Hölder estimates
Holder continuity of the solutions can sometimes be proved only from ellipticity assumptions on the equation, without depending on smoothness of the coefficients. This allows great flexibility in terms of applications of the result. The corresponding result for elliptic equations of second order is the Krylov-Safonov theorem in the non-divergence form, or the De Giorgi-Nash-Moser theorem in the divergence form.
The holder estimates are closely related to the Harnack inequality.
There are integro-differential versions of both De Giorgi-Nash-Moser theorem and Krylov-Safonov theorem. The former uses variational techniques and is stated in terms of Dirichlet forms. The latter is based on comparison principles.
A Holder estimate says that a solution to an integro-differential equation $L_x u(x) = f(x)$ in $B_1$, is $C^\alpha$ in $B_{1/2}$ for some $\alpha>0$ (small). It is very important to allow for a very rough dependence of $L_x$ with respect to $x$, since the result then applies to the linearization of fully nonlinear equations without any extra a priori estimate. On the other hand, the linearization of a fully nonlinear equation (for example the Isaacs equation) would inherit the initial assumptions regarding for the kernels with respect to $y$. Therefore, smoothness (or even structural) assumptions for the kernels with respect to $y$ can be made keeping such result useful.
In the non variational setting the integro-differential operators $L_x$ will be assumed to belong to some family, but no continuity is assumed for its dependence with respect to $x$. Typically, $L_x u(x)$ has the form $$ L_x u(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \, \chi_{B_1}(y)) K(x,y) \, dy$$ Since integro-differential equations allow for a great flexibility of equations, there are several variations on the result: different assumptions on the kernels, mixed local terms, evolution equations, etc. The linear equation with rough coefficients is equivalent to the function $u$ satisfying two inequalities for the extremal operators corresponding to the family of operators $L$, which stresses the nonlinear character of the estimates.
Some estimates blow up as the order of the equation converges to two, and others pass to the limit. In the latter case the estimates are a true generalization of either De Giorgi-Nash-Moser or Krylov-Safonov]
Estimates which blow up as the order goes to two
Non variational case
The first Holder estimates were obtained using probabilistic techniques[citation needed]. The first purely analytic proof was given in[citation needed]. The assumptions are that for each $x$ the kernel $K(x,.)$ belongs to a family satisfying certain set of assumptions. No regularity of any kind is assumed for $K$ with respect to $x$. The assumption for the family of operators are
- Scaling: If $L$ belongs to the family, then so does its scaled version $L_r u(x) = C_{r,L} L [u(x/r)] (x)$ for any $r<1$ and some $C_{r,L}<1$ which could depend on $L$, but $C_{r,L} \to 0$ as $r \to 0$ uniformly in $L$.
- Nondegeneracy: If $K$ is the kernel associated to $L$, $\frac{\int_{\R^n} \min(y^2,y^\alpha) K(y) \, dy} {\sup_{B_1} K} \leq C_1$ for some $C_1$ and $\alpha>0$ independent of $K$.
The right hand side $f$ is assumed to belong to $L^\infty$.
A particular cases in which this result applies is the uniformly elliptic case. $$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$ where $s$ is bounded below and above: $0 < s_0 \leq s(x) \leq s_1 < 2$, but no continuity of $s$ respect to $x$ is required. The kernel $K$ is assumed to be symmetric with respect to $y$: $K(x,y)=K(x,-y)$. However this assumption can be overcome in the following two situations.
- For $s<1$, the symmetry assumption can be removed if the equation does not contain the drift correction term: $\int_{\R^n} (u(x+y) - u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
- For $s>1$, the symmetry assumption can be removed if the drift correction term is global: $\int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) K(x,y) \, dy = f(x)$ in $B_1$.
The reason for the symmetry assumption, or the modification of the drift correction term, is that in the original formulation the term $y \cdot \nabla u(x) \, \chi_{B_1}(y)$ is not scale invariant.
Variational case
A Dirichlet forms is a quadratic functional of the form $$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.
Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in De Giorgi-Nash-Moser theorem.
The symmetry assumption $K(x,y)=K(y,x)$ is natural since the skew-symmetric part of $K$ would be ignored by the quadratic functional.
In[citation needed] it is shown that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous. The method of the proof builds an integro-differential version of the parabolic De Giorgi technique that was developed for the study of critical surface quasi-geostrophic equation.
At some point in the original proof of De Giorgi, it is used that the characteristic functions of a set of positive measure do not belong to $H^1$. Moreover, a quantitative estimate is required about the measure of intermediate level sets for $H^1$ functions. In the integro-differential context, the required statement to carry out the proof would be the same with the $H^{s/2}$ norm. This required statement is not true for $s$ small, and would even require a non trivial proof for $s$ close to $2$. The difficulty is bypassed though an argument that takes advantage of the nonlocal character of the equation, similarly as in[citation needed].
Estimates which pass to the second order limit
Non variational case
In[citation needed], an integro-differential generalization of Krylov-Safonov theorem is proved. The assumption on the kernels are
- Symmetry: $K(x,y) = K(x,-y)$.
- Uniform ellipticity: $\frac{(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac{(2-s) \Lambda}{|y|^{n+s}}$ for some fixed value $s \in (0,2)$.
The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Holder estimate do not blow up as $s \to 2$.
Variational case
In[citation needed], it is shown that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of De Giorgi-Nash-Moser to the nonlocal setting.