By plantri, we know that the following graph is the unique planar graph with 15 vertices of minimum degree 5. I add the graph in The House of Graphs (Graph 49406).
But I do not know if it has a name.
We note that the unique planar graph with $12$ vertices with minimum degree $5$ is the icosahedral graph and there is no planar graph with $13$ vertices with minimum degree $5$.