Paola Vernole

Irregular Heat Flow Problems

We study a nonsteady transmission problem across either a fractal layer S or the corresponding prefractal layer

Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

S_h

Fractal snowflake domain diffusion with boundary and interior drifts

. The transmission condition is of order two. Existence, uniqueness, and regularity results for the strict solution, in both cases, are established as well as convergence results for the solutions of the approximating problems in varying Hilbert spaces.

Venttsel’ problems in fractal domains

A semilinear parabolic transmission problem with Ventcels boundary conditions on a fractal interface or the corresponding prefractal interface is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed.

Semilinear evolution transmission problems across fractal layers

Abstract We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector analysis for local Dirichlet forms and PDE on fractals, coercive closed forms, and the analysis of Lipschitz functions.