Nonlocal electrostatics and De Giorgi-Nash-Moser theorem: Difference between pages

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(Created page with "Nonlocal electrostatics is a technique currently under development which may turn into a powerfull tool for drug design <ref name="ICH"/> <ref name="HBRK"/> <ref name="SBRF"/>. ...")
 
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Nonlocal electrostatics is a technique currently under development which may turn into a powerfull tool for drug design <ref name="ICH"/> <ref name="HBRK"/> <ref name="SBRF"/>.
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 idea is that when computing the electric potential around a protein, which is surrounded by water, this potential interacts with the ions in the water, which affect the potential effectively transforming it from the classical coulomb potential (i.e. the fundamental solution of the Laplacian) to the potential of an integral operator (the fractional Laplacian in the simplest case). Experimentally, this has shown to provide a much more accurate model to predict protein docking (if two proteins will stuck together). When seeking drug which would interact with certain protein, the first step is to look for molecule which will stick to the desired protein, and that is when this methods become very useful.
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.


== Links ==
The corresponding result in non divergence form is [[Krylov-Safonov theorem]].
There is a group in the center for Bioinformatics in Saarland University doing research in this field actively. They have a webside describing the project
http://bioinf-www.bioinf.uni-sb.de/projects/solvation


== References ==
For nonlocal equations, there are analogous results both for [[Holder estimates]] and the [[Harnack inequality]].
{{reflist|refs=
 
<ref name="ICH">{{Citation | last1=Ishizuka | first1=R | last2=Chong | first2=S-H | last3=Hirata | first3=F | title=An integral equation theory for inhomogeneous molecular fluids: the reference interaction site model approach. | url=http://www.ncbi.nlm.nih.gov/pubmed/18205507 | publisher=AIP | year=2008 | journal=The Journal of Chemical Physics | volume=128 | issue=3 | pages=034504}}</ref>
== Elliptic version ==
<ref name="HBRK">{{Citation | last1=Hildebrandt | first1=A. | last2=Blossey | first2=R. | last3=Rjasanow | first3=S. | last4=Kohlbacher | first4=O. | last5=Lenhof | first5=H.P. | title=Electrostatic potentials of proteins in water: a structured continuum approach | publisher=Oxford Univ Press | year=2007 | journal=Bioinformatics | volume=23 | issue=2 | pages=e99}}</ref>
For the result in the elliptic case, we assume that the equation
<ref name="SBRF">{{Citation | last1=Scott | first1=R. | last2=Boland | first2=M. | last3=Rogale | first3=K. | last4=Fernández | first4=A. | title=Continuum equations for dielectric response to macro-molecular assemblies at the nano scale | publisher=IOP Publishing | year=2004}}</ref>
\[ \mathrm{div} A(x) \nabla u(x) = 0 \]
}}
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.

Revision as of 15:14, 14 March 2012

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.