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

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The celebrated "Aleksandrov-Bakelman-Pucci Maximum Principle" (often abbreviated as "ABP Estimate") is a pointwise estimate for weak solutions of elliptic equations.  The strength of the ABP estimate is that it is the main tool in the theory of non-divergence elliptic equations which gives a pointwise bound on solutions in terms of a measure theoretic quantity of the equation.  It is not only 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"/>, but it also is essential in stochastic homogenization of uniformly elliptic equations in both the second order and integro-differential setting. 
{{stub}}


== The classical Aleksandrov-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*}
Let $u$ be a viscosity supersolution of the linear equation:
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
 
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega
\[ Lu \leq f(x) \;\; x \in B_1\]
\end{align*}
\[ u \geq 0 \;\; x \in \partial B_1\]
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).
\[ Lu:=a_{ij}(x) u_{ij}(x)\]
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 coefficients $a_{ij}(x)$ are only assumed to be measurable functions such that for positive constants $\lambda<\Lambda$ we have
Non-local aspects of the equation can be appreciated by noticing that a given 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.
 
\[ \lambda |\xi|^2 \leq a_{ij}(x) \xi_i\xi_j \leq \Lambda |\xi|^2 \;\;\forall \; \xi \in \mathbb{R}^n \]
 
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 "convex envelope" of $u$: it is the largest non-positive convex 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 convex 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
 
\[ Lu \leq f(x) \;\; x \in B_1\]
\[ u \geq 0 \;\; x \in  B_1^c\]
\[ Lu:= (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{a(x,y)}{|y|^{n+\sigma}} dy\]
 
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
 
\[ \lambda \leq  a(x,y) \leq  \Lambda \;\;\forall\;x,y\in \R^n\]
 
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
 
\[ \inf \limits_{B_1} \; |u_-|^n \leq  C_{n,\lambda,\Lambda,\sigma}\sum \limits_{i=1}^m  ( \sup \limits_{Q_i^*} |f|^n) |Q^*_i|  \]
 
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 "convex 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.
 
So far, only very little is known about an ABP result which can capture the more refined measure theoretic information of the right hand side of the equation for such integro-differential $L$ as above.  The only known result applies to a very restricted family of $L$ which are indexed by degenerate elliptic matrices:


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
\[
\[
L(u,x):=  (2-\sigma)\int_{\R^n} \delta u(x,y)\frac{y^TA(x)y}{|y|^{n+\sigma+2}} dy,
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\]
\]
where $\text{Tr}(A(x))\geq \lambda$.  For these operators, Guillen and Schwab proved that
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 $\mathbb R^{n-1} = \{x_n=0\}$,
\[
\[
\sup_{B_1}|u_-|\leq \frac{C(n,\sigma)}{\lambda}\lVert f^+\rVert^{(2-\sigma)/2}_{L^\infty(\mathcal C)}\Vert f^+\rVert^{\sigma/2}_{L^n(\mathcal C)},
\partial_t w = a\Delta_{\mathbb R^{n-1}}^{1/2} w
\]
\]
where $\mathcal C$ is the contact set between $u$ and a $\sigma$-order replacement for the convex envelope.


== References ==  
== References ==
{{reflist|refs=
 
<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>


{{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 12:20, 29 July 2016

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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 } \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 \[ P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 \]

Non-local aspects of the equation can be appreciated by noticing that a given 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