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Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$.

As one could find in G.M. Troianello "Elliptic Differential Equations and Obstacles Problems" (lem. 1.5) there exists a linear and continuos extension operator of $C^{k+\alpha}(\partial\Omega)$ to $C^{k+\alpha}(\bar B)$, with $B$ open ball of $\mathbb{R}^N$ containing $\bar\Omega$.

I need a similar result for parabolic Hölder spaces.

I mean, there exists a linear and continuos extension operator of $C^{\frac{k+\alpha}{2};k+\alpha}([0,T]\times\partial\Omega)$ to $C^{\frac{k+\alpha}{2};k+\alpha}([0,T]\times\bar B)$, with $B$ open ball containing $\bar\Omega$?

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It is possible to extend the function first to $ Q=[0,T]\times \Omega$ and then to $\mathbb{R}^{N+1}$. The second step is covered by a general statement for anisotropic Besov spaces, theorem 18.5 in [O. V.Besov, V. I.Il'in, S. M.Nikolskii, Integral representations of functions and imbedding theorems].

To extend a function $\varphi\in C^{\frac{k+\alpha}{2};k+\alpha}([0,T]\times\partial\Omega)$ to $C^{\frac{k+\alpha}{2};k+\alpha}(\bar Q)$ one can use a solution of some auxiliary BVP problem for the heat operator $L=\partial_t-\Delta$, since for solutions the required smooothnes is known. First let's extend $\varphi$ to $\tilde\varphi\in C^{\frac{k+\alpha}{2};k+\alpha}([-T,T]\times\partial\Omega)$ for $t<0$ as a linear combination of functions of the form $\varphi(x,-t/i)$, $i=1,\ldots,k+1$. This construction is in effect one-dimensional and is described in Lemma 6.37 in [D. Gilbarg, N. Trudinger. Elliptic Partial Differential Equations of Second Order]. Let $\varkappa\in C^\infty_0(\mathbb R) $ be s.t. $\varkappa(t)=1$ for $t\ge0$ and $\varkappa(t)=0$ for $t\le -T/2$. Put $\hat \varphi=\tilde\varphi\varkappa$. Now the solution of thw BVP $$\left\{ \begin{eqnarray} Lu&=&0,\\ u|_{[-T,T]\times\partial\Omega}&=&\hat\varphi,\\ u|_{t=-T}&=&0, \end{eqnarray} \right. $$ belongs to $C^{\frac{k+\alpha}{2};k+\alpha}([-T,T]\times\Omega)$. The compatibility conditions hold because initial and boundary conditions (for $t<-T/2$) are zero.

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  • $\begingroup$ Thank you. When you say that the solution $u$ of the BVP belongs to $C^{\frac{k+\alpha}{2};k+\alpha}(\bar Q)$, you use the parabolic Schauder esitimates? $\endgroup$
    – foo90
    Nov 17, 2014 at 12:44
  • $\begingroup$ The existence result for the first BVP is used. If $\varphi\in C^{\frac{k+\alpha}{2};k+\alpha}([0,T]\times\partial\Omega)$, the compatibility conditions hold etc, then there exists a solution from $C^{\frac{k+\alpha}{2};k+\alpha}(\bar Q)$. $\endgroup$
    – Andrew
    Nov 17, 2014 at 13:05
  • $\begingroup$ @foo90 Yes, you are right. So here is another way to satisfy compatibility conditions. I've rewritten the answer, it was too long for a comment. $\endgroup$
    – Andrew
    Nov 20, 2014 at 13:27
  • $\begingroup$ Thank you. And last thing, do you have a reference the existence result for the first BVP you invoke? $\endgroup$
    – foo90
    Nov 20, 2014 at 13:56
  • $\begingroup$ For $k\ge2$ it's theorem 5.2 from ch. 4 of [O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasi–Linear Equations of Parabolic Type]. For $k=1$ [E. A. Baderko, Mathematical Methods in the Applied Sciences, V. 20, #5], $k=0$ [A.N. Konenkov, Differential Equations, 2004, V. 40, #3]. $\endgroup$
    – Andrew
    Nov 20, 2014 at 14:25

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