{"id":16496,"date":"2022-12-11T12:49:55","date_gmt":"2022-12-11T11:49:55","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=16496"},"modified":"2022-12-29T18:31:14","modified_gmt":"2022-12-29T17:31:14","slug":"equilibrium-measures-and-obstacle-problems","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2022\/12\/11\/equilibrium-measures-and-obstacle-problems\/","title":{"rendered":"Equilibrium measures and obstacle problems"},"content":{"rendered":"
\"Gilbert<\/a>
Gilbert Agnew Hunt, Jr. (1916 - 2008) Mathematician and tennis player. Famous notably for his work connecting potential theory and Markov processes.<\/figcaption><\/figure>\n

\nThis post is devoted to a reformulation of the equilibrium measure problem of potential theory<\/strong> into an obstacle problem for minimal surfaces<\/a>, advertised by Sylvia Serfaty<\/a>.<\/p>\n

Riesz kernel<\/strong>. On $\\mathbb{R}^d$, for $-2 < s < d$,\n$$\nK_s=\\frac{1}{s\\left|\\cdot\\right|^s}\n\\quad\\text{with the convention}\\quad\nK_0=-\\log\\left|\\cdot\\right|.\n$$ The special case $K_{d-2}$ is the Newton or Coulomb kernel<\/strong>. The lower bound $-2 < s$ ensures the strict convexity of an energy functional (see below) while the upper bound $s< d$ ensures integrability of the kernel and well definiteness of the potential $K_s*\\mu$ of natural measures.\n<\/p>\n

Riesz energy<\/strong>. The Riesz energy of a probability measure $\\mu$ on $\\mathbb{R}^d$ is
\n$$
\n\\mathcal{E}_s(\\mu)=\\iint K_s(x-y)\\mu(\\mathrm{d}x)\\mu(\\mathrm{d}y).
\n$$ The Riesz energy with external field<\/strong> $V:\\mathbb{R}^d\\to\\mathbb{R}$ (taken continuous) is
\n$$
\n\\mathcal{E}^V_s=\\iint(K_s(x-y)+V(x)+V(y))\\mu(\\mathrm{d}x)\\mu(\\mathrm{d}y)
\n=\\mathcal{E}_s(\\mu)+2\\int V\\mathrm{d}\\mu.
\n$$ This functional is strictly convex and has compact sub-level sets with respect to the topology of weak convergence of probability measures, in particular it is lower semi-continuous.\n<\/p>\n

Equilibrium measure and Frostman conditions<\/strong>. The equilibrium measure $\\mu_{\\mathrm{eq}}$ is
\n$$
\n\\mu_{\\mathrm{eq}}=\\arg\\min_\\mu\\mathcal{E}^V_s(\\mu).
\n$$ It depends on $d,s,V$. Since $\\mathcal{E}^V_s$ is a nice quadratic form, its minimizer is characterized by nice Euler-Lagrange equations, which are known in this context as Frostman conditions<\/strong>. Namely $\\mu_{\\mathrm{eq}}$ is the unique probability measure $\\mu$ for which there exists a constant $c$ such that
\n$$ K_s*\\mu+V
\n\\begin{cases}
\n=c&\\text{on $\\mathrm{supp}(\\mu)$}\\\\
\n\\geq c&\\text{everywhere}
\n\\end{cases}.
\n$$<\/p>\n

Fundamental solution.<\/strong> There exists a positive linear operator $L_s$ such that
\n$$
\nL_s K_s=\\delta_0,\\quad K_s=(L_s)^{-1}.
\n$$ More generally, for a probability measure $\\mu$ on $\\mathbb{R}^d$, the Riesz potential $U:=K_s*\\mu$ satisfies
\n$$
\nL_s U=\\mu.
\n$$ The operator $L_s$ is essentially equal to a (possibly fractional) power of the Laplacian
\n$$
\nL_s=c_{s,d}(-\\Delta)^{\\frac{d-s}{2}},
\n\\quad\\text{in particular}\\quad
\nL_{d-2}=-c_d\\Delta.
\n$$ The Coulomb case $s=d-2$ is the unique value of $s$ for which the operator $L_s$ is local<\/strong> namely $L_sf(x)$ depends on $f$ locally at $x$ only. In this case we can apply $L_{d-2}$ to the Frostman variational characterization of $\\mu_{\\mathrm{eq}}$ to get that in the sense of distributions,
\n$$
\n\\mu_{\\mathrm{eq}}=\\frac{\\Delta V}{c_d}
\n\\quad\\text{on the interior of }\\mathrm{supp}(\\mu_{\\mathrm{eq}})
\n$$ (in particular $\\mu_{\\mathrm{eq}}$ has no support on $\\{\\Delta V<0\\}$). When $s=d-2$ and $V=\\left|\\cdot\\right|^2$ then this can be used to prove that $\\mu_{\\mathrm{eq}}$ is the uniform probability measure on the unit ball<\/strong>. Beyond the Coulomb case, the Riesz kernel is no longer local, and the computation of the equilibrium measure becomes more difficult.<\/p>\n

Obstacle problem.<\/strong> Let us express the Frostman conditions in terms of the Riesz potential $U=K_s*\\mu$, the operator $L_s$, and the function $\\varphi=c-V$ :
\n$$
\n\\begin{cases}
\nU & \\geq\\varphi\\\\
\nL_s U & \\geq0,\\text{ and $=0$ on $\\{U>\\varphi\\}$}
\n\\end{cases}
\n$$ in other words we have the ``dichotomy''
\n$$
\n\\min(U-(c-V),L_s U)=0.
\n$$ This is called an obstacle problem<\/strong>, where $U$ is modelling the height of a membrane and $\\varphi$ an obstacle. Since $\\mathrm{supp}(\\mu)$ is unknown, it is a free boundary problem<\/strong>. The analysis of such problems in terms of existence, uniqueness, and regularity is still an active subject of research. It has also a probabilistic side, explored notably by Getoor, Chung, and others, related to first and last passage times<\/strong> of L\u00e9vy processes. The obstacle problem offers an alternative way to compute equilibrium measures. Last but not least, the obstacle problem admits a variational formulation<\/strong> with a Dirichlet energy, which is actually the convex dual<\/strong> problem of the equilibrium measure problem, and the probabilistic side corresponds in a way to a probabilistic interpretation of the modified Robin problem<\/strong> versus the Dirichlet problem<\/strong>.<\/p>\n

About the links between potential theory and Markov processes, let us recall, following Hunt, that the potential kernel associated to a Markov semigroup ${(P_t)}_{t\\geq0}$ is $$\\int_0^\\infty P_s\\mathrm{d}s,$$ well understood when writing $P_s=\\mathrm{e}^{-sL}$ where $-L$ is the infinitesimal generator, since formally $$\\int_0^\\infty\\mathrm{e}^{-sL}\\mathrm{d}s=L^{-1}\\circ(\\mathrm{id}-\\pi_{\\ker L}).$$<\/p>\n

Further reading.<\/strong><\/p>\n