Ngô Quốc Anh

May 20, 2012

The Wolff potential

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:26

The Wolff potential probably first appeared in a joint paper between L.I. Hedberg and Th.H. Wolff in 1983 in relation to the spectral synthesis problem for Sobolev spaces. Generally speaking, it is defined for any non-negative Borel measure $\mu$ as follows $\displaystyle {\mathbf W_{\beta ,\gamma }}\mu (x) = \int_0^\infty {{{\left[ {\frac{{\mu ({B_t}(x))}}{{{t^{n - \beta \gamma }}}}} \right]}^{\frac{1}{{\gamma - 1}}}}\frac{{dt}}{t}}$

where $1<\gamma<\infty$, $\beta>0$, $\beta \gamma, and $B_t(x)$ is the ball of radius $t$ centered at the point $x$.

If $d\mu=f dx$ with $f \geqslant 0$ and $f \in L^1_{loc}(\mathbb R^n)$, we write $\displaystyle {\mathbf W_{\beta ,\gamma }}(f)(x) = \int_0^\infty {{{\left[ {\frac{{\int_{{B_t}(x)} {f(y)dy} }}{{{t^{n - \beta \gamma }}}}} \right]}^{\frac{1}{{\gamma - 1}}}}\frac{{dt}}{t}} .$

There are several cases

• If $\beta=1$ and $\gamma=2$, we have $\displaystyle {\mathbf W_{1,2}}(f)(x) = \int_0^\infty {\left( {\int_{{B_t}(x)} {f(y)dy} } \right)\frac{{dt}}{{{t^{n - 3}}}}} .$

Clearly, this is the well-known Newton potential. Indeed, by exchanging the order of the integral variables, we have $\displaystyle \begin{gathered} {\mathbf W_{1,2}}(f)(x) = \int_0^\infty {\left( {\int_{{B_t}(x)} {f(y)dy} } \right)\frac{{dt}}{{{t^{n - 3}}}}} \hfill \\ \qquad= \int_{{\mathbb{R}^n}} {f(y)\left( {\int_{|x - y|}^\infty {\frac{{dt}}{{{t^{n - 3}}}}} } \right)dy} \hfill \\ \qquad= \int_{{\mathbb{R}^n}} {\frac{{f(y)}}{{|x - y{|^{n - 2}}}}dy}. \hfill \\ \end{gathered}$

• If $\beta=\frac{\alpha}{2}$ and $\gamma=2$ with $0<\alpha, we have $\displaystyle {\mathbf W_{\frac{\alpha }{2},2}}(f)(x) = \int_0^\infty {\left( {\int_{{B_t}(x)} {f(y)dy} } \right)\frac{{dt}}{{{t^{n - \alpha - 1}}}}} .$

Clearly, this is the well-known Riesz potential of order $\alpha$ since $\displaystyle\begin{gathered} {\mathbf W_{\frac{\alpha }{2},2}}(f)(x) = \int_0^\infty {\left( {\int_{{B_t}(x)} {f(y)dy} } \right)\frac{{dt}}{{{t^{n - \alpha - 1}}}}} \hfill \\ \qquad= \int_{{\mathbb{R}^n}} {f(y)\left( {\int_{|x - y|}^\infty {\frac{{dt}}{{{t^{n - \alpha - 1}}}}} } \right)dy} \hfill \\ \qquad= \int_{{\mathbb{R}^n}} {\frac{{f(y)}}{{|x - y{|^{n - \alpha }}}}dy}. \hfill \\ \end{gathered}$

In this case, the following PDE $\displaystyle u(x)=\mathbf W_{\frac{\alpha}{2},2}(u^\frac{n+\alpha}{n-\alpha})(x)$

is equivalent to the following integral equation $\displaystyle u(x)=\int_{\mathbb R^n}\frac{1}{|x-y|^{n-\alpha}}u^\frac{n+\alpha}{n-\alpha}(y)dy.$

• In the case $\beta=1$ and $\gamma=p$, the following PDE $\displaystyle u(x)=\mathbf W_{1,p}(u^q)(x)$

is equivalent to the $p$-Laplacian equation $\displaystyle -\text{div}(|\nabla u|^{p-1}\nabla u)=u^q(x).$

• In the case $\beta=\frac{2k}{k+1}$ and $\gamma=k+1$, the following PDE $\displaystyle u(x)=\mathbf W_{\frac{2k}{k+1},k+1}(u^q)(x)$

is equivalent to the well-known $k$-Hessian equation $\displaystyle F_k[-u]=u^q(x)$

where $F_k[u]=S_k(\lambda(D^2u))$ the $k$-th symmetric function and $\lambda(D^2u)=(\lambda_1,...,\lambda_n)$ the set of eigenvalues of the Hessian matrix $(D^2u)$. A good reference can be found here.