Uniform weighted estimates on pre-fractal domains

Density and trace results in generalized fractal networks

“…As underlined before, such a trace result has potential applications to inverse problem. Similar questions at a discrete level or on higher-dimensional domains are also considered, let us quote discrete laplacian on infinite networks [11,13,15,18,[20][21][22][23], two-dimensional domains with a fractal boundary [1][2][3][4], pre-fractal domains approximating the Koch snowflake [10]. Let us finally mention some related problems such as the Hamilton-Jacobi equation [5] and the Gauss-Bonnet operator on infinite graphs [6].…”

Weighted Estimates on Fractal Domains

“…Hence, Theorem 4.1 improves Theorem 3.1 also for the case α = 3. In §5, by combining the tools and methods of the paper [5] with the results of §4, we establish uniform estimates that are more accurate than those established in [5], (compare Theorems 5.1 and 3.2). Lastly, in §6 we briefly discuss how we can extend the results of previous sections to the solutions of obstacle problems.…”

“…As the boundaries are the union of an increasing number of graphs and develop at the limit a fractal geometry, then the sharp regularity result (3.4) involves constants that might diverge as the number of graphs becomes infinite. In [5] we proved that there exists a suitable value of µ * depending on the structural parameter of the limit fractal domain 3 for which uniform weighted estimates hold. More precisely, the following result holds.…”

“…Introduction. In a previous paper [5], we established uniform estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on polygonal domains approximating the snowflake domain. In the present article, we deduce from the aforementioned estimates a regularity result for the Dirichlet problem on snowflake domain 3 .…”

“…In order to prove Theorem 3.1, we consider the solutions of the Dirichlet problems on polygonal domains n 3 approximating the snowflake domain 3 and we use the uniform estimates in weighted Sobolev spaces established in [5]. We recall that as the domain n 3 is not convex, then the solution u n of problem (2.4) does not belong to the space H 2 ( n 3 ).…”