Interior regularity results (local) and Hölder estimates: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Luis
No edit summary
 
No edit summary
 
Line 1: Line 1:
Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).
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]].


== Linear equations ==
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.


Regularity results for linear equations are applicable to nonlinear equations as well through the linearization of the equation. However, this process requires some initial regularity knowledge on the solution (since the coefficients of the linearization depend on the solution itself). Therefore, the less regularity required for the coefficients, the more useful the theorem is.
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.


All regularity results that require some modulus of continuity or smallness condition for the coefficients rely on the idea that the solution is locally close to a solution to an equation with constant coefficients. The proof is based on an estimate on how far these two solutions are at small scales. These type of arguments are often called [[perturbation methods]].
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.


From the results below for linear equations, [[De Giorgi-Nash-Moser]] and [[Krylov-Safonov]] are the only non perturbative results. Their assumptions are scale invariant in the sense that a rescaling of the solution ($u_r(x) = u(rx)$) would solve an elliptic equation with the same bounds as the original.
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]]]


* [[De Giorgi-Nash-Moser]]
== Estimates which blow up as the order goes to two ==
<math> {\rm div \,} A(x) Du + b(x) \cdot \nabla u = 0 </math>


then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
=== Non variational case ===
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>{\rm div \,} b=0</math>).


* [[Krylov-Safonov]]
The Holder estimates were first obtained using probabilistic techniques <ref name="BL"/> <ref name="BK"/> , and then using purely analytic methods <ref name="S"/>. 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
<math>a_{ij}(x) u_{ij} + b \cdot \nabla u = f </math>
# '''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 $K$.


with <math>a_{ij}</math> unif elliptic, <math>b \in L^n</math> and <math>f \in L^n</math>, then the
The right hand side $f$ is assumed to belong to $L^\infty$.
solution is <math>C^\alpha</math>


* [[Calderon-zygmund]]
A particular cases in which this result applies is the uniformly elliptic case.
<math>a_{ij}(x) u_{ij} = f</math>
$$\frac{\lambda}{|y|^{n+s(x)}} \leq K(x,y) \leq \frac{\Lambda}{|y|^{n+s(x)}}.$$
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 $B_1$.


with <math>a_{ij}</math> close enough to the identity (or continuous) and <math>f \in L^p</math>, then <math>u</math> is in <math>W^{2,p}</math>.
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.


* [[Cordes-Nirenberg]]
=== Variational case ===
<math>a_{ij}(x) u_{ij} = f</math>


with <math>a_{ij}</math> close enough to the identity uniformly and f in L^infty,
A [[Dirichlet forms]] is a quadratic functional of the form
then $u$ is in $C^{1,\alpha}$
$$ J(u) = \iint_{\R^n \times \R^n} |u(x)-u(y)|^2 K(x,y) \, dx \, dy $$.


* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
Minimizers of Dirichlet forms are a nonlocal version of minimizers of integral functionals as in [[De Giorgi-Nash-Moser]] theorem.
<math>a_{ij}(x) u_{ij} = f</math>


with $a_{ij}$ close enough to the identity in a scale invariant Morrey
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.
norm in terms of $L^n$ and $f \in L^n$, then $u$ is in $C^{1,\alpha}$.


($a_{ij} \in VMO$ is a particular case of this)
It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous <ref name="CCV"/>. 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]].


*[[Schauder]]
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, and hence the estimate blows up as the order approaches two.
<math>a_{ij}(x) u_{ij} = f</math>


with $a_{ij}$ in $C^\alpha$ and $f \in C^\alpha$, then $u$ is in $C^{2,\alpha}$
== Estimates which pass to the second order limit ==


== Non linear equations ==
=== Non variational case ===


* [[De Giorgi-Nash-Moser]]
An integro-differential generalization of [[Krylov-Safonov]] theorem is available <ref name="CS"/>. The assumption on the kernels are
For any smooth strictly convex Lagrangian $L$, minimizers of functionals
# '''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 $s \in (0,2)$.


$ \int_D L(\nabla u) \ dx $
The right hand side $f$ is assumed to be in $L^\infty$. The constants in the Holder estimate do not blow up as $s \to 2$.


are smooth (analytic if $L$ is analytic).
=== Variational case ===


* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
In the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of [[De Giorgi-Nash-Moser]] to the nonlocal setting <ref name="K"/>.


Any continuous function $u$ such that
== References ==
 
{{reflist|refs=
$M^+(D^2 u) \geq 0 \geq M^-(D^2 u)$
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
 
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
in the viscosity sense (where $M^+$ and $M^-$ are the Pucci operators), is Holder continuous.
<ref name="CCV">{{Citation | last1=Caffarelli | first1=Luis | last2=Chan | first2=Chi Hin | last3=Vasseur | first3=Alexis | title= | doi=10.1090/S0894-0347-2011-00698-X | year=2011 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | issue=24 | pages=849–869}}</ref>
 
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Hölder continuity of harmonic functions with respect to operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 | doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial Differential Equations | issn=0360-5302 | volume=30 | issue=7 | pages=1249–1259}}</ref>
 
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
* [[Ishii-Lions]]
}}
 
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, Du, u, x) = 0$
 
which is degenerate elliptic but satisfies some structure conditions and some smoothness assumptions respect to $x$, then $u$ is Lipschitz.
 
(The proof of this is based on the uniqueness technique for viscosity solutions)
 
* [[Lin]]
 
Any continuous function $u$ such that
 
$0 \geq M^-(D^2 u)$
 
in the viscosity sense, is twice differentiable almost everywhere and $D^2 u \in L^\varepsilon$.
 
(Note that any solution to a fully nonlinear uniformly elliptic equation satisfies this, even with rough coefficients)
 
* [[Krylov-Safonov|Krylov-Safonov-Caffarelli]]
If $u$ solves a fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and continuous respect to $x$ ($VMO$ is actually enough), then $u \in C^{1,\alpha}$.
 
* [[Evans-Krylov]]
If $u$ solves a convex (or concave) fully nonlinear equation
 
$F(D^2 u, x) = 0$
 
which is uniformly elliptic and $C^\alpha$ respect to $x$, then $u \in C^{2,\alpha}$.

Revision as of 20:14, 23 May 2011

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 Holder estimates were first obtained using probabilistic techniques [1] [2] , and then using purely analytic methods [3]. 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

  1. 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 $L$.
  2. 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.

It is known that the gradient flow of a Dirichlet form (parabolic version of the result) becomes instantaneously Holder continuous [4]. 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, and hence the estimate blows up as the order approaches two.

Estimates which pass to the second order limit

Non variational case

An integro-differential generalization of Krylov-Safonov theorem is available [5]. The assumption on the kernels are

  1. Symmetry: $K(x,y) = K(x,-y)$.
  2. 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 the stationary case, it is known that minimizers of Dirichlet forms are Holder continuous by adapting Moser's proof of De Giorgi-Nash-Moser to the nonlocal setting [6].

References

  1. 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 
  2. Bass, Richard F.; Kassmann, Moritz (2005), "Hölder continuity of harmonic functions with respect to operators of variable order", Communications in Partial Differential Equations 30 (7): 1249–1259, doi:10.1080/03605300500257677, ISSN 0360-5302, http://dx.doi.org/10.1080/03605300500257677 
  3. 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 
  4. Caffarelli, Luis; Chan, Chi Hin; Vasseur, Alexis (2011), Journal of the American Mathematical Society (24): 849–869, doi:10.1090/S0894-0347-2011-00698-X, ISSN 0894-0347 
  5. 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 
  6. 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