On the heat content functional and its critical domains

Abstract We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times $$t>0$$ t > 0 . do that by first computing variation of content, then showing $$\Omega $$ Ω is if only it has so-called constant flow property, so we can use a previous classification result established [33] [34]. The outcome time t , admits isoparametric foliation, is, foliation whose leaves parallel to boundary have mean curvature. Then, consider sequence functionals given exit-time moments $$T_1(\Omega ),T_2(\Omega ),\dots T 1 ( ) , 2 ⋯ generalize torsional rigidity $$T_1$$ prove $$T_k$$ k every get as well. main purpose paper understand variational properties general foliations their role PDE’s theory; some respects they $$\mathbf{R}^{n}$$ R n Euclidean spheres.