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Parabolic measure and the Dirichlet problem for the heat equation in two dimensions. (English) Zbl 0698.35068
This proves three theorems which characterize those domains D defined below for which parabolic measure, adjoint parabolic measure and Lebesgue measure are mutually absolutely continuous. This results in Theorem 1. That this is nearly the best possible result is shown by Theorem 2. The proof of Theorem 1 depends on the solution of the Dirichlet problem for the heat equation in D as stated in Theorem 3. Theorems 1-3 form part of the second author’s doctoral dissertation under the direction of the first author. Authors are careful and thorough in giving credit to other authors. Paper is detailed but easy to read.
Theorem 1. Let f be a continuous function on the real line \({\mathbb{R}}\), with \(f(t)=0\), \(| t| \geq 1\), and put \(D=\{(x,t):\) \(x>f(t)\}\). If \(E\subseteq {\mathbb{R}}\) is Lebesgue measurable, let \(| E|\) be the Lebesgue measure of E and set \(\rho(E)=\{f(t),t):\) \(t\in E\}.\)
If \(F\subseteq \partial D\) is a Borel set, let \(\omega^+(F)\) denote the parabolic measure of F with respect to (10,10), \(\omega^-(F)\) the adjoint parabolic measure of F with respect to (10,-10). Suppose that \(\psi\) is a nondecreasing continuous function on \([0,\infty)\) with \(\psi(0)=0\) and \(\psi>0\) on \((0,\infty)\), \[ \int^{\infty}_{0}\tau^{-2}\psi (\tau)^ 2d\tau =B^ 2<\infty, \] such that \[ | f(t)-f(s)| \leq \psi (| s-t|),\quad s,t\in {\mathbb{R}}. \] Then \[ \psi(\tau)^ 2/\tau \leq 2\int^{2\tau}_{\tau} \psi(\tau)^ 2 \tau^{-2} d\tau \to 0 \] as \(\tau\to 0\) and \[ d\omega^+\circ \rho =h^+dt,\quad d\omega^-\circ \rho =h^-dt \] on \([-1,1]\), and for a given interval \(I\subset [-1,1]\), and \(1<p<\infty\) \[ | I|^{-1}\int_{I}(h^+)^ p d\tau \leq [A| I|^{- 1}\int^{I}h^+d\tau]^ p;\quad | I|^{-1}\int_{I}(h^- )^ p d\tau \leq [A| I|^{-1}\int^{I}h^-d\tau]^ p, \] where \(A>0\) is a constant depending on p,B.
Theorem 2. Let \(\psi\) be a nondecreasing continuous function on \([0,\infty)\) with \(\psi(0)=0\), \(\psi=1\) on \([1,\infty)\) and \(\int^{1}_{0}\tau^ 2\psi (\tau)^ 2d\tau =+\infty.\)
Suppose for some \(\tau_ 0>0\) that \(\psi(2\tau)\leq 2\psi(\tau)\), \(0\leq \tau \leq \tau_ 0\). Then there exists an f such that \[ | f(t)- f(s)| \leq \psi (| s-t|),\quad s,t\in {\mathbb{R}} \] and \(\omega^+\circ \rho\), \(\omega^-\circ \rho\), and Lebesgue measure are singular with respect to each other on \([-1/2/1/2].\)
Theorem 3. Let f, \(\psi\), D be as in Theorem 1. Let \[ W(x,t)=(4\pi)^{- 1/2} xt^{-3/2} \exp (-x^ 2/4t),\quad t>0;\quad W(x,t)=0,\quad t\leq 0. \] Given \(A_ 0>0\) let \(\Gamma (s)=\Gamma (s,A_ 0)=\{(x,t):\) \(x- f(s)>0\), \(| t-s| <A_ 0(x-f(s))^ 2\}\). Fix \(A_ 0\). Fix p, \(1<p<\infty\). Given \(g\in L_ p({\mathbb{R}})\), then there exists \(k\in L_ p({\mathbb{R}})\) with \(\| k\|_ p\leq A_ 0\| g\|_ p\) and such that if \[ u(x,t)=\int_{{\mathbb{R}}}W(x-f(\tau),t-\tau)k(\tau)d\tau, \] then u is the unique solution to the heat equation in D for which \(\lim_{(s,t)\to (f(s),s)}u(x,t)=g(s)\) almost everywhere with respect to Lebesgue measure on \({\mathbb{R}}\), and \[ u^*(s)=\sup_{(x,t)\in \Gamma (s)\cap D}\in L_ p({\mathbb{R}}),\quad s\in {\mathbb{R}} \] for some \(A_ 0\). Also k is the solution of the integral equation \[ k(t)+\lim_{\epsilon \to 0^+}\int^{t-\epsilon}_{-\infty}W(f(t)-f(\tau),t- \tau)k(\tau)d\tau =g(t). \]
Reviewer: J.J.Cross

35K05 Heat equation
28A12 Contents, measures, outer measures, capacities
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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