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Based on the domain variational point of view, we carry on stability analysison two shape optimization problems from thermal insulation background. Thenovelty is that, we do not require that the second variation is normal to theboundary. For example, translation variation is not normal, but as one can seein our work, it not only plays a role in obtaining the necessary and sufficientcondition for stability of ball shape in the first problem when heat source isradial, but also is essential in deriving the precise value of symmetrybreaking threshold of insulation material in the second problem, which turnsout to be related to isoperimetric constant and in turn implies that ballshapes are stable in two dimensions.

Stability analysis on the thermal insulation problems