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First I apologize for my bad English and for any error: this is my first question.

I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental solution of the heat equation, the single and double layer potentials are defined as follows:

(SLP) $u(t,x)= \int_{0}^{t}\int_{\partial \Omega}\Gamma(t-\tau,x-y)\varphi(\tau,y)d\sigma(y) d\tau$

(DLP) $v(t,x)= \int_{0}^{t}\int_{\partial \Omega}\frac{\partial\Gamma(t-\tau,x-y)}{\partial\nu(y)}\varphi(\tau,y)d\sigma(y) d\tau$.

In the cylinder $Q=\Omega\times[0,T]$, $\Omega\subset\mathbb{R}^n$, $\Omega$ bounded domain regular enough and $\nu$ in the unit outer normal to $\partial \Omega$

My questions are:

1) It is true that if $\varphi\in C^{1+\alpha/2,2+\alpha}(Q)$ then $v$ extends with continuity to $\bar{\Omega}\times[0,T]$ and this extension stay in $C^{1+\alpha/2,2+\alpha}(\bar{Q})$?

2)It is true that if $\varphi\in C^{(1+\alpha)/2,1+\alpha}(Q)$ then $u$ extends with continuity to $\bar{\Omega}\times[0,T]$ and this extension stay in $C^{1+\alpha/2,2+\alpha}(\bar{Q})$?

Where the functions spaces $C^{1+\alpha/2,2+\alpha},C^{(1+\alpha)/2,1+\alpha}$ are the parabolic holder spaces for example defined in "Lectures on Elliptic and Parabolic Equations in Holder Spaces, N.V. Krylov, pg 117,118"

(Is possible that the parameters of the Holder spaces are different.)

These questions are well known for the armonic potentials and I want to know if similar results hold true for the heat potentials.

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  • $\begingroup$ You take $\varphi\in C^{1+\alpha/2,2+\alpha}(Q)$ but densities should be defined on $\partial\Omega\times[0,T]$. Does it means that the trace of $\varphi$ is taken? $\endgroup$
    – Andrew
    Commented Sep 23, 2014 at 12:33
  • $\begingroup$ You're right, i think that the density $\varphi$ must stay in $C^{1+\alpha/2,2+\alpha}(\partial\Omega\times[0,T])$ $\endgroup$
    – foo90
    Commented Sep 23, 2014 at 12:49

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The answer is yes for the smooth enough boundary, provided $\varphi|_{t=0}=0$ and for 2) additionally $\partial_t\varphi|_{t=0}=0$. Say, for the SLP from $C^{1+\alpha/2,2+\alpha}(\bar{Q})$ it should hold $u|_{t=0}=\partial_tu|_{t=0}=0$. From here the mentioned conditions on $\varphi$ at $t=0$ follow.

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  • $\begingroup$ Thank you.Can you give me some references for these results? $\endgroup$
    – foo90
    Commented Sep 23, 2014 at 12:54
  • $\begingroup$ When $\Omega$ is a half-space it is written in O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type". For general case I havn't a reference handy, but smoothness of the heat potentials was studied in the works of Kamynin. $\endgroup$
    – Andrew
    Commented Sep 23, 2014 at 13:10
  • $\begingroup$ Thank you. I think that the works of Kamynin are what I was looking for. $\endgroup$
    – foo90
    Commented Sep 24, 2014 at 13:08
  • $\begingroup$ In the Kamynin's works I think that the SLP and the DLP don't reach the regularity $C^{1+\alpha/2,2+\alpha}$ but $C^{1+\alpha'/2,2+\alpha'}$ for all $\alpha'<\alpha$. If you recall some references for the result with $\alpha$ with holder coefficient, came you give me? Thank you in advance $\endgroup$
    – foo90
    Commented Sep 25, 2014 at 11:06
  • $\begingroup$ It's very kind of you. I can't read Russian, but possibly a friend of mine can traslate it to me. My email address is [email protected] $\endgroup$
    – foo90
    Commented Sep 26, 2014 at 15:36

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