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We study properties of the boundary trace operator on the Sobolev space $$W^1_1(\Omega )$$ . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator $$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$ , where $$\Omega _K$$ is von Koch’s snowflake and $$X(\Omega _K)$$ is a trace space with the quotient norm. Since $$\Omega _K$$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator $$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$ such that $$Tr \circ S= Id_{X(\Omega _K)}$$ . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as $$\ell _1$$ . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain $$\Omega $$ with regular boundary, which explains Banach space geometry cause for this phenomenon.

In this article, we apply the parametrix method in order to obtain the existence and the regularity properties of the density of a skew diffusion and provide a Gaussian upper bound. This expansion leads to a probabilistic representation.

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order $\beta \in (\frac13,\frac12)$ . We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H $\in (\frac13,\frac12)$ .

This paper is mainly concerned with the local behavior of singular solutions of the biharmonic equation $$\Delta ^2 u = |x|^\sigma u^p $$ with u ≤ 0 , $$- \Delta u $$ ≤ 0 in $$\Omega \backslash \{ 0\} \subset \mathbb{R}^N ,N$$ ≥ 4, and Ω = B(0, R) is a ball centered at the origin of radius R > 0. The complete description of the singularity together with an existence result will be given when $$$$ ≤ 0, for N > 4, or 1 < p < +∞, for N = 4. Moreover, an a priori estimate of the radially symmetric solutions will be established when p ≥ $$\frac{{N+ \sigma}}{{N-4}}, -4 < \sigma$$ ≤ 0, N > 4. This paper generalizes the results in Brezis and Lions (1981) and Lions (1980) for the corresponding Laplace equation.

We study weak and strong convergence of the stochastic parallel transport for time t→∞ on Euclidean space. We show that the asymptotic behavior can be controlled by the Yang–Mills action and the Yang–Mills equations. For open paths we show that under appropriate curvature conditions there exits a gauge in which the stochastic parallel transport converges almost surely. For closed paths we show that there exists a gauge invariant notion of a weak limit of the random holonomy and we give conditions that insure the existence of such a limit. Finally, we study the asymptotic behavior of the average of the random holonomy in the case of t'Hooft's 1-instanton.

In this paper, we study regular sets in metric measure spaces with Ricci curvature bounded from below. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also we define the dimension of RCD spaces and prove the lower semicontinuity of that under the Gromov-Hausdorff convergence.

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