Energy Stability for a Class of Semilinear Elliptic Problems

Danilo Gregorin Afonso1, Alessandro Iacopetti2, Filomena Pacella1
1Dipartimento di Matematica Guido Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185, Rome, Italy
2Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123, Turin, Italy

Tóm tắt

AbstractIn this paper, we consider semilinear elliptic problems in a bounded domain $$\Omega $$ Ω contained in a given unbounded Lipschitz domain $${\mathcal {C}} \subset {\mathbb {R}}^N$$ C R N . Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $$\Omega $$ Ω inside $${\mathcal {C}}$$ C . Once a rigorous variational approach to this question is set, we focus on the cases when $${\mathcal {C}}$$ C is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.

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