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| The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic [[elliptic linear integro-differential operator]] of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^n)$ we can write the operator as | | The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. |
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| \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\] | | The equation is |
| | \[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \] |
| | in the elliptic case, or |
| | \[ u_t = \mathrm{div} A(x,t) \nabla u(x). \] |
| | Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$, |
| | \[ \langle A v,v \rangle \geq \lambda |v|^2,\] |
| | for every $v \in \R^n$, uniformly in space and time. |
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| where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a [[comparison principle]] can be derived for equations involving the fractional Laplacian.
| | The corresponding result in non divergence form is [[Krylov-Safonov theorem]]. |
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| == Definitions ==
| | For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]]. |
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| All the definitions below are equivalent.
| | == Elliptic version == |
| | For the result in the elliptic case, we assume that the equation |
| | \[ \mathrm{div} A(x) \nabla u(x) = 0 \] |
| | is satisfied in the unit ball $B_1$ of $\R^n$. |
| | ===Holder estimate=== |
| | The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and |
| | \[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\] |
| | The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$. |
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| === As a Fourier multiplier ===
| | The result can be scaled to balls of arbitrary radius $r>0$ to obtain |
| The fractional Laplacian is the operator with symbol $|\xi|^{2s}$. In other words, the following formula holds | | \[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\] |
| \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\] | |
| for any function (or tempered distribution) for which the right hand side makes sense.
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| This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
| | Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$. |
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| === Via the heat semigroup === | | ===Harnack inequality=== |
| Recall the numerical formula
| | The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$: |
| $$\lambda^s=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{-t\lambda}-1\big)\,\frac{dt}{t^{1+s}},$$ | | \[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\] |
| valid for $\lambda\geq0$ and $0<s<1$. By taking $\lambda=|\xi|^2$ and using the Fourier transform definition of the fractional Laplacian we get
| | The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only. |
| \begin{equation}\label{semigroup} | |
| (-\Delta)^sf(x)=\frac{1}{\Gamma(-s)}\int_0^\infty\big(e^{t\Delta}f(x)-f(x)\big)\,\frac{dt}{t^{1+s}}.
| |
| \end{equation} | |
| Here $v(x,t)=e^{t\Delta}f(x)$ is the solution to the heat equation on $\mathbb{R}^n$ with initial temperature $f$, namely,
| |
| $$\begin{cases} | |
| \partial_tv=\Delta v,&\hbox{for}~(x,t)\in\mathbb{R}^n\times(0,\infty)\\
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| v(x,0)=f(x),&\hbox{for}~x\in\mathbb{R}^n.
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| \end{cases}$$
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| This semigroup language approach was introduced in <ref name="Stinga"/> and <ref name="Stinga-Torrea"/>.
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| === From functional calculus === | | ===Minimizers of convex functionals=== |
| Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.
| | The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals |
| | \[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\] |
| | are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by [[bootstrapping]] with the [[Schauder estimates]] and the smoothness of $F$. |
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| This definition is not as useful for practical applications, since it does not provide any explicit formula.
| | Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made. |
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| === As a singular integral === | | == Parabolic version == |
| If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
| | For the result in the parabolic case, we assume that the equation |
| \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\] | | \[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \] |
| | is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$. |
| | ===Holder estimate=== |
| | The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and |
| | \[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\] |
| | The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$. |
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| Where $c_{n,s}$ is a constant depending on dimension and $s$.
| | ===Harnack inequality=== |
| | The [[Harnack inequality]] says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time: |
| | \[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \] |
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| One way of deducing this formula is the following: write down the heat kernel in the semigroup formula \eqref{semigroup}, use that $e^{t\Delta}1\equiv1$ and apply Fubini's Theorem. See <ref name="Stinga"/> and <ref name="Stinga-Torrea"/> for the details of this computation. Notice that this derivation gives the explicit value of the constant:
| | ===Gradient flows=== |
| $$c_{n,s}=\frac{4^s\Gamma(n/2+s)}{\pi^{n/2}|\Gamma(-s)|}.$$
| | The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth. |
| | | \[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\] |
| This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
| | The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients. |
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| === As a generator of a [[Levy process]] ===
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| The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then | |
| \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]
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| This definition is important for applications to probability.
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| == Inverse operator ==
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| The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'':
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| \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
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| which holds as long as $u$ is integrable enough for the right hand side to make sense.
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| == Heat kernel ==
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| The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
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| \begin{align*}
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| p(0,x) &= \delta_0 \\
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| p_t(t,x) + (-\Delta)^s p &= 0
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| \end{align*}
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| The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$
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| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \] | |
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| Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
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| \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \] | |
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| == Poisson kernel ==
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| Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
| |
| \begin{align*}
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| u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
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| (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1.
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| \end{align*}
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| The solution can be computed explicitly using the Poisson kernel
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| \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
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| where<ref name="R"/>
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| \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
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| The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
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| == Green's function for the ball ==
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| For a function $g \in L^2(B_1)$, there exists a unique function $u \in H^s(\R^n)$ such that
| |
| \begin{align*}
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| u(x) &= 0 && \text{if } x \notin B_1 \\ | |
| (-\Delta)^s u &= g(x) && \text{if } x \in B_1.
| |
| \end{align*}
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| The solution is given explicitly using the Green's function,
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| \[ u(x) = \int_{B_1} G_{B_1}(x, y) g(y) \mathrm d y, \]
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| where<ref name="R"/>
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| \[ G_{B_1}(x, y) = C_{n,s} |x - y|^{2 s - n} \int_0^{r_0(x, y)} \frac{r^{s-1}}{(r+1)^{n/2}} \, \mathrm d r \]
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| with
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| \[ r_0(x, y) = \frac{(1 - |x|^2) (1 - |y|^2)}{|x - y|^2} . \]
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| The above formula holds for all $s \in (0, 1)$ also for $n = 1$.<ref name="BGR"/> | |
| | |
| == Regularity issues ==
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| Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls.
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| More generally, one has the estimate
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| \[\|u\|_{C^{\alpha+2s}(B_{1/2})}\leq C\left(
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| \|(-\Delta)^s u\|_{C^{\alpha}(B_1)}+\|u\|_{L^{\infty}(B_1)}+\int_{\R^n\setminus B_1}|u(y)|\frac{dy}{|y|^{n+2s}}\right)\]
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| for any $\alpha\geq0$ such that $\alpha+2s$ is not an integer.
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| === Full space regularization of the Riesz potential ===
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| If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein<ref name="Stein"/>.
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| === Boundary regularity ===
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| From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).
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| Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem
| |
| \[\left\{ \begin{array}{rcll}
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| (-\Delta)^s u &=&g&\textrm{in }\Omega \\
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| u&=&0&\textrm{in }\R^n\backslash \Omega.
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| \end{array}\right.\]
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| If $\Omega$ is $C^\infty$, then
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| \[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\]
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| where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see <ref name="Grubb"/> and also <ref name="RS"/>.
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| If $\Omega$ is $C^{2,\alpha}$ and $g$ is $C^\alpha$, then $u/d^s$ is $C^{\alpha+s}$ up to the boundary <ref name="RS-K"/>.
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| Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in <ref name="RS2"/>.
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| == References ==
| |
| {{reflist|refs=
| |
| <ref name="BGR">{{Citation | last1=Blumenthal | first1=R. M. | last2=Getoor | first2=R. K. | last3=Ray | first3=D. B. | title=On the distribution of first hits for the symmetric stable processes | url=http://www.jstor.org/stable/1993561 | year=1961 | journal=Trans. Amer. Math. Soc. | issn=0002-9947 | volume=99 | pages=540–554}}</ref>
| |
| <ref name="L">{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}</ref>
| |
| <ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref>
| |
| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
| |
| <ref name="RS">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[J. Math. Pures Appl.]] | volume=101 | pages=275-302 }}</ref>
| |
| <ref name="Stein">{{Citation | last1=Stein | first1=E. | title=Singular Integrals And Differentiability Properties Of Functions | publisher=[[Princeton Mathematical Series]] | year=1970}}</ref>
| |
| <ref name="RS2">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The extremal solution for the fractional Laplacian | url=http://arxiv.org/abs/1305.2489 | year=2013 | journal=[[Calc. Var. Partial Differential Equations]] | pages=to appear }}</ref>
| |
| <ref name="Grubb">{{Citation | last1=Grubb | first1=G. | title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators | url=http://arxiv.org/abs/1310.0951 | year=2014 | journal=[[arXiv]] | pages=1-43 }}</ref>
| |
| <ref name="RS-K">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=Boundary regularity for fully nonlinear integro-differential equations | year=2014 | journal=[[preprint arXiv]] }}</ref>
| |
| <ref name="Stinga">{{Citation | last1=Stinga | first1=P. R.| title=Fractional powers of second order partial differential operators: extension problem and regularity theory | url=https://www.ma.utexas.edu/users/stinga/Stinga%20-%20PhD%20Thesis.pdf | year=2010 | journal=[[PhD. Thesis - Universidad Aut\'onoma de Madrid]]}}
| |
| </ref><ref name="Stinga-Torrea">{{Citation | last1=Stinga | first1=P. R.| last2=Torrea | first2=J. L. | title=Extension problem and Harnack's inequality for some fractional operators | url=http://www.tandfonline.com/doi/abs/10.1080/03605301003735680#.VcJb9UWmR1Q | year=2010 | journal=[[Comm. Partial Differential Equations]] | volume=35 | issue=11 | pages=2092-2122}}</ref>
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| }}
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The De Giorgi-Nash-Moser theorem provides Holder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form.
The equation is
\[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \]
in the elliptic case, or
\[ u_t = \mathrm{div} A(x,t) \nabla u(x). \]
Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$,
\[ \langle A v,v \rangle \geq \lambda |v|^2,\]
for every $v \in \R^n$, uniformly in space and time.
The corresponding result in non divergence form is Krylov-Safonov theorem.
For nonlocal equations, there are analogous results both for Holder estimates and the Harnack inequality.
Elliptic version
For the result in the elliptic case, we assume that the equation
\[ \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit ball $B_1$ of $\R^n$.
Holder estimate
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $B_{1/2}$ and
\[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.
The result can be scaled to balls of arbitrary radius $r>0$ to obtain
\[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]
Moreover, by covering an arbitrary domain $\Omega$ with balls, one can show that a solution to the equation in $\Omega$ is $C^\alpha$ in the interior of $\Omega$.
Harnack inequality
The Harnack inequality says that if $u$ is a non negative solution of the equation in $B_1$, then its minimum controls its maximum in $B_{1/2}$:
\[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\]
The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.
Minimizers of convex functionals
The theorem of De Giorgi, Nash and Moser was used originally to solve one of the famous Hilbert problems. The question was whether the minimizers of Dirichlet integrals
\[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\]
are always smooth if $F$ is smooth and strictly convex. The theorem of De Giorgi-Nash-Moser in its elliptic form can be applied to the differential quotients of the minimizer of $J$ to show that the solution is $C^{1,\alpha}$. Once that initial regularity is obtained, further regularity follows by bootstrapping with the Schauder estimates and the smoothness of $F$.
Note that in order to apply the theorem to these nonlinear equations, it is very important that no smoothness assumption on the coefficients $A(x)$ is made.
Parabolic version
For the result in the parabolic case, we assume that the equation
\[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \]
is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.
Holder estimate
The Holder estimate says that if $u$ is an $L^2$ solution to a uniformly elliptic divergence form equation as above, then $u$ is Holder continuous in $[1/2,1] \times B_{1/2}$ and
\[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\]
The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.
Harnack inequality
The Harnack inequality says that if $u$ is a non negative solution of the equation in $[0,1] \times B_1$, then its minimum controls its maximum in a previous time:
\[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,0] \times B_{1/2}} u. \]
Gradient flows
The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
\[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
The idea of the proof is that the derivatives of $u$ (or its differential quotients) satisfy an equation with rough but uniformly elliptic coefficients.