Higher regularity of weak solutions on low-dimensional structures

The talk is devoted to a regularity of weak solutions of elliptic PDEs on irregular, lowdimensional subsets of the Euclidean space. Due to the singular nature of considered structures and technicalities related to functional spaces, it is not possible to apply standard regularity theorems in a straightforward way. We construct specific, well-behaved approximations of solutions that play the role analogous to difference quotients. This key result is a source of numerous significant implications connected with various aspects of the regularity of solutions.

By combining the mentioned result with a detailed analysis of properties of the second derivative operator, we establish the correspondence between weak solutions and higher-order functional spaces introduced by Bouchitté. Another consequence of the proposed approximation theorem is that by applying certain facts from the capacity theory, we establish the characterization of a global continuity of solutions. The presented results are based on the joint work with M. Fabisiak.

 

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link: https://us02web.zoom.us/j/82881783943?pwd=TUp3OGNvZ3R5akpXclZESmx2aGVkUT09

Meeting ID: 828 8178 3943
Passcode: 435791