \] 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 $s_2 \to 2$ ^{[6]}.
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^{[7]}.
- An equation where the Harnack inequality fails.
There exist integro-differential equations of the form \[ \int_{\R^n} (u(x+y)-u(x)) K(y) \dd y = 0,\] for which the Harnack inequality fails. In this case the kernels $K$ can be nonnegative, stable (i.e $K(ry) = r^{-n-s} K(y)$), symmetric (i.e. $K(y) = K(-y)$), independent of $x$, bounded above by \[ K(y) \leq \Lambda |y|^{-n-s},\] and satisfying the bound below only in a cone of directions \[ K(y) \geq \lambda |y|^{-n-s} \mathbb{1}_S(y),\] where $S$ is a cone with positive measure on $\partial B_1$.
The construction was first given in ^{[8]}. See also the discussion in ^{[9]}. The example is particularly relevant because Holder estimates hold for this equation.
References
- ↑ Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20274
- ↑ Chang-Lara, Hector; Davila, Gonzalo, "Holder estimates for non-local parabolic equations with critical drift", arXiv preprint arXiv:1408.0676
- ↑ Bass, Richard F.; Kassmann, Moritz (2005), "Harnack inequalities for non-local operators of variable order", Transactions of the American Mathematical Society 357 (2): 837–850, doi:10.1090/S0002-9947-04-03549-4, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-04-03549-4
- ↑ Bass, Richard F.; Levin, David A. (2002), "Harnack inequalities for jump processes", Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis 17 (4): 375–388, doi:10.1023/A:1016378210944, ISSN 0926-2601, http://dx.doi.org/10.1023/A:1016378210944
- ↑ Silvestre, Luis (2006), "Hölder estimates for solutions of integro-differential equations like the fractional Laplace", Indiana University Mathematics Journal 55 (3): 1155–1174, doi:10.1512/iumj.2006.55.2706, ISSN 0022-2518, http://dx.doi.org/10.1512/iumj.2006.55.2706
- ↑ Barlow, Martin T.; Bass, Richard F.; Chen, Zhen-Qing; Kassmann, Moritz (2009), "Non-local Dirichlet forms and symmetric jump processes", Transactions of the American Mathematical Society 361 (4): 1963–1999, doi:10.1090/S0002-9947-08-04544-3, ISSN 0002-9947, http://dx.doi.org/10.1090/S0002-9947-08-04544-3
- ↑ Kassmann, Moritz (2009), "A priori estimates for integro-differential operators with measurable kernels", Calculus of Variations and Partial Differential Equations 34 (1): 1–21, doi:10.1007/s00526-008-0173-6, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-008-0173-6
- ↑ Bogdan, Krzysztof; Sztonyk, Pawe\l (2005), "Harnack’s inequality for stable Lévy processes", Potential Analysis 22: 133--150
- ↑ Rang, Marcus; Kassmann, Moritz; Schwab, Russell W, "H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence", arXiv preprint arXiv:1306.0082