Aleksandrov-Bakelman-Pucci estimates and Hele-Shaw: Difference between pages

From nonlocal pde
(Difference between pages)
Jump to navigation Jump to search
imported>Nestor
 
imported>Hector
No edit summary
 
Line 1: Line 1:
The celebrated "Alexandroff-Bakelman-Pucci Maximum Principle" (often abbreviated often as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations. It is the backbone of the regularity theory of fully nonlinear second order elliptic equations <ref name="CC"/> and more recently for [[Fully nonlinear integro-differential equations]] <ref name="CS"/>.
{{stub}}


== The classical Alexsandroff-Bakelman-Pucci Theorem ==  
The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain,
\begin{align*}
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega
\end{align*}
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).
Particular solutions are given for instance by the planar profiles
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]


Let $u$ be a viscosity supersolution of the linear equation:
The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.


\[ Lu \leq f(x) \;\; x \in B_1\]
Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
\[ u \leq 0 \;\; x \in \partial B_1\]
\[
\[ Lu:=a_{ij}(x) u_{ij}(x)\]
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies
\begin{align*}
\Delta w &= 0 \text{ in } \{x_n>0\}\\
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\end{align*}
Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]


The coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have
== References ==
 
{{reflist|refs=
\[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \xi \in \mathbb{R}^n \]
 
Moreover, the function $f$ is assumed to be continuous. Then, the ABP Theorem says that
 
\[ \sup \limits_{B_1}\; u^n \leq C_{n,\lambda,\Lambda} \int_{\{ u=\Gamma_u \} } f_+^n dx \]
 
Where $\Gamma_u$ is the "concave envelope" of $u$: it is the smallest non-nonnegative concave function in $B_2$ that lies above $u$ in $B_1$. The fact that the integration on the right hand side takes place only on the set where $u$ agrees with its concave envelope is an important feature of the estimate and it is not to be overlooked <ref name="CC"/>.
 
== ABP-type estimates for integro-differential equations ==
 
The setting for integro-differential equations is similar, what changes are the operators: let $u$ be a viscosity supersolution of the equation


\[ Lu \leq f(x) \;\; x \in B_1\]
<ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref>
\[ u \leq 0 \;\; x \in  B_1^c\]
\[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{|y|^{n+\sigma}} dy\]


Here $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$ and $\sigma \in (0,2)$. The function $a(x,y)$ is only assumed to be measurable and such that for some $\Lambda\geq\lambda>0$ we have
\[ (\lambda \leq  a(x,y) \leq  \Lambda\]
As in the second order ABP, the function $f$ is assumed to be continuous. Then, Caffarelli and Silvestre proved <ref name="CS"/> there is an estimate
\[ \sup \limits_{B_1} \; u^n \leq  C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m  ( \sup \limits_{Q_i^*} |f|^n) |Q^*_i|  \]
For some finite collection of non-overlapping cubes $\{Q_i \}_{i=1}^m$ that cover the set $\{ u=\Gamma_u\}$, each cube having non-zero intersection with this set.  Moreover, all the cubes have diameters $d_i \lesssim 2^{-\frac{1}{2-\sigma}}$. As before, $\Gamma_u$ denotes the "concave envelope" of $u$ in $B_2$.
Furthermore,  although the sharp constant may depend on $\sigma$, it is uniformly bounded for all $\sigma$ bounded away from zero. In particular, as $\sigma \to 2^-$ the above constant does not blow up, and since the diameter of the cubes goes to zero as $\sigma$ approaches $2$, one can check that this last estimate implies the second order ABP in the limit.
== References ==
{{reflist|refs=
<ref name="CC">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Cabré | first2=Xavier | title=Fully nonlinear elliptic equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-0437-7 | year=1995 | volume=43}}</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>
}}
}}

Revision as of 12:23, 29 July 2016

This article is a stub. You can help this nonlocal wiki by expanding it.

The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain, \begin{align*} \Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\ \frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega \end{align*} The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side). Particular solutions are given for instance by the planar profiles \[ P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 \]

The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.

Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives \[ \frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2 \] By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies \begin{align*} \Delta w &= 0 \text{ in } \{x_n>0\}\\ \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} \end{align*} Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$, \[ \partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w \]

References

  1. Saffman, P. G.; Taylor, Geoffrey (1958), "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. London. Ser. A 245: 312--329. (2 plates), ISSN 0962-8444