Birds and frogs2021-08-26T21:18:19+00:00http://birdsnfrogs.github.ioBirds and frogsnguillen@math.umass.eduHarmonic Functions in Lipschitz domains2020-06-29T00:00:00+00:00http://hankquinlan.github.io//2020/06/29/HarmonicFunctionsLipschitz<p>When you start studying certain free boundary problems you soon appreciate the importance of understanding the boundary behavior of harmonic functions, especially harmonic functions which vanish on a portion of the boundary. Motivated by questions in the Hele-Shaw problem and related free boundary problems, I am going to review several important theorems proved by Dahlberg in the late 1970’s.</p>
<p>Let $D\subset \mathbb{R}^n$ be a Lipschitz domain. For such a domain it is well known that for any continuous function $f:\partial D\to\mathbb{R}$ there is a unique function $U_f$ which is continuous in $\overline D$ and twice differentiable in $D$ solving</p>
\[\left \{ \begin{array}{rl}\Delta U_f = 0 & \text{ in } D \\ U_f = f & \text{ on } \partial D\end{array} \right.\]
<p>This defines an operator $f \mapsto U_f$ which is linear, it is also monotone in the sense that if $f\leq g$ in $\partial D$ then $U_f \leq U_g$ in $D$. In particular, for every $X \in D$ the linear functional</p>
\[f \mapsto U_f(X)\]
<p>is positive. By the representation theorem for each $X \in D$ there is a Borel measure $\omega_X$ in $\partial D$ such that</p>
\[U_f(X) = \int_{\partial D}f(y)d\omega_X(y).\]
<p>The measure $\omega_X$ is called the harmonic measure associated with the domain $D$ at the point $X$. When $D$ is a Lipshitz domain, the measure $\omega_X$ is absolutely continuous with respect to the surface measure of $\partial D$, so</p>
\[d\omega_X(y) = k(X,y)\;d\sigma(y)\]
<p>The function $K$ is called the Poisson kernel of the domain $D$. The best known instances of the Poisson kernel are those where $D$ is a ball or a half space. For the unit ball in $\mathbb{R}^n$ the kernel takes the form</p>
\[k(X,y) = \frac{1}{|\partial B_1|}\frac{1-|X|^2}{|X-y|^n}\]
<p>Dahlberg showed that for a general Lipschitz domain the Poisson kernel for each fixed $X$ yields a function on the boundary which satisfies a reverse Harnack inequality.</p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong> Theorem. </strong>
<br />
There is a constant $C$ such that given $X\in D$, $y_0 \in \partial D$, and $r\in(0,1)$ we have
$$ \left ( \frac{1}{\sigma(\Delta)}\int_{\Delta} k(X,y)^2\;d\sigma(y) \right )^{\frac{1}{2}} \leq \frac{C}{\sigma(\Delta)}\int_\Delta k(X,y)\;d\sigma(y),\;\;\Delta := B_r(y_0) \cap \partial D.$$
</p>
<p>There is the Boundary Harnack Principle. It says that if $U$ is a positive harmonic function in $D$ which vanishes on a portion of $\partial D$, then the value of $U$ at a point properly place away from said portion will control the values of $U$ everywhere.</p>
<p>Before we state the principle, let us state two preliminary facts about harmonic functions.</p>
<p>$\bullet$ (De Giorgi’s oscillation lemma) Fix $\delta \in(0,1)$. Let $U$ be a subharmonic function $U$ in some ball $B_{2r}(X_0)$, and suppose that $U \leq 1$ in $B_{2r}(X_0)$ and</p>
\[|\{ U\leq 0 \} \cap B_{2r}(X_0)| \geq \delta |B_r|\]
<p>then \(U\leq \mu\) in $B_r(X_0)$, where $\mu = \mu(\delta) \in (0,1)$.</p>
<p>$\bullet$ (Harnack inequality) If $P \in D$ is such that $d(P,\partial D) = \rho$, then</p>
\[U(P) \leq C \left ( \frac{r}{\rho}\right )^\alpha U(X_1)\]
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em">
<strong> Theorem. </strong>
<br />
Let $U$ be a positive harmonic function in $B_r(X_0) \cap D$ such that $U \equiv 0$ on $B_r(X_0) \cap \partial D$ and let $X_1$ be a point in $B_{r/2}$ such that $B_{r/4}(X_1) \subset B_R(X_0) \cap D$. Then
$$U(X) \leq CU(X_1) \text{ for any } X \in B_{r/2}(X_0) \cap D$$
where the constant $C$ is determined by $D$.
</p>
<p><strong>Proof.</strong></p>
<p>For every $X$, we shall denote by \(X^*\) a point on \(\partial D\) such that \(d(X,\partial D) = d(X^*,\partial D)\).</p>
<p>The proof hinges on a conflict between two effects:</p>
<ol>
<li>The positivity and harmonicity of $U$ force it to grow, meaning if $U$ reaches a certain value $h$ in a ball $B_r$ then in $B_{2r}$ it must reach at least a value above $\kappa h$, where $\kappa>1$ is a dimensional constant.</li>
<li>The Harnack inequality prevents $U(X)$ from being much larger than $U(X_1)$ if the distance from $X$ to $X_1$ is comparable to the distance from $X$ to $\partial D$.</li>
</ol>
<p>Let $P_0$ be the point where $U$ is the largest in $B_{r/2} \cap D$. Using the two effects above we are going to show that if the value $U(P_0)$ is much larger than $U(X_1)$, then there is a sequence of points $P_k$ all lying in $B_r \cap D$ with the property $U$ takes arbitrarily large values along this sequence.</p>
<p>The sequence \(\{P_k\}_k\) is built recursively, starting from $P_0$. Having determined the sequence up to some $P_k$, we determine the next element $P_{k+1}$. Let us write</p>
\[\rho_k := d(P_k,\partial D)\]
<p>The oscillation lemma says that</p>
\[\sup \limits_{B_{\rho_k}(P_k)} U \leq \theta \sup \limits_{B_{2\rho_k}(P_k)} U\]
<p>Applying the oscillation lemma in $B_{2\rho_k}(P_k^*)$ we obtain a new point $P_{k+1} \in D$ such that</p>
\[|P_{k+1}-P_k| \leq 2\rho_k,\;\; U(P_{k+1}) \geq \theta^{-1} U(P_k)\]
<p>This can be done as long as … so the sequence is either finite and ends at some $k=k_0$ or it goes on indefinitely.</p>
<p>In either case, the second inequality applied recursively means that for every $P_k$ in the sequence we have</p>
\[U(P_k) \geq \theta^{-k} U(P_0)\]
<p>We now use the Harnack inequality. From the H</p>
<p>Note there is a ball of radius $r+\rho_k$ ….containing $X_1$ and $P_k$… and which is contained in $D$, then, thanks to Harnack’s inequality</p>
\[U(P_k) \leq C \left (\frac{r}{\rho_k} \right )^\alpha U(X_1)\]
<p>In other words,</p>
\[\rho_k \leq C r\left (\frac{U(X_1)}{U(P_k)}\right )^{1/\alpha}\]
\[\rho_k \leq C r \left (\frac{U(X_1)}{U(P_0)}\right )^{1/\alpha}(\theta^{1/\alpha})^k\]
<p>In summary, the sequence ${P_k}_k$ has the following properties</p>
\[U(P_k) \geq \theta^{-k}U(P_0)\]
\[|P_k-P_{k+1}| \leq Cr \left (\frac{U(X_1)}{U(P_0)} \right )^{1/\alpha} (\theta^{1/\alpha} )^{k}\]
<p>Observe that</p>
\[\sum \limits_{k=0}^\infty |P_k-P_{k+1}| \leq Cr\left (\frac{U(X_1)}{U(p_0)} \right )^{1/\alpha} \frac{1}{1-\theta^{1/\alpha}}\]
<p align="right"> ∎</p>
<p>Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a bounded, Lipschitz function,</p>
\[\|\phi\|_\infty \leq C,\; \|\nabla \phi\|_\infty \leq \delta\]
<p>and consider the domain</p>
\[D(\phi) = \{ X = (x,x_{n+1})\in \mathbb{R}^{n+1} \mid x_{n+1} > \phi(x) \}\]
<p><strong>Proof.</strong></p>
<p>Let $V$ be superharmonic and such that $CV(y) \geq k(P,y)^q$ for $y\in \partial D$. Then</p>
\[CV(P) \geq \int_{\partial D}k(P,y)^{1+q}\;d\sigma(y)\]
Boltzmann2019-05-08T00:00:00+00:00http://hankquinlan.github.io//2019/05/08/Saint_Raymond_Lectures<h3 id="hard-sphere-dynamics">Hard Sphere Dynamics</h3>
<ol>
<li>
<p>(Alexander) For almost every initial data the dynamics are globally well defined (i.e. nothing but simultaneous binary collisions)</p>
</li>
<li>
<p>Qualitative behavior of the system:</p>
<ol>
<li>
<p>Reversibiliy</p>
</li>
<li>
<p>Poincaré recurrence theorem</p>
</li>
<li>
<p>If $\varepsilon«1$ stability deteriorates</p>
</li>
</ol>
</li>
</ol>
<p>We consider $N$ particles of size $\varepsilon$, with $N\to \infty$ and $\varepsilon \to 0$.</p>
<p>If $\varepsilon = 0$ and we take $N\to \infty$ or $\varepsilon\to 0$ much faster compared to $N\to\infty$ then particles never or rarely collide. In the limit we end up with the equation for free transport</p>
\[\partial_t f + v\cdot \nabla_x f = 0\]
<p>We are interested in a regime where there are not too many collisions but not so few that they become meaningless in the limit. The specific regime we will study is what is known as the low density regime.</p>
<p>Maxwell did a computation regarding this regime: you consider a $N$ particles layed out on a lattice (and we see them as fixed) and we have another particle which is freely moving initial with velocity $v$</p>
<p>This leads to the relation</p>
\[N\varepsilon^{d-1} \approx 1\]
<p>In particular, $N\varepsilon^d \to 0$ as $N\to \infty$, so the particle density is small in the limit. This is what is known as <strong>the low density regime</strong>.</p>
\[f_N^0(x_1,v_1,\ldots,x_N,v_N) \leftarrow \text{Initial distribution of states}\]
\[f_N(t,x_1,v_1,\ldots,x_N,v_N) \leftarrow \text{Distribution of states at time } t\]
<p>The Liouville equation</p>
\[\partial_t f_N + \sum \limits_{i=1}^{N} v_i \cdot \nabla_{x_i} f_N = 0 \text{ in } (0,\infty)\times \mathcal{D}_N\]
<p>where</p>
\[\mathcal{D}_N = \{ (X_N,V_N) \mid |x_i-x_j| > 2\varepsilon \;\forall\;i\neq j\}\]
<p>the boundary conditions (#what was the name of this, specular reflection?)</p>
\[{f_N}_{\mid \Sigma_{ij}^-} = R_{ij} {f_N}_{\mid \Sigma_{ij}^+}\]
<p>We now work instead in the grand canonical setting (explain what this means)</p>
<h3 id="boltzmanns-equation">Boltzmann’s equation</h3>
<ol>
<li>Exchangeability (we don’t care about labels on the particles)</li>
</ol>
<p>(this translates to symmetry of $f_N$ with respect to the $(x_i,v_i)$)</p>
<ol>
<li>Averaging
\(f(t,x,v) \leftarrow \text{probability density for a single particule}\)</li>
</ol>
<p>(this means among other things that all observables shouldbe in terms of this)</p>
<p>Moreover, $f(t,v,x)$ can be interpreted as a marginal distribution</p>
\[f(t,v,x) = \int f_N(t,x,v,x_2,v_2,\ldots,x_N,v_N)\;dx_2dv_2 \ldots dx_Ndv_N\]
<p>What about the equation for $f$? We can advance that</p>
\[\partial_tf +v\cdot \nabla_x f = \left ( \begin{array}{l} \text{boundary terms} \\ \text{in Green's formula} \end{array}\right )\]
\[\sum \limits_{i=2}^N \int_{\{|x_i-x_1|=\varepsilon\}} f_N(t,X_N,V_N)(v_i-v_1) \cdot n_{i1}dv??? dv_1dx_2\ldots dx_{n}dv_n \\
= (n-1)\varepsilon^{d-1}\int f_n^{(2)}(t,x,v,x_2,v_2)(v_2-v_1)\cdot n_{12}dn_{12}dv_2\]
<p>Here is we bring in the <strong>Boltzmann-Grad scaling</strong>.</p>
<p>Then,</p>
\[\begin{array}{rl}
\partial_t f + v\cdot \nabla_x f \sim & \int f_N^{(2)}(t,x_1,v_1',x_1+\varepsilon??,v_2')((v_2-v_1)\cdot n_{12})_+\\
& -\int f_N^{(2)}(t,x_1,v_1,x_1+\varepsilon??,v_2')((v_2-v_1)\cdot n_{12})_+\end{array}\]
<p>This system is not closed!! To compute the dynamics for $f = f^{(1)}_N$ we need to know $f^{(2)}_N$ $\ldots$</p>
<p>Here comes a fundamental reduction in the model: <strong>Boltzmann’s chaos assumption</strong>.</p>
\[(*) \;\;\;\;\; f_N^{(2)}(t,x_1,v_1,x_2,v_2) \approx f_N^{(1)}(t,x_1,v_1)f_N^{(1)}(t,x_2,v_2)\]
<p>(note this states an independence relation between particles)</p>
<p>Then, the equation becomes</p>
\[\partial_tf + v\cdot \nabla_x f = C(f,f)\]
<p>Notes:</p>
<ol>
<li>
<p>Even at $t=0$, assumption $(*)$ cannot be correct.</p>
</li>
<li>
<p>For $t>0$, there is some chance that $(*)$ is approximately true.</p>
</li>
<li>
<p>This last model, unlike the original equation for $f_N$, is not <strong>reversible in time</strong>. So, understanding the validity of Boltzmann’s chaos assumption requires understanding the loss of time reversibility in the approximation.</p>
</li>
</ol>
<p><strong>Theorem (Lanford)</strong></p>
<p>Consider a system of $N$ hard spheresof size $\varepsilon$ which are initially independent and identically distributed, i.e.</p>
\[f_N^0 = \frac{1}{Z_N}(f^0)^{\otimes N} \chi_{\mathcal{D}_N}\]
<p>Then, in the low density limit as $N\to \infty$ we have</p>
\[\frac{1}{N}\sum \limits_{i=1}^N \delta_{(x_i(t),v_i(t))} \to f \text{ (almost surely)}\]
<p>where $f$ is a solution of the Boltzmann equation on a time interval $[0,T_0]$. Moreover, we have the convergence of the marginals</p>
\[f^{(1)}_N \to f,\; f^{(2)}_N \to f^{\otimes 2}\]
<p>Note: this last limit indicates the validity of the Boltzmann chaos assumption in the limit.</p>
<h3 id="strategy-of-the-proof">Strategy of the proof</h3>
\[\partial_t f_N^{(1)} + v_1 \cdot \nabla_{x_1} f_{N}^{(1)} = C(f_N^{(2)})\]
\[\partial_t f_N^{(2)} + (v_1,v_2)\cdot \nabla_{(x_1,x_2)} f_{N}^{(2)} = C_{2,3} (f_N^{(3)})\]
<p>$(\ldots)$</p>
<p>and that is how we arrive at what is known as the BBGKY hierarchy.</p>
\[\partial_t f_N^{(k)} + \sum \limits_{i=1}^k v_i \cdot \nabla_{x_i} f_N^{(k)} = C_{k,k+1}(f_N^{(k+1)})\]
<p>Combinatorics of collisions (collision diagram)</p>
<p>$(\ldots)$</p>
<p>We will take this combinatorial data and create a series expansion</p>
\[\sum\limits_k \sum \limits_{A_k} \int dt_{i}dx_idv_i S_1(t-t_2)C_{1,2}(S_2(t_2-t_3))\ldots\]
<p>The proof boils down to showing the “bad trees” (those corresponding to “recollisions”) do not occur very often.</p>
<p><strong>Step 1</strong> Convergence of the series</p>
<p><strong>Step 2</strong> Controlling contribution of recollisions (i.e. show they contribute zero in the limit)</p>
<p><strong>Step 3</strong> If you are not in the bad set of parameters then the small shifts are controlled.</p>
<p>The reason you cannot go back in time is that you are losing <strong>information</strong> (the bad trajectories) when averaging. This information is being carried by a set of vanishing measure (this encodes the deviation of $f_N^{(2)}$ away from $f^{\otimes (2)}$).</p>
Generated Jacobian Equations, part 3/n2019-02-26T00:00:00+00:00http://hankquinlan.github.io//2019/02/26/Generated_Jacobian_Equations_3<p><strong>The exponential map</strong></p>
<p><strong>Example 5 (the standard gradient map):</strong></p>
<p><strong>Example 7 (the exponential in Riemannian geometry)</strong></p>
<p><strong>The Jacobian equation</strong></p>
<p><strong>Example 8 (Far field reflector)</strong></p>
<p><strong>Example 9 (Near field reflector)</strong></p>
<p><strong>The (real) Monge-Ampère equation</strong></p>
<p>The most basic Generated Jacobian Equation, and also one of the most important elliptic equations, is the real Monge-Ampère equation</p>
\[\det(D^2u) = f(x)\]
<p>where one only considers convex solutions $u$, and $f\geq 0$.</p>
<p>If we denote the $d$ eigenvalues of $D^2u(x)$ by $0\leq \lambda_1(x)\leq \ldots \leq \lambda_d(x)$ then the equation reads</p>
\[\prod \limits_{i=1}^d \lambda_i(x) = f(x)\]
<p>If we knew a priori that our solution satisfied the estimate $\inf \lambda_1(x) \geq c_0>0$, then</p>
\[\lambda_d \leq c_0^{-(d-1)} f(x)\]
<p>In which case $D^2u \in L^p$ if $f \in L^p$.</p>
<p>Since we are looking at $u$ which is convex, it can be represented by</p>
\[u(x) = \max \limits_{y \in \overline{\Omega}} \{ x\cdot y - v(y) \}\]
<p>for some function $v(y)$ such that</p>
\[v(y) = \max \limits_{x \in \Omega} \{ x\cdot y - u(x) \}\]
<p>Suppose, for moment, that for a given $x$ there is exactly one $y$ realizing the maximum, then this $y$ must be $y = \nabla u(x)$.</p>
<p>Add discussion of 1) continuity method, 2) Urbas notes 3) Caffarelli’s estimate.</p>
<p><br /></p>
<p><strong>Optimal transport and the $c$-Monge-Ampére equation</strong></p>
\[J(T) = \int_{X}c(x,T(x))f(x)\;dx\]
<p>A theorem of Brenier and Gangbo-McCann says that</p>
\[Dc(x,T(x)) = Du(x)\]
\[T(x) = \text{exp}_x^{c}(Du(x))\]
\[u(x) = \max \limits_{y \in \overline{\Omega}} \{c(x,y)-v(y) \}\]
Stable matchings, generating functions, and optimal transport2019-02-24T00:00:00+00:00http://hankquinlan.github.io//2019/02/24/Stable_matchings<p>Left to do here:</p>
<ol>
<li>Explain why quasi-linear utilities are called “transferable”</li>
<li>Clean up the discussion of the basic model</li>
<li>Prove the first proposition</li>
<li>Prove the second proposition</li>
<li>Discussion about what could happen in the non quasi-linear case</li>
<li>Add references</li>
</ol>
<h2 id="the-stable-matching-model"><strong>The Stable Matching model</strong></h2>
<p>In an idealized (that is, simplified) labor market situation, each worker chooses among all firms which would be <em>the best for them</em> to work, conversely, each firm chooses which workers <em>would be the best for them</em> to hire. The <strong>Stable Matching Model</strong> is used by economists to model such a situation. The stable model is not exclusive to employment markets, it is used to model any situation where you have two types of agents deciding whether to pair up according to what partner is best for each of them. What is the essential for the model is the assumption that agents make rational decisions and have definite and complete information about the utility of each of their possible choices.</p>
<p>This as much detail about the model I can give without using more notation. So let’s get to it. It will be helpful to keep using the interpretation of a labor market, so I will be referring to the two types of agents as workers and firms, even if all what follows will apply to many other situations. I will deal only with the continuum setting, and simply note here that most (if not all) of what I say here extends in some way or another the discrete setting. Lastly, a simplifying assumption in this model is that each worker chooses just one firm, and each firm chooses just one worker.</p>
<p>The <strong>basic assumptions</strong> of the model are:</p>
<p>$\bullet$ A set $X \subset \mathbb{R}^p$, representing workers.</p>
<p>$\bullet $ A set $Y \subset \mathbb{R}^q$, representing firms.</p>
<p>$\bullet$ A <em>utility function</em> given by $G:X\times Y \times \mathbb{R}\to \mathbb{R}$, which is assumed to be continuous. The meaning of $G(x,y,v)$ is the following: if worker $x$ works for firm $y$ and produces for them a utility of $v$, then $x$ gets a utility of $G(x,y,v)$. Lastly, the function $G$ is strictly decreasing with respect to the variable $v$.</p>
<p><strong>Definition</strong>: The fac that $G$ is strictly decreasing in $v$ means that for each $x,y$ the function $v\to G(x,y,v)$ is invertible and has a unique inverse which is also decreasing. This defines a function $H(x,y,u)$, which can be easily be seen to be continuous (thanks to the continuity assumption on $G$). In particular,</p>
\[G(x,y,H(x,y,u)) = u \text{ and } H(x,y,G(x,y,v)) = v.\]
<p>Since $H(x,y,\cdot)$ is defined as an inverse to $G(x,y,\cdot)$ it is easy to see that it represents the following: if firm $y$ hires worker $x$ and gives them a utility of $u$, then $y$ will get a utility of $H(x,y,u)$.</p>
<p>The utility functions $G$ and $H$ is how we will encode the assumption that each worker and each firm has, as I said earlier, ‘‘definite and complete information’’. Each worker and each firm knows how much utility they will get based on who they match with and what utility transfer takes place (respectively, $G(x,y,v)$ and $H(x,y,u)$).</p>
<p>Now, how does the model work? It does not concern with the extended process by which workers and firms choose one another and choose how much utility they are willing to transfer (i.e. how much a worker wants to make, how much a firm wants a worker to pruduce), but instead studies a final matching of workers and agents. This means each worker has decided on $u(x)$, the utility they would be willing to work for, and conversely each firm $y$ has decided on $v(y)$, the utility they would be willing to hire someone for. So we have a pair of scalar functions</p>
\[u:X\to\mathbb{R},\; v:Y\to\mathbb{R},\]
<p>moreover, each worker $x$ has chosen a firm $T(x)$ and each firm $y$ has chosen a worker $S(y)$. This means that $u,v$ and the functions $T(x)$ and $S(y)$ satisfy the functional equations</p>
\[u(x) = G(x,T(x),v(T(x))),\; v(x) = H(S(y),y,u(S(y)))\]
<p>This leads to the following definition.</p>
<p><strong>Definition:</strong> A <em>matching</em> or <em>outcome</em> is a pair of scalar functions $u,v$ and a pair of maps $T,S$</p>
\[u:X\to\mathbb{R},\;v:Y\to\mathbb{R},\;T:X\to Y,\; S:Y\to X,\]
<p>all such that for every $x\in X$ and every $y\in Y$ we have</p>
\[u(x) = G(x,T(x),v(T(x))),\; v(x) = H(S(y),y,u(S(y)))\]
<p>A matching is stable if no worker or firm would be better off if they switched their respective choices.</p>
<p>Therefore, we see that stable matchings are the same as matchings where the utility functions form are conjugate.</p>
<hr />
<p><strong>Proposition:</strong> A matching $(u,v,T)$ is stable if and only if $u= v^* $ and $v = u^*$.</p>
<hr />
<p><br /></p>
<p>Moreover</p>
\[T(x) \in \partial u(x) \text{ for every } x \in X\]
<h2 id="perfectly-transferable-utilities"><strong>Perfectly transferable utilities</strong></h2>
<p>The case that has been studied most widely is the case of perfectly transferable utilities. In this case the respective generating function $G(x,y,v)$ is quasi-linear, that is linear in the $v$-variable, so</p>
\[G(x,y,v) = b(x,y)-v,\]
<p>for some function $b:X\times Y\to\mathbb{R}$.</p>
<p><strong>The central planner problem</strong></p>
<p>Then, the planner is interested in maximizing the total welfare</p>
\[\int_{X\times Y} b(x,y) \;d\pi(x,y)\]
<p><strong>The dual problem</strong></p>
<p>The problem is now to minimize</p>
\[\int_{X}u(x)\;d\mu(x) + \int_{Y}v(y)\;d\nu(y)\]
<p>among all pairs $(u,v)$ satisfying the pointwise constraint</p>
\[u(x)+v(y) \geq b(x,y)\]
<hr />
<p><strong>Proposition</strong>: There is at least one pair $(u,v)$ minimizing the dual functional, moreover</p>
\[u = v^* \text{ and } v = u^*.\]
<hr />
<p><br /></p>
<p>The dual problem is, as its name suggests, closely related to the central planner problem.</p>
<hr />
<p><strong>Theorem</strong>: The optimal solution to the dual problem agrees with the solution to the central planner problem</p>
\[\max \int_{X\times Y} b(x,y) \;d\pi(x,y) = \min \int_{X\times Y} b(x,y) \;d\pi(x,y)\]
<p>Moreover, any solution $\pi$ to the Central Planner problem and any solution $(u,v)$ to the Dual Problem must satisfy the relation</p>
\[u(x)+v(y) = b(x,y) \;\;\text{ for}\;\pi\text{-a.e. } (x,y)\]
<hr />
<p><br /></p>
<p>A corollary of this theorem is that for quasi-linear utilities there is always at least one stable outcome: a triad $(u,v,T)$ resulting from the theorem.</p>
<h2 id="when-utilities-are-not-quasi-linear"><strong>When utilities are not quasi-linear</strong></h2>
Generated Jacobian Equations, part 2/n2019-02-21T00:00:00+00:00http://hankquinlan.github.io//2019/02/21/Generated_Jacobian_Equations_2<p>Today I am going to explain how generating functions define a duality structure, discuss the first few examples, and introduce the subdifferential.</p>
<p><strong>A Generating Function gives a duality structure between $\Omega$ and $\overline{\Omega}$</strong></p>
<p>In the previous post I gave a definition of a Generating Function $G(x,y,z)$. However, in less precise terms, a Generating Function is best thought of as a “duality structure” between real valued functions in $\Omega$ and real valued functions in $\overline{\Omega}$. By this I mean an operation that transforms a real valued function in $\Omega$ into a real valued function in $\overline{\Omega}$, and viceversa.</p>
<p>The <em>transform</em> of a function $v:\overline{\Omega}\to\mathbb{R}$ is a new function, $v^*:\Omega\to\mathbb{R}$ defined by</p>
\[v^*(x) = \sup \limits_{y\in \overline{\Omega}} G(x,y,v(y)),\]
<p>while the <em>transform</em> of a function $u:\Omega\to\mathbb{R}$ yields a function $u^*:\overline{\Omega}\to \mathbb{R}$ given by the formula</p>
\[u^*(y) = \sup \limits_{x \in \Omega} H(x,y,u(x)).\]
<p>A pair of functions $u,v:\Omega,\overline{\Omega}\to\mathbb{R}$ will be called a <em>conjugate pair</em> if $u= v^* $ and $v=u^*$, note that in this case</p>
\[G(x,y,v(y)) \leq u(x) \text{ and } H(x,y,u(x)) \leq v(y) \text{ for all } x\in\Omega,\;y\in\overline{\Omega},\]
<p>where for fixed $x$, there is at least one $y$ for which the first inequality becomes an equality, and for fixed $y$, there is at least one $x$ for which the second inequality becomes an equality.</p>
<p>Now, let us discuss some concrete (and not so concrete) examples.</p>
<p><strong>Example 1 (Standard Legendre duality):</strong> The most popular instance of this is the Legendre transform between functions in Euclidean space,</p>
\[v^*(x) := \sup \limits_{y\in \mathbb{R}^d} \{x\cdot y-v(y)\},\]
<p>which corresponds in the definition above to the special case where $G(x,y,z) = x\cdot y -z$ (note that in this case $H(x,y,u) = G(x,y,u)$).</p>
<p><strong>Example 2 (Polar Dual):</strong> A convex body is a bounded convex set $K\subset \mathbb{R}^d$ with non-empty interior. The <em>polar dual</em> of a convex body is the convex set</p>
\[K^* := \{ \xi \in \mathbb{R}^d \mid \eta \cdot \xi \leq 1 \;\text{ for all } \eta \in K \}.\]
<p>It is well known that $(K^* )^* = K$, so this defines a duality between convex bodies. This duality can be captured in terms of a generating function when expressed in the right variables.</p>
<p>First, we must describe convex bodies in terms of a function. Suppose that the origin of the system of coordinates lies in the interior of $K$, then $K$ is described by the radial function</p>
\[r_K:\mathbb{S}^{d-1}\to \mathbb{R},\; r_K(x) := \sup \{ t \mid tx \in K\}.\]
<p>In words, $r_K$ is the function on $\mathbb{S}^{d-1}$ whose radial subgraph is equal to $K$.</p>
<p>A straithforward calculation one leads to the following formula expressing $r_{K^*}$ in terms of $r_K$, namely</p>
\[r_{K^*}(y) = \frac{1}{\sup \limits_{x\in \mathbb{S}^{d-1}} (x \cdot y )r_K(x)}.\]
<p>Let $u(x)= \frac{1}{r_K(x)}$ and $v(y) = \frac{1}{r_{K^*}(y)}$. The above formula implies that $u$ and $v$ form a conjugate pair with respect to the generating function</p>
\[G:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1} \times (0,\infty) \to \mathbb{R},\;\;G(x,y,z) := \frac{x\cdot y}{z}.\]
<p><strong>Example 3 (The shipper’s dilemma)</strong></p>
<p>The following is the shipper’s dillemma, perhaps better known as the <em>dual problem</em> in optimal transportation. In this problem we are given</p>
<ol>
<li>Two domains $\Omega,\overline{\Omega}$</li>
<li>A probability measure $\mu$ in $\Omega$, and a probability measure $\nu$ in $\overline{\Omega}$.</li>
<li>A continuous function $c:\Omega\times \overline{\Omega} \to \mathbb{R}$ known as the <em>cost function</em>.</li>
</ol>
<p>Then the dual optimal transportation problem amounts to finding a pair of functions $u:\Omega\to\mathbb{R}$, $v:\overline{\Omega}\to \mathbb{R}$ maximizing</p>
\[\int_{\Omega}u(x)\;d\mu(x)+\int_{\overline{\Omega}}v(y)\;d\nu(y),\]
<p>among all pairs satisfying the pointwise constraint</p>
\[u(x)+v(y) \leq c(x,y).\]
<p>A basic fact in optimal transportation is that this problem admits at least one solution, and the solution is given by a pair of functions $\phi,\psi$ such that</p>
\[u(x) = \inf \limits_{y\in\overline{\Omega}}\{c(x,y)-v(x)\},\;\;v(y) = \inf \limits_{x\in\Omega}\{c(x,y)-u(x)\}.\]
<p>In other words, the functions $-u$ and $-v$ form a conjugate pair with respect to the generating function $G(x,y,z) = -c(x,y)-z$.</p>
<p><strong>Example 4 (stable matchings in economics)</strong></p>
<p>Consider a situation where we have a set of types of buyers (represented by $\Omega$) and a set of types of sellets (represented by $\overline{\Omega}$). A type of buyer and a type of seller can decide to engage in a transaction resulting in a utility for each of them. Here the generating function $G(x,y,v)$ will represent this utility, as follows: if a buyer $x$ decides to engage in a transaction with seller $y$ with the seller obtaining a utility of $v$, then $x$ obtains a utility of $G(x,y,v)$.</p>
<p>Lastly, we are given a probability measure $\mu$ representing the distribution of buyers, and a probability measure $\nu$ representing the distribution of sellers.</p>
<p>We are interested in studying <em>outcomes</em>. This refers to three things: a function $u(x)$, a function $v(y)$, and a measure $\lambda$ over $\Omega\times \overline{\Omega}$ with marginals $\mu$ and $\nu$, respectively, such that</p>
\[u(x) = G(x,y,v(y)) \text{ and } v(y) = H(x,y,u(x)) \text{ for } \lambda-\text{a.e.} (x,y)\]
<p>All of this simply says that pairs $(x,y)$ given by $\lambda$ describe buyers and sellers engaging in a transaction resulting for each in a utility of $u(x)$ and $v(y)$, respectively.</p>
<p>An <em>stable outcome</em> is an outcome where each buyer $x$ is doing the best they can do given the utility profile of sellers, and conversely each seller is doing the best they can do given the utility profile of buyers. This means that in addition to the condition above, we have</p>
\[u(x) \geq G(x,y,v(y)) \;\forall\;x\in\Omega,y\in\overline{\Omega},\\
v(y) \geq H(x,y,u(y)) \;\forall\;y\in\Omega,x\in\overline{\Omega}.\]
<p>In other words, if $(u,v,\lambda)$ is an stable outcome, then in particular $(u,v)$ must be a conjugate pair.</p>
<p><strong>Note:</strong> This problem has been studied by economists and mathematicians for many decades. For utility functions of the form $G(x,y,z) = b(x,y)-z$, the problem falls within the framework of optimal transporation, a fruitful fact that has led to many contributions by Carlier, Ekeland, McCann and many others. Such utility functions are known in the economics literature as <em>quasi-linear utility functions</em>, and correspond to what is known as <em>perfectly transferable utility</em>. Problems with <em>imperfectly transferabe utility</em> which are used to model e.g. high income and taxation effects require utility functions which are not quasi-linear. I am only barely acquainted with the economic literature but I hope to be able to write later about the works of Nöldeke and Samuelson; McCann and Zhang; and Galichon, Kominers, and Webers. Nöldeke and Samuelson in particular show the existence of stable matchings for non quasilinear utilities using the formalism of generating functions (which, they note corresponds to what is known as a Galois connection), while also studying a principal-agent model with non-quasilinear utility.</p>
<p>One last point about this example. If one has a stable outcome $(u,v,\lambda)$, then for each buyer $x_0$ there is at least one $y_0$ achieving the maximum of</p>
\[G(x_0,y,v(y))\]
<p>The set of such sellers represent those which $x$ could buy from to realize their maximum utility. Observe that finding such a $y_0$ is the same as finding, $y_0$ such that the function $G(x,y_0,v(y_0))$ touches $u(x)$ from below at $x_0$. If there was a map $x\to y(x)$ with the property that $y(x)$ maximizes $G(x,y,v(y))$, then we say that $y(x)$ is a map implemented by $v(y)$ (note that there could be more than one map).</p>
<p>This lead us to our next topic.</p>
<p><strong>$G$-convex functions and the subdifferential</strong></p>
<p>A function $u:\Omega\to\mathbb{R}$ will be called $G$ - <em>convex</em> or simply <em>convex</em> if for every $x_0\in \Omega$ there is a $y_0 \in \overline{\Omega}$ and $z_0 \in \Omega$ such that</p>
\[\begin{array}{rl} u(x) & \geq G(x,y_0 ,z_0) \;\forall\;x\in\Omega, \\ u(x_0 ) & = G(x_0 ,y_0 , z_0) \end{array}\]
<p>In a similar manner we shall talk of a function $v: \overline{\Omega}\to \mathbb{R}$ being $H$ - <em>convex</em> or simply convex. Observe that if $u$ and $v$ form a conjugate pair, then $u$ is $G$ - <em>convex</em> and $v$ is $H$ - <em>convex</em>.</p>
<p>Consider a $G$-convex function, $u$ its subdifferential at $x_0 \in \Omega$ is the set</p>
\[\partial u(x_0) = \{y \in \overline{\Omega}\;\mid\; u(x) \geq G(x,y,H(x_0,y,u(x_0))) \;\forall\;x\in\Omega \}.\]
<p>Likewise we define the subdifferential $\partial v(y_0)$ of a $H$-convex function $v$. Often for emphasis people write $\partial_G$ or $\partial_H$ to clarify which type of subdifferential is being talked about, but in these posts I will not use such sub indices to minimize notation.</p>
<p>The subdifferential is always non-empty, and it could be multivalued, the best known and most used one is the standard subdifferential in convex analysis which for a convex function $u$ and point $x_0$ gives the set of all vectors $p$ representing the slopes of hyperplanes which are supporting to the graph of $u$ at $x_0$.</p>
<p><br /></p>
<p><strong>(Updated February 21st)</strong></p>
<p>A few relevant references I forgot to add:</p>
<p>Caffarelli’s summer school notes on optimal transport (relevant to Example #3) <a href="https://link.springer.com/content/pdf/10.1007/978-3-540-44857-0_1.pdf">Link</a></p>
<p>Nöldeke and Samuelson’s implementation duality paper (relevant to Example #4) <a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA13307">Link</a></p>
<p>McCann and Zhang’s monopolist problem with nonlinear price prefs (relevant to Example #4) <a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.21817">Link</a></p>
<p><br /></p>
<p>For the next post in this series I <em>plan</em> to talk about the exponential map, discuss a few more examples from Riemannian geometry and geometric optics, and maybe start talking about the Monge-Ampère equation. I might make a post first about Nöldeke and Samuelson’s implementation duality paper and related literature.</p>
Generated Jacobian Equations, part 1/n2019-02-12T00:00:00+00:00http://hankquinlan.github.io//2019/02/12/Generated_Jacobian_Equations_1<!--idea you are going to talk broadly about Generated Jacobian Equations first, then switch to the real Monge Ampere equation
and progress from it to the General Theory-->
<p><strong>Prescribed Jacobians</strong></p>
<p>Often in mathematics we find ourselves looking for transformations between geometric shapes that send a given mass distribution into another. Concretely, given two mass distributions $f$ and $g$ in $\mathbb{R}^d$, we seek a map $T$ such that</p>
\[\int_{T(E)}g(y)\;dy = \int_{E}f(x)\;dx,\;\forall\;E\subset \mathbb{R}^d.\]
<p>That is, we want to find a transformation $T$ that spreads the mass $f$ into the mass $g$. There are many contexts where one needs to find such transformation or study its properties, and we will go over a list of examples later.</p>
<p>If the transformation $T$ satisfies the above condition and is also differentiable the change of variables formula tells us that the Jacobian of $T$ (the determinant of the derivative of $T(x)$) must satisfy the equation</p>
\[\det (DT(x))g(T(x)) = f(x),\;\forall\;x.\]
<p>In other words, the Jacobian of the map $T$ is prescribed in terms of $x$ and its image under $T$. Conversely, if a differentiable map $T$ has a Jacobian satisfying the above relation for all $x$, then it will map $f$ into $g$.</p>
<p>This can be considered as a first order partial differential equations which is (very) nonlinear, with $T(x)$ being the unknown. However this is too broad and difficult for our purposes. We are interested in situations where the mapping $T$ is determined from the gradient of a scalar field, a potential. This means a rule by which we can associate to every scalar function $u(x)$ a map $T_u(x)$, and in this case we are only interested in maps $T$ that arise from such a scalar $u$.</p>
<p>Concretely, we are interested in situations where we are given a function</p>
\[Y:\Omega\times \mathbb{R}\times \mathbb{R}^d \to \overline{\Omega}\]
<p>and then one looks for $T$ of the form</p>
\[T_u(x) := Y(x,u(x),Du(x))\]
<p>for some scalar function $u$. Note that in this case we chain rule gives us the formula</p>
\[DT_u (x) = Y_x + Y_z \otimes Du(x) + Y_p D^2u(x)\]
<p>Therefore, the first order equation for $T_u$ becomes a second order for $u$, and it has the form</p>
\[\det ( D^2u(x) +A(x,u(x),Du(x))) = \frac{f(x)}{g(Y(x,u(x),Du(x)))}\frac{1}{\det(Y_p(x,u(x),Du(x)))}\]
<p>This type of equation is known as a <strong>prescribed Jacobian equation</strong>.</p>
<p>The simplest such equation is the <strong>Monge-Ampère equation</strong>, corresponding to $Y(x,u,Du) = Du$,</p>
\[\det (D^2u) = \frac{f(x)}{g(Du(x))},\]
<p>solving this equation with the right boundary conditions gives a function whose gradient $Du$ maps $f$ into $g$.</p>
<p><strong>Generating Jacobian Equations</strong></p>
<p>We are only going to be interested in prescribed Jacobian equations arising from a <em>generating function</em> and known as Generated Jacobian Equations (GJE). This is a broad class of nonlinear equations covering optimal transportation, optimal surfaces design, differential geometry, economics, and more.</p>
<p>In each case, the following three ingredients are present</p>
<p>1) A map $T$ between two $d$-dimensional manifolds, whose Jacobian is prescribed</p>
\[\det(DT(x)) = f(x)/g(T(x)).\]
<p>2) A structure that produces $T$ from a scalar potential $u$, meaning there is some $\Psi$ such that for some scalar field $u$ we have</p>
\[T(x) = Y(x,u(x),Du(x)).\]
<p>3) The scalar potential $u(x)$ satisfies a generalized convexity condition.</p>
<p>As it will turn out the second and third ingredients are two aspects of the same structure, and behind that structure is a <em>generating function</em>. One may think of the generating function as determining the type of GJE we are considering in the same way a Riemannian metric determines a linear elliptic equation (i.e. $\Delta_gu = 0$ where $\Delta_g$ is Laplace-Beltrami operator for the metric).</p>
<p>Consider two domains $\Omega,\overline{\Omega} \subset \mathbb{R}^d$ (open, bounded subsets), a <em>generating function</em> $G$ is a function</p>
\[G: \Omega\times \overline{\Omega}\times \mathbb{R} \to \mathbb{R},\]
<p>which has the following two properties</p>
<p>1) $G$ s differentiable and strictly decreasing in its third argument, i.e. $G(x,y,z)>G(x,y,z’)$ if $z < z’$. In particular, this defines a smooth function $H$ such that $G(x,y,H(x,y,u)) = u$ for all $x,y,u$.</p>
<p>2) For fixed $x\in \Omega$, the map $ (y,z) \mapsto (D_xG(x,y,z),G(x,y,z))$ is injective, with a smooth inverse defined on its image.</p>
<p>The simplest example of a generating function is the linear generating function,</p>
\[G(x,y,z) = x\cdot y-z\]
<p>In the following posts I will explain how <strong>associated to every</strong> generating function there is a notion of gradient/subdifferential, a transform between functions, a notion of convex functions, and a notion of segments and convex sets. These objects in the case of the linear generating function reduce to the usual notion of a gradient/subdifferential, the Legendre transform, and the usual notions from convex geometry. This rich geometric structure is intrinsic to the equation, be it from optimal transform or from geometric geometrics.</p>
<p><br /></p>
<p>In the next post I will review other, more interesting examples of generating functions, discuss more about the elements of generating functions, later on I will introduce the Ma-Trudinger-Wang tensor and the notion of weak solutions to the GJE.</p>
Thoughts about nonlocal elliptic operators2018-12-20T00:00:00+00:00http://hankquinlan.github.io//2018/12/20/Thoughts_about_nonlocal_operators<p>This is my first post in two years (some other day I can ponder again about why I keep failing at blogging regularly). Perhaps unsurprisingly, two years later I still find myself thinking about the structure of nonlocal operators! What I want to discuss here is a very basic observation that I came across while working on my latest paper with Russell Schwab (where we revisit our 2016 preprint on min-max formulas for nonlocal operators). As such, this post can be seen as a kind of follow up to this <a href="/2016/10/30/Min_max_formulas.html">one</a>, where the Global Comparison Property and other basic definitions are discussed.</p>
<p>The idea is how to describe elliptic integro-differential equations in a way that is entirely analogous to</p>
\[F(D^2u,\nabla u(x),u(x),x) = 0,\]
<p>which is how local non-divergence equations are often considered. That is, the operator in the equation depends on the Hessian, gradient, and value of the function at $x$, as well as the location $x$ itself. To put it differently, a local fully nonlinear equation is described by a real valued function</p>
\[F: \text{Sym}(\mathbb{R}^d) \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]
<p>which is monotone with respect to its $\text{Sym}(\mathbb{R}^d)$ and $\mathbb{R}$ arguments –here $\text{Sym}(\mathbb{R}^d)$ denotes the space of symmetric real matrices of dimension $d$.</p>
<p>The way to achieve a similar description for nonlocal operators involves a functional following space, which arises naturally when dealing with elliptic integro-differential operators. The space is defined by</p>
\[L^\infty_\beta = \{ h \in L^\infty(\mathbb{R}^d) \text{ such that } |h(y)| = O(|y|^\beta) \text{ as } |y| \to 0 \},\]
<p>when $\beta \in (1,2)$ –the discussion below can be extended to all $\beta \in [0,2]$, but we focus on this range for the sake of a simplified presentation. The space $L^\infty_\beta$ has a topology given by the norm</p>
\[\| h \|_{L^\infty_\beta} := \sup \limits_{y \in \mathbb{R}^d} |h(y)| (\min\{1,|y|^{\beta} \})^{-1}.\]
<p>Now, suppose we are given a continuous, real valued function</p>
\[F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]
<p>which is monotone increasing with respect to its $L^\infty_\beta$ and $\mathbb{R}$ arguments. To such a function $F$ we can associate an operator $I = I_F$, as follows</p>
\[I(u,x) := F(\delta_x u,\nabla u(x),u(x),x),\]
<p>here, $\delta_x u$ is the function in $L^\infty_\beta$ defined by</p>
\[\delta_x u(y) := u(x+y)-u(x)-\chi_{B_1}(y)\nabla u(x)\cdot y.\]
<p>It follows, thanks to the monotonicity assumed on $F$, that $I_F$ satisfies the Global Comparison Property (GCP).</p>
<p>All operators with the GCP arise in this fashion. This can be seen easily, in fact. First, note that given</p>
\[(h,p,z,x_0) \in L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d,\]
<p>we can define a function $u_{h,p,z,x_0}:\mathbb{R}^d\to \mathbb{R}$ by</p>
\[u_{h,p,z,x_0}(x) = z+\chi_{B_1}(x-x_0)p\cdot (x-x_0)+ h(x-x_0),\;\;\forall\;x\in\mathbb{R}^d.\]
<p>This function may not be of class $C^\beta$ in all of $\mathbb{R}^d$, but it has enough regularity at $x_0$ for our purposes.</p>
<p>Now with this definition at hand, suppose $I:C^\beta(\mathbb{R}^d)\to C^0(\mathbb{R}^d)$ is an operator satisfying the GCP, then we may define $ F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ via</p>
\[F(h,p,z,x) := I( u_{h,p,z,x},x).\]
<p>Since $u_{h,p,z,x}$ is sufficiently regular, the operator is clasically defined for $u_{h,p,z,x}$ at the point $x$ be clasically defined. Furthermore, by construction, we have</p>
\[u_{h,p,z,x_0}(x_0) = z,\; \nabla u_{h,p,z,x_0}(x_0) = p,\; \delta_{x_0} u_{h,p,z,x_0} = h,\]
<p>so it follows that for $F$ thus constructed we have $I = I_F$, and that $F$ is monotone increasing with respect to $h$ and $z$ as long as $I$ satisfies the GCP.</p>
<p>Therefore, when thinking of a nondivergence integro-differential equation, a good way to think about it is as a function</p>
\[F:L^\infty_\beta \times \mathbb{R}^d \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\]
<p>As one last remark: note that the (positive) elements in the dual of $L^\infty_\beta$ are simply the Levy measures that are integrable against</p>
\[\min\{1,|y|^\beta\}.\]
Min max formulas for nonlocal equations2016-10-30T00:00:00+00:00http://hankquinlan.github.io//2016/10/30/Min_max_formulas<p>Recently, with <a href="http://users.math.msu.edu/users/rschwab/">Russell Schwab</a>, we finished project related to the question of which operators satisfy the global comparison principle. The preprint can be found <a href="https://arxiv.org/abs/1606.08417#">here</a>.</p>
<p>The manuscript turned out to be longer than we had anticipated. In part, this is due to our having to revisit several facts about Whitney extensions for functions in a Riemannian manifold, which took considerable extra space. We could not find a reference for Whitney extensions on manifolds that stated what we needed explicitly, but our proofs followed closely the very well known ideas for the case of $\mathbb{R}^d$. In any case, we intend to write a shorter paper reviewing our result in the case of $\mathbb{R}^d$ only, where several technical matters, including the Whitney extension, become much simpler.</p>
<p>This post will be an even shorter discussion of the ideas in the paper. I might do a later post discussing matters in greater generality (operators in a manifold or in a metric space). For now, this post will deal only with operators acting on $C^2$ functions in $\mathbb{R}^d$.</p>
<p>(Let me stress that the case of operators on a Riemannian manifold merits attention for several reasons, one is the study of Dirichlet to Neumann maps for elliptic equations, and another is that many free boundary problems can be posed as parabolic integro-differential equations in a manifold, both topics for another post).</p>
<p><strong>(1) The global comparison property</strong></p>
<p>We are considering scalar equations in $\mathbb{R}^d$, concretely, functions $u:\mathbb{R}^d\to\mathbb{R}$ which solve the equation</p>
\[I(u,x) = 0 \;\;\text{ for } x\in \Omega,\]
<p>where $I$ is some (possibly nonlinear) mapping between functions. The operators $I$ we are interested are (heuristically) those for which one expects the comparison principle to hold, which roughly speaking says that</p>
\[u \leq v \text{ in } \mathbb{R}^d\setminus \Omega \text{ and } I(u,x) \geq I(v,x) \text{ in } \Omega \Rightarrow u\leq v \text{ everywhere.}\]
<p>This is the case, for instance, for the Laplace operator. If one looks at the proof of the comparison principle for harmonic functions, then one sees that the crucial fact is the following:</p>
\[u\leq v \text{ everywhere, with } u=v \text{ at } x_0 \Rightarrow \Delta u \leq \Delta v \text{ at } x_0\]
<p>This motivates the following definition.</p>
<p><strong>Definition:</strong> An operator $I:C^2_b(\mathbb{R}^d)\to C^0_b(\mathbb{R}^d)$ is said to have the <strong>global comparison property</strong> (GCP) if whenever $u,v$ are such that $v$ touches $u$ from above at some $x_0\in\mathbb{R}^d$, we have $I(u,x_0)\leq I(v,x_0)$.</p>
<p>Recall that "$v$ touches $u$ from above at $x_0$" means that</p>
\[u(x) \leq v(x) \text{ for all } x\in\mathbb{R}^d \text{ and } u(x_0) = v(x_0) \text{ at some } x_0.\]
<p>By its very definition, the class of equations having the GCP is the class of equations that are amenable to treatment by methods based on the comparison property (i.e. barrier arguments, viscosity solutions, Perron's method, etc).</p>
<p><strong>Question:</strong> Is there a simple characterization for the class of operators which have the GCP?.</p>
<p><strong>(2) A few Examples</strong></p>
<p>1) $I u= \Delta u(x) $.</p>
<p>2) $Iu = H(\nabla u(x)) $, for some differentiable function $H:\mathbb{R}^d\to\mathbb{R}$.</p>
<p>3) The operator known as "the fractional Laplacian ", $Lu = -(-\Delta)^{\alpha/2}u$ with $\alpha \in [0,2]$, also written for $u\in C^2_b(\mathbb{R}^d)$ by the formula</p>
\[Lu (x) = C(d,\alpha)\int_{\mathbb{R}^d} \frac{1}{2}(u(x+y)+u(x-y)-2u(y))|y|^{-d-2s}\;dy.\]
<p>4) Any given a Borel measure $\mu$ defines such an operator, via</p>
\[Lu(x) = \int_{\mathbb{R}^d} u(x+y)-u(x)\;d\mu(y).\]
<p>5) If $L_1$ and $L_2$ are two operators having the GCP, then</p>
\[\min\{ L_1(u,x), L_2(u,x) \} \text{ and } \max\{ L_1(u,x), L_2(u,x) \}\]
<p>also have the GCP.</p>
<p>6) Given $u \in C^2_b(\mathbb{R}^{d-1})$, let $U_u$ denote be the unique bounded solution to the elliptic Dirichlet problem</p>
\[F(D^2U) = 0 \;\; \text{in } \{ (x,x_d) \in \mathbb{R}^d \mid \;0<x_d< 1 \},\; U = u \;\text{on } \{ x_d =0\},\; U= 0 \;\text{on } \{x_d=1\}\]
<p>Then, $I(u,x):= \partial_{d} U_u(x,0)$ satisfies the global comparison property.</p>
<p><strong>(3) A warm up exercise</strong></p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Lemma</strong>: Suppose $L$ is a bounded linear map $L:C^2(\mathbb{R}^d)\to C_b^0(\mathbb{R}^d)$ such that $Lu(x_0)\leq 0$ for any $u\in C^2_b(\mathbb{R}^d)$ having a nonnegative local maximum at $x_0$. Then
$$Lu(x) = \text{tr}(A(x)D^2u(x))+b(x)\cdot Du(x)+c(x)u(x)$$
where $A(x)\geq 0$ and $c(x)\leq 0$.
</p>
<p><strong>Proof</strong>: Let $P(x)$ denote the second order Taylor polynomial for $u$ at $x_0$.</p>
<p>For any $\delta>0$ one can construct a function $\eta \in C^2_b(\mathbb{R}^d)$ with $\| \eta\|_{C^2(\mathbb{R}^d)} \leq \delta$ such that $\eta(x_0)=0$ and </p>
\[P(x)+\eta(x) \geq u(x) \geq P(x)-\eta(x) \text{ in some neighborhood of } x_0\]
<p>In which case</p>
\[L(P,x_0) +C_0 \delta \geq L(u,x_0) \geq L(P,x_0) -C_0\delta\]
<p>Since $\delta>0$ was arbitrary, it follows that</p>
\[L(u,x_0) = L(P,x_0),\]
<p>in other words, for every $x \in \mathbb{R}^d$ we have that $L(u,x)$ is a (linear) function of the second order polynomial of $u$ at $x$. In particular, there must be a symmetric matrix $A(x)$, a vector $b(x)$ and a scalar $c(x)$ such that</p>
\[L(u,x) = \text{tr}(A(x)D^2u(x))+b(x)\cdot D u(x) + c(x) u(x).\]
<p>From here it is not difficult to see that $A(x)\geq 0$ and $c(x)\leq 0$.
∎</p>
<p>Keeping in mind the proof of the above lemma, think about the following:</p>
<p>-What can be said if instead of asking $L(u,x_0)\leq 0$ at every local maximum $x_0$, we only assume that this happens at <strong>global</strong> maxima?.</p>
<p>-What if the operator is not linear?</p>
<p>The first question was answered by P. Courrege in the 60’s, and the answer to this (semingly) purely analytical question leads to an important class of operators from the theory of stochastic processes.</p>
<p><strong>Definition:</strong> By a Levy measure we will refer to a Borel measure $\nu$ in $\mathbb{R}^d \setminus \{ 0 \}$ which may not have finite total mass, but is at least such that</p>
\[\int_{\mathbb{R}^d\setminus\{0\}} \min\{ |x|^2,1\}\;\nu(dy)<\infty\]
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Theorem</strong> (Courrege): A linear operator $L:C^2_b(\mathbb{R}^d)\to C_b^0(\mathbb{R}^d)$ has the global comparison property if and only if it is of the form
$$L = L_{\text{loc}}+L_{\text{Levy}},$$
where the operators $L_{\text{loc}}$ and $L_{\text{Levy}}$ are given by
$$ L_{\text{loc}}(u,x) = \text{tr}(A(x)D^2u(x))+b(x)\cdot D u(x) + c(x) u(x),$$
$$ L_{\text{Levy} }= \int_{\mathbb{R}^d\setminus \{ 0\}} u(x+y)-u(x)-\chi_{B_1(0)}(y) \nabla u(x)\cdot y \;\nu(x,dy).$$
for $A(x)\geq 0$, $A,b,c\in L^\infty$ and $\{ \nu(x,dy) \}_{x\in\mathbb{R}^d}$ is a family of Levy measures.
</p>
<p><strong>(4) A new min-max formula</strong></p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Theorem</strong> (joint with Russell Schwab): Let $I:C^2_b(\mathbb{R}^d)\to C_b^\gamma(\mathbb{R}^d)$ ($\gamma\in (0,1)$) be a Lipschitz continuous map which satisfies the GCP, and such that
$$\exists \text{ modulus of continuity } \omega(\cdot) \text{ and a constant } C \text{ such that:}$$
$$\|I(u)-I(v)\|_{L^\infty(B_r)} \leq C\|u-v\|_{C^2(B_{2r})}+C\omega(r)\|u-v\|_{L^\infty(\mathbb{R}^d)}.$$
Then, then there exists
i) a (uniformly continuous) family of linear operators $$ L_{ab}:C^2_b(\mathbb{R}^d) \to C^\gamma_b(\mathbb{R}^d)$$, each having the global comparison property,
ii) a (bounded) family of functions $f_{ab}\in C^\gamma_b(\mathbb{R}^d)$,
and these are such that for any function $u \in C^2_b(\mathbb{R}^d)$ we have the formula
$$ I(u,x) = \min\limits_{a} \max \limits_{b} \{ f_{ab}(x)+ L_{ab}(u,x)\}.$$
</p>
<p><strong>Remark:</strong> If one asks that $I$ be Lipschitz as a map between the spaces $C^\beta_b(\mathbb{R}^d)$ and $C^\gamma_b(\mathbb{R}^d)$ (where now $\beta,\gamma\in(0,1)$), then one can say more about the terms appearing in the min-max formula, in fact, in that case the theorem says that</p>
\[I(u,x) = \min\limits_{a} \max \limits_{b} \left \{ f_{ab}(x)+c_{ab}(x)u(x)+ \int_{\mathbb{R}^d\setminus \{0 \}}u(x+y)-u(x) \;\nu_{ab}(x,dy)\right \}.\]
<p>«««< HEAD
<strong>(3) Elementary proof when $I$ is Fréchet differentiable</strong></p>
<h1 id="let-us-suppose-that-ic2_bmathbbrd-to-c0_bmathbbrd-is-fréchet-differentiable">Let us suppose that $I:C^2_b(\mathbb{R}^d) \to C^0_b(\mathbb{R}^d)$ is Fréchet differentiable.</h1>
<p><strong>(3) Elementary proof when $I$ is Frechet differentiable</strong></p>
<p>Let us suppose that $I:C^2_b(\mathbb{R}^d) \to C^0_b(\mathbb{R}^d)$ is Frechet differentiable.</p>
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<p>1.Fix $u,v\in C^2_b(\mathbb{R}^d)$ and let</p>
\[u_t := v+t(u-v)\]
<p>Then</p>
\[I(u)-I(v) = \int_0^1 \frac{d}{dt} I(u_t)\;dt\]
<p>Then, the chain-rule says that</p>
\[I(u)-I(v) = \int_0^1 (DI(u_t))(u-v)\;dt\]
\[\;\;\;\;\;= \left ( \int_0^1 DI(u_t)\;dt\right )(u-v)\]
<p>That is, if we define an operator $L$ by $\int_0^1 DI(u_t)\;dt$ then</p>
\[I(u)-I(v) = L(u-v)\]
<p>2.For any $u\in C^2_b(\mathbb{R}^d)$, the linear operator $DI(u)$ is a continuous linear map from $C^2_b(\mathbb{R}^d)$ to $C^0_b(\mathbb{R}^d)$ which was the GCP.</p>
<p>3.The GCP is closed under convex combinations. Therefore, if we define</p>
\[\mathcal{D}(I) := \text{hull} \{ DI(u) \mid u \in C^2_b(\mathbb{R}^d \},\]
<p>then every element of $\mathcal{D}(I)$ has the GCP.</p>
<p>4.Thanks to to step 1), for any $u,v\in C^2_b(\mathbb{R}^d)$</p>
\[I(u,x) \leq \max \limits_{L \in \mathcal{D}(I)} \{ I(v,x) +L(u-v,x) \}.\]
<p>Since we have equality for $v=u$, it follows that</p>
\[I(u,x) = \min \limits_{v \in C^2_b(\mathbb{R}^d))} \max \limits_{L \in \mathcal{D}(I)} \{ I(v,x) +L(u-v,x) \}.\]
<p><strong>(4) A finite dimensional version</strong></p>
<p>Let $G$ be a finite set, and let $C(G)$ denote the space of real valued functions in $G$.</p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Lemma:</strong> Let $I:C(G)\to C(G)$ be a Lipschitz map satisfying the GCP, then,
$$ I(u,x) = \min \limits_{v \in C(G)} \max \limits_{L \in \mathcal{D}I} \{ I(v,x) + L(u-v,x)\},$$
where each $L$ is a linear map from $C(G)$ to $C(G)$ having the form
$$ L(u,x) = c(x)u(x) + \sum \limits_{y\in G} (u(y)-u(x))k(x,y),$$
for some $c\in C(G)$ and some $k:G\times G\to\mathbb{R}$ with $k(x,y)\geq 0$ for all $x$ and $y$ in $G$.
</p>
<p>The key difference now is that in this case $I:C(G)\to C(G)$ amounts to a Lipschitz map between two finite dimensional vector spaces, and here we have Clarke’s non-smooth calculus at our disposal.</p>
<p><strong>Definition</strong> (Clarke Jacobian): Let $I:C(G)\to C(G)$ be a Lipschiz continuous map and $f \in C(G)$. The <em>Clarke Jacobian</em> of $I$ at $f$ is defined as the set</p>
\[\mathcal{D}_f(I) := \text{hull}\{ L = \lim\limits_n L_n \mid \exists f_n \to f \text{ s.t. } I \text{ differentiable at } f_n \text{ and } L_n = DI(f_n) \forall n \}.\]
<p>Finally, we will consider the total Clarke Jacobian of $I$, denoted $\mathcal{D}I$, and defined by</p>
\[\mathcal{D}(I) := \text{hull} ( \bigcup_f \mathcal{D}_f(I) ).\]
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Lemma</strong> (mean value theorem): Let $I:C(G)\to C(G)$ be Lipschitz mapping. For any $u,v\in C(G)$, there exists some $L\in \mathcal{D}(I)$ such that
$$ I(u)-I(v) = L(u-v).$$
</p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<strong>Corollary</strong>: Let $I$ be as before, then for any $u\in C(G)$ and any $x\in G$, we have
$$ I(u,x) \leq I(v,x) + \max \limits_{L \in \mathcal{D}(I)} L(u-v,x).$$
</p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;"> <b>Proposition:</b> If $I:C(G)\to C(G)$ is Lipschitz and has the GCP, then each $L \in \mathcal{D}(I)$ has the GCP.
</p>
<p>«««< HEAD
The proof of this last proposition is quite simple. First, let $u\in C(G)$ be a point of differentiability for $I(\cdot)$, and let $L_u$ denote the Fréchet derivative of $I(\cdot)$ at $u$.
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The proof of this last proposition is quite simple. First, let $u\in C(G)$ be a point of differentiability for $I(\cdot)$, and let $L_u$ denote the Frechet derivative of $I(\cdot)$ at $u$.</p>
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<p>Then, let $v_1,v_2 \in C(G)$ be such that $v_1\leq v_2$ in $G$ with $=$ at some $x_0 \in G$, then for every $t>0$ we have</p>
\[u+tv_1 \leq u+tv_2 \text{ in } G,\;\; u+tv_1 = u+tv_2 \text{ at } x_0\]
<p>Then</p>
\[I(u+tv_1,x_0)\leq I(u+tv_2,x_0)\]
<p>«««< HEAD
Using the fact that $I(\cdot)$ is Fréchet differentiable at $t$, it follows that
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Using the fact that $I(\cdot)$ is Frechet differentiable at $t$, it follows that</p>
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\[L(v_1,x_0)\leq L(v_2,x_0)\]
<p><strong>(5) The min-max formula via finite dimensional approximations.</strong></p>
<p>With this finite dimensional result at hand, one can try to prove the result by approximating the space $C^2_b(\mathbb{R}^d)$ by finite dimensional subspaces, obtained roughly as follows: one constructs an increasing sequence of finite graphs $G_n$, which converge to $\mathbb{R}^d$.</p>
<p>Consider the following increasing sequence of discrete sets in $\mathbb{R}^d$:</p>
\[\tilde G_n := 2^{-n}\mathbb{Z}^d,\;\; G_n := [-2^n,2^n]^d \cap (2^{-n}\mathbb{Z}^d),\]
<p>in terms of these sets, we define projection operators</p>
\[\pi_n: C^2_b(\mathbb{R}^d) \to X_n \subset C^2_b(\mathbb{R}^d),\;\;\;\pi_n^0: C^0_b(\mathbb{R}^d) \to X_n ^0\subset C^0_b(\mathbb{R}^d).\]
<p>Then, for each $n$, we define a finite dimensional approximation to $I$, $ I_n :C^2_b(\mathbb{R}^d) \to C^0_b(\mathbb{R}^d)$, by</p>
\[I_n : = \pi_n^0 \circ I \circ \pi_n.\]
<p>Now, we can think of $I_n$ also as a (Lipschitz) map $C(G) \to C(G)$, and apply the min-max formula for this case,</p>
<p style="border:1.5px; border-style:solid; border-color:#000000; padding: 1em;">
<b>Lemma</b>: For every $n \in \mathbb{N}$ and $x\in G_n$, we have, for any $u \in C^2_b(\mathbb{R}^d)$,
$$ I_n(u,x) = \min\limits_{v \in C^2_b(\mathbb{R}^d)} \max \limits_{L \in \mathcal{D}(I_n)} \{ I_n(v,x) + L(u-v,x) \},$$
moreover, for each $L \in \mathcal{D}(I_n)$, there is some $c\in C(G_n)$ and measures $\{ m(x,dy) \}_{x\in G_n}$ such that
$$L(v,x) = c(x)v(x) + \int_{\mathbb{R}^d} v(y)-v(x)\; m(x,dy),\;\;\;\forall\;x \in G_n$$
Moreover, for each $x\in G_n$, we have some nonnegative function $k:G_n\times G_n \to \mathbb{R}$ such that
$$ m(x,dy) = \sum \limits_{z \in G_n\setminus \{x\}} k(x,z)\delta_z(dy).$$
</p>
Why write about mathematics?2016-10-25T00:00:00+00:00http://hankquinlan.github.io//2016/10/25/Why_write<p>Over the years, I have tried again and again to mantain a math blog. In all those instances, I would ultimately plateau after a flurry of early posts, often overwhelmed by more urgent matters (teaching duties, research, job applications, fixing fatal mistake in a paper). I would eventually abandon the blog –who knows, maybe the same will happen this time.</p>
<p>But now I believe I understand the main reason for why these attempts failed, ultimately, I would not hit “publish” on a post until I felt it was significantly polished (which is not the same as actually polished). I mean, I already feel I am over analizing this very first post!. Thinking too much before publishing defeats the purpose of blogging: a feature, not a bug of blogging is the ability to write often, and post things without agonizing whether the thing really is perfect (and yes, to any of my coauthors reading this: I appreciate the irony of me writing these words).</p>
<p>What I envision in the short term is to make this blog a universal placeholder for notes on topics I am currently learning or thinking about, and the occasional rant about topics related to mathematics. The exposition will therefore be mostly technical (often confusing or downright incorrect), and it will deal almost exclusively with analysis and partial differential equations.</p>
<p>Another aim of the blog is to experiment and simply see what happens. How can technical discussion of mathematics be done online, in blog format? Can short posts I write here be of use to others? Can I develop a blogging habit which may turn out to be an appropriate use of my time? Can it serve as a place to speculate about approaches to problems, and to gain useful feedback? Hopefully the answer to some of these questions will turn out to be positive.</p>
<p>In short, this blog’s mantra will be: <strong>write a lot of crappy posts about math, and hope for the best.</strong></p>
<p><em>About the blog’s name</em>: The name of the blog was inspired by a well known lecture by Freeman Dyson, where he uses birds and frogs as metaphor for different styles or approaches to mathematical research. A written version can be found <a href="/pdf/Dyson.pdf">here</a>.</p>
<p><em>About the blog’s layout:</em> For now I intend to keep the website layout minimal (this website was built from scratch on github pages using jekyll, for which <a href="http://jmcglone.com/guides/github-pages/">this</a> is a good tutorial). The only other feature beside the post feed is the RSS feed linked above, I will add features later as needed.</p>