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<n, 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<n, 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.

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