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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 result was first obtained independently by Ennio De Giorgi <ref name="DG"/> and John Nash <ref name="N"/>. Later, a different proof was given by Jurgen Moser <ref name="M"/>.
| | I've just wrote a simple page about this theorem to start with. It would be desirable to later add the required assumptions on lower order terms. At least the full theorem from Gilbarg-Truding book should be described. We also need to fill the literature. [[User:Luis|Luis]] 15:13, 14 March 2012 (CDT) |
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| The equation is
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| \[ \mathrm{div} A(x) \nabla u(x) = \partial_i a_{ij}(x) \partial_j u(x) = 0, \]
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| in the elliptic case, or
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| \[ u_t = \mathrm{div} A(x,t) \nabla u(x). \]
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| Here $A = \{a_{ij}\}$ is a matrix valued function in $L^\infty$ satisfying the uniform ellipticity condition for some $\lambda>0$,
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| \[ \langle A v,v \rangle \geq \lambda |v|^2,\]
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| for every $v \in \R^n$, uniformly in space and time.
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| The corresponding result in non divergence form is [[Krylov-Safonov theorem]].
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| For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].
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| == Elliptic version ==
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| For the result in the elliptic case, we assume that the equation
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| \[ \mathrm{div} A(x) \nabla u(x) = 0 \]
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| is satisfied in the unit ball $B_1$ of $\R^n$.
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| ===Holder estimate===
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| 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
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| \[ ||u||_{C^\alpha(B_{1/2})} \leq C ||u||_{L^2(B_1)}.\]
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| The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.
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| The result can be scaled to balls of arbitrary radius $r>0$ to obtain
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| \[ [u]_{C^\alpha(B_{r/2})} \leq C \frac{||u||_{L^2(B_r)}}{r^\alpha}.\]
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| 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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| ===Harnack inequality===
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| 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 $B_{1/2}$:
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| \[ \max_{B_{1/2}} u \leq C \min_{B_{1/2}} u.\]
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| The constant $C$ depends on $n$, $\lambda$ and $||A||_{L^\infty}$ only.
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| ===Minimizers of convex functionals===
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| 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
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| \[ J(u) := \int_{\Omega} F(\nabla u) \mathrm{d} x,\]
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| 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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| 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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| == Parabolic version ==
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| For the result in the parabolic case, we assume that the equation
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| \[ u_t - \mathrm{div} A(x) \nabla u(x) = 0 \]
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| is satisfied in the unit cylinder $(0,1] \times B_1$ of $\R \times \R^n$.
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| ===Holder estimate===
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| 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
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| \[ ||u||_{C^\alpha([1/2,1] \times B_{1/2})} \leq C ||u||_{L^2([0,1] \times B_1)}.\]
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| The constants $C$ and $\alpha>0$ depend on $n$ (dimension), $\lambda$ and $||A||_{L^\infty}$.
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| ===Harnack inequality===
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| 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:
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| \[ \sup_{[1/4,1/2] \times B_{1/2}} u \leq \inf_{[3/4,1] \times B_{1/2}} u. \]
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| ===Gradient flows===
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| The parabolic version of the theory can be used to show that the solutions to gradient flow equations with strictly convex energies are smooth.
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| \[ u_t + \partial_u J[u] = u_t - \mathrm{div} \left( (\partial_i F)(\nabla u) \partial_i u \right) = 0.\]
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| 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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| == References ==
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| {{reflist|refs=
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| <ref name="DG">{{Citation | last1=De Giorgi | first1=Ennio | title=Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari | year=1957 | journal=Memorie della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematicahe e Naturali. (3) | volume=3 | pages=25–43}}</ref>
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| <ref name="N">{{Citation | last1=Nash | first1=John | author1-link=John Forbes Nash | title=Parabolic equations | year=1957 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=43 | pages=754–758}}</ref>
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| <ref name="M">{{Citation | last1=Moser | first1=Jürgen | title=A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations | year=1960 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=13 | pages=457–468}}</ref>
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| }}
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