In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator Δpu=∑i∂∂xi|∇u|p−2-0.16em∂u∂xi, that is the p‐Laplacian, and trueΔ̃pu=∑i∂∂xi|∂u∂xi|p−2∂u∂xi, namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set K , which represents a conductor of constant temperature, say 1, is thermally insulated by surrounding it with a layer of thermal insulator, the open set \K with K ⊂¯. The heat dispersion is then obtained as inf |∇ϕ| 2 dx + β ∂ * ϕ 2 dH n−1 , ϕ ∈ H 1 (R n), ϕ ≥ 1 in K , for some positive constant β. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set K vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.

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We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

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